Transport of Energetic Electrons in Solids: Computer Simulation with Applications to Materials Analysis and Characterization (Springer Tracts in Modern Physics, 290) 3031372417, 9783031372414

Table of contents :
Preface to the Fourth Edition
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Acknowledgements
Contents
1 Electron Transport in Solids
1.1 Motivation: Why Are Electrons Important
1.2 The Monte Carlo Method
1.3 The Monte Carlo Ingredients
1.4 Electron-Beam Interactions with Solids
1.5 Electron Energy-Loss Peaks
1.6 Auger Electron Peaks
1.7 Secondary Electron Peak
1.8 Characterization of Materials
1.9 Summary
References
2 Computational Minimum
2.1 Numerical Differentiation
2.2 Numerical Quadrature
2.2.1 Trapezoidal Rule, Simpson's Rule, Bode's Rule
2.2.2 Gaussian Quadrature
2.3 Ordinary Differential Equations
2.4 Special Functions of Mathematical Physics
2.4.1 Legendre Polynomials and Associated Legendre Functions
2.4.2 Bessel Functions
2.5 Summary
References
3 Cross-Sections. Basic Aspects
3.1 Cross-Section and Probability of Scattering
3.2 Stopping Power and Inelastic Mean Free Path
3.3 Range
3.4 Energy Straggling
3.5 Summary
References
4 Scattering Mechanisms
4.1 Elastic Scattering
4.1.1 Mott Cross-Section Verses Screened Rutherford Cross-Section
4.1.2 Polarized Electron Beams Elastically Scattered by Atoms
4.1.3 Effect of the Sherman Function on the Differential Elastic Scattering Cross-Section
4.1.4 Electron-Molecule Elastic Scattering
4.2 Quasi-elastic Scattering
4.2.1 Electron-Phonon Interaction
4.3 Inelastic Scattering
4.3.1 Stopping: Bethe-Bloch Formula
4.3.2 Stopping: Semi-empirical Formulas
4.3.3 Dielectric Theory
4.3.4 Sum of Drude Functions
4.3.5 The Mermin Theory
4.3.6 Ab-initio Methods for Calculating the Complex Dielectric Function
4.3.7 Exchange Effects
4.3.8 Polaronic Effects
4.4 Surface Phenomena
4.4.1 Bulk and Surface Plasmon Losses
4.5 Summary
References
5 Random Numbers
5.1 Generating Pseudo-Random Numbers
5.2 Testing Pseudo-Random Number Generators
5.3 Pseudo-Random Numbers Distributed According to a Given Probability Density
5.4 Pseudo-Random Numbers Uniformly Distributed in the Interval [a, b]
5.5 Pseudo-Random Numbers Distributed According …
5.6 Pseudo-Random Numbers Distributed According to the Gauss Density of Probability
5.7 Summary
References
6 Monte Carlo Strategies
6.1 The Continuous-Slowing-Down Approximation
6.1.1 The Step-Length
6.1.2 Interface Between Over-Layer and Substrate
6.1.3 The Polar Scattering Angle
6.1.4 Direction of the Electron After the Last Deflection
6.1.5 Electron Position in Three Dimensional Cartesian Coordinates
6.1.6 The Energy Loss
6.1.7 End of the Trajectory and Number of Trajectories
6.2 The Energy-Straggling Strategy
6.2.1 The Step-Length
6.2.2 Elastic and Inelastic Scattering
6.2.3 Energy Loss
6.2.4 Electron-Atom Collisions: Scattering Angle
6.2.5 Electron–Electron Collisions: Scattering Angle
6.2.6 Electron-Phonon Collisions: Scattering Angle
6.2.7 Direction of the Electron After the Last Deflection
6.2.8 The First Step
6.2.9 Transmission Coefficient
6.2.10 How Inelastic Scattering Depends on the Distance from the Surface
6.2.11 End of the Trajectory and Number of Trajectories
6.3 Summary
References
7 Electron Beam Interactions with Solid Targets and Thin Films. Basic Aspects
7.1 Definitions, Symbols, Properties
7.2 Unsupported Thin Films
7.3 Supported Thin Films
7.4 Multilayers
7.4.1 Conservation of the Total Number of Particles
7.5 Summary
References
8 Backscattering Coefficient
8.1 Electrons Backscattered from Bulk Targets
8.1.1 The Backscattering Coefficient: The Everhart Model
8.1.2 The Backscattering Coefficient: The Archard Model
8.1.3 The Backscattering Coefficient: The Vicanek and Urbassek Model
8.1.4 The Backscattering Coefficient: Monte Carlo Simulation
8.2 Electrons Backscattered from One Layer Deposited on Semi-infinite Substrates
8.2.1 Carbon Overlayers
8.2.2 Gold Overlayers
8.3 Electrons Backscattered from Two Layers Deposited on Semi-infinite Substrates
8.4 A Comparative Study of Electron and Positron Backscattering …
8.5 Summary
References
9 Secondary Electron Yield
9.1 Secondary Electron Emission
9.2 Monte Carlo Approaches to the Study of Secondary Electron Emission
9.3 Specific MC Methodologies for SE Studies
9.3.1 Continuous-Slowing-Down Approximation (CSDA Scheme)
9.3.2 Energy-Straggling (ES Scheme)
9.4 Secondary Electron Yield of Metals
9.5 Secondary Electron Yield: PMMA and Al2O3
9.5.1 Secondary Electron Emission Yield as a Function of the Energy
9.5.2 Comparison Between ES Scheme and Experiment
9.5.3 Comparison Between CSDA Scheme and Experiment
9.5.4 CPU Time
9.6 Summary
References
10 Electron Energy Distributions
10.1 Monte Carlo Simulation of the Spectrum
10.2 Plasmon Losses and Electron Energy Loss Spectroscopy
10.2.1 Plasmon Losses in Graphite
10.2.2 Plasmon Losses in Silicon Dioxide
10.3 Surface Phenomena
10.4 Energy Losses of Auger Electrons
10.5 Elastic Peak Electron Spectroscopy (EPES)
10.6 Spectrum of Secondary Electrons
10.6.1 Wolff Theory
10.6.2 Other Formulas Describing the Spectrum of the Secondary Electrons
10.6.3 Initial Polar and Azimuthal Angle of the Secondary Electrons
10.6.4 Comparison with Theoretical and Experimental Data
10.7 Summary
References
11 Applications
11.1 Linewidth Measurement in Critical Dimension SEM
11.1.1 Critical Dimension SEM
11.1.2 Lateral and Depth Distributions
11.1.3 Linescan of a Silicon Step
11.1.4 Linescan of PMMA Lines on a Silicon Substrate
11.2 Application to Energy Selective Scanning Electron Microscopy
11.2.1 Doping Contrast
11.2.2 Energy Selective Scanning Electron Microscopy
11.3 Energy Density Radially Deposited Along Ion Tracks
11.3.1 Ion Track Simulation and Bragg Peak
11.3.2 Damage in the Biomolecules by Ionizations, Excitations, and Dissociative Electron Attachment
11.3.3 Simulation of Electron Transport and Further Generation
11.3.4 Radial Distribution of the Energy Deposited in PMMA by Secondary Electrons Generated by Energetic Proton Beams
11.3.5 Radial Distribution of the Energy Deposited in Liquid H2O by Secondary Electrons Generated by Energetic Carbon Ion Beams
11.4 Summary
References
Appendix A The First Born Approximation and the Rutherford Cross-Section
A.1 The Elastic Scattering Cross-Section
A.2 The First Born Approximation
A.3 Integral-Equation Approach
A.4 The Rutherford Formula
A.5 Summary
Appendix B Partial Wave Expansion Method
B.1 The Radial Equation
B.2 Expansion of the Plane Wave
B.3 Scattering Amplitude
B.4 Differential and Total Elastic Scattering Cross-Section. Optical Theorem
B.5 Summary
Appendix C The Mott Theory
C.1 The Dirac Equation in a Central Potential
C.2 Relativistic Partial Wave Expansion Method
C.3 Phase Shift Calculations
C.4 Analytical Approximation of the Mott Cross-Section
C.5 Positron Differential Elastic Scattering Cross-Section
C.6 Summary
Appendix D The Atomic Potential
D.1 The Atomic Screening Function
D.2 Electron Exchange
D.3 Charge Cloud Polarization Effects
D.4 Solid-State Effects
D.5 Summary
Appendix E The Fröhlich Theory
E.1 Electrons in Lattice Fields. Interaction Hamiltonian
E.2 Electron-Phonon Scattering Cross-Section
E.3 Summary
Appendix F The Ritchie Theory
F.1 Energy Loss and Dielectric Function
F.2 Homogeneous and Isotropic Solids
F.3 Summary
Appendix G The Chen and Kwei and the Li et al. Theory
G.1 Outgoing and Incoming Electrons
G.2 Probability of Inelastic Scattering
G.3 Summary
Appendix H The Mermin Theory and the Generalized Oscillator Strength Method
H.1 The Mermin Theory
H.2 The Mermin Energy Loss Function-Generalized Oscillator Strength Method (MELF-GOS)
H.3 Summary
Appendix I The Kramers–Kronig Relations and the Sum Rules
I.1 Linear Response to External Perturbations
I.2 The Kramers–Kronig Relations
I.3 Sum Rules
I.4 Summary
Appendix J From the Electron Energy Loss Spectrum to the Dielectric Function
J.1 From the Single-Scattering Spectrum to the Energy Loss Function
J.2 Summary
Index

Citation preview

Springer Tracts in Modern Physics 290

Maurizio Dapor

Transport of Energetic Electrons in Solids Computer Simulation with Applications to Materials Analysis and Characterization Fourth Edition

Springer Tracts in Modern Physics Volume 290

Series Editors Mishkatul Bhattacharya, Rochester Institute of Technology, Rochester, NY, USA Yan Chen, Department of Physics, Fudan University, Shanghai, China Atsushi Fujimori, Department of Physics, University of Tokyo, Tokyo, Japan Mathias Getzlaff, Institute of Applied Physics, University of Düsseldorf, Düsseldorf, Nordrhein-Westfalen, Germany Thomas Mannel, Emmy Noether Campus, Universität Siegen, Siegen, Nordrhein-Westfalen, Germany Eduardo Mucciolo, Department of Physics, University of Central Florida, Orlando, FL, USA William C. Stwalley, Department of Physics, University of Connecticut, Storrs, USA Jianke Yang, Department of Mathematics and Statistics, University of Vermont, Burlington, VT, USA

Springer Tracts in Modern Physics provides comprehensive and critical reviews of topics of current interest in physics. The following fields are emphasized: – – – –

Particle and Nuclear Physics Condensed Matter Physics Light Matter Interaction Atomic and Molecular Physics

Suitable reviews of other fields can also be accepted. The Editors encourage prospective authors to correspond with them in advance of submitting a manuscript. For reviews of topics belonging to the above mentioned fields, they should address the responsible Editor as listed in “Contact the Editors”.

Maurizio Dapor

Transport of Energetic Electrons in Solids Computer Simulation with Applications to Materials Analysis and Characterization Fourth Edition

Maurizio Dapor European Centre for Theoretical Studies in Nuclear Physics and Related Areas Trento, Italy

ISSN 0081-3869 ISSN 1615-0430 (electronic) Springer Tracts in Modern Physics ISBN 978-3-031-37241-4 ISBN 978-3-031-37242-1 (eBook) https://doi.org/10.1007/978-3-031-37242-1 1st edition: © Springer International Publishing Switzerland 2014 2nd edition: © Springer International Publishing AG 2017 3rd edition: © Springer Nature Switzerland AG 2020 4th edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer 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, expressed 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. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my dearest father, always alive in my mind and in my heart

Preface to the Fourth Edition

This book describes the computational methods that are most frequently used to deal with the interaction of charged particles, notably electrons, with condensed matter. Both elastic and inelastic scattering phenomena are discussed and methods for calculating the relevant cross-sections are explained in a rigorous but simple way. The book provides readers with all the information they need to write their own Monte Carlo code and simulate the transport of fast particles in condensed matter. Many numerical and experimental examples are presented throughout the book. The updated and extended fourth edition features ab-initio methods for calculating dielectric and energy loss functions. The non-relativistic partial wave expansion method for calculating the differential elastic scattering cross-section is also included in this new edition. It represents a very useful introduction to the relativistic partial wave expansion method, i.e., to the Mott theory, already discussed in the previous editions of this book. Further details about the effects of spin-polarization on the differential elastic scattering cross-section are included in this new edition. The multiple reflection method is applied also to the general case of a system composed of a set of layers of different materials and varying in thickness. Analytical expressions are provided for calculating the backscattering coefficient of multilayers. New results are presented, notably about Monte Carlo simulations of reflection electron energy loss spectra and the radial dose deposited along the track of ions impinging on materials. Villazzano, Italy

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Preface to the Third Edition

This volume is an extended edition of a book published in 2014 (first edition) and 2017 (second edition). It provides all the information necessary for the reader to write his/her own Monte Carlo code. The possibility to compare the obtained results with the many numerical and experimental examples presented through the text can help the reader to better describe and understand the processes and the phenomena concerning the transport of electrons and positrons in solid targets. The updated content of this third edition includes the theory of the spin-polarized electron beams, the study of elastic scattering by molecules, a simple derivation of the Bethe-Bloch stopping power formula, the f-sum rule and the ps-sum rule, the Vicanek and Urbassek formula for the calculation of the backscattering coefficient, and the Wolff theory of the secondary electron spectra. A new chapter is devoted to the basic concepts of computational physics. Another chapter is dedicated to the elementary aspects of the interaction of beams of electrons and positrons with thin solid films (the so-called multiple reflection method). Further comparisons between theory and experimental data are presented here as well. Villazzano, Italy December 2019

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Preface to the Second Edition

This book describes all the mechanisms of scattering (elastic and inelastic) of electrons with the atoms of the target in the simplest possible way. The use of techniques of quantum mechanics is described in detail for the investigation of interaction processes of electrons with matter. The book presents the strategies of the Monte Carlo method, as well as numerous comparisons among the results of the simulations with the experimental data available in the literature. The new content of this extended and updated second edition includes the derivation of the Rutherford formula, details about the calculation of the phase shifts that are used in the relativistic partial wave expansion method, and the description of the Mermin theory. The role of secondary electrons in the proton cancer therapy is discussed as well in the chapter devoted to applications: In this context, Monte Carlo results about the radial distribution of the energy deposited in solids by secondary electrons generated by energetic proton beams are presented. Povo, Italy August 2016

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Preface to the First Edition

In modern physics we are interested in systems with many degrees of freedom. Let us consider, for example, the number of atoms in a solid, the number of electrons in an atom, or the number of electrons of a beam interacting with the many atoms and electrons of a solid. In many situations, these systems can be described by the calculations of definite integrals of very high dimension. An example is the evaluation of the classical partition function for gas of many atoms at a temperature T. The Monte Carlo method gives us a very accurate way to calculate high dimensional definite integrals: It evaluates the integrand at a random sampling of abscissa. The Monte Carlo method is also used for evaluating the many physical quantities necessary for the study of the interactions of particle-beams with solid targets. The simulation of the involved physical processes, by random sampling, allows to solve many particle transport problems. Letting the particles carry out an artificial, random walk—taking into account the effect of the single collisions—it is possible to accurately evaluate the diffusion process. This book is devoted to electron-beam interactions with solid targets. As a researcher in this field, I am persuaded that a book on kV electron transport in solids can be very useful. It is difficult, for the newcomer, to find this topic exhaustively treated. It is not easy for the beginner to sort out the importance of the great number of published papers. The Monte Carlo simulation is the most powerful theoretical method for evaluating the physical quantities related to the interaction of electrons with a solid target. A Monte Carlo simulation can be considered an idealized experiment. The simulation does not investigate the fundamental mechanisms of the interaction. It is necessary to know them—in particular the energy loss and angular deflection phenomena—to produce a good simulation.

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Preface to the First Edition

This book is complementary with respect to many other texts dedicated to similar subjects (including my book entitled Electron-Beam Interactions with Solids, published by Springer-Verlag in 2003) for the following two aspects. 1. I have, on the one hand, systematically minimized the mathematical contents of the more difficult theoretical parts. Since the essential concept to digest is the meaning of the various cross-sections, the mathematical details, for the sake of clarity, have been deliberately avoided. I have reduced the theoretical part to the presentation of the energy loss and the angular deflections, providing simple recipes to calculate the stopping power, the differential inverse inelastic mean free path, and the differential elastic scattering cross-section. This allowed me to avoid describing in-depth quantum theory. Mathematical contents and details can be found in the Appendices, in my previous book, and in many other books dedicated to modern physics and quantum mechanics. 2. In the derivation and use of the simpler theoretical transport models, I have, on the other hand, included many details. I think, indeed, there is a need for a basic physical picture in order to provide the beginner with a solid background about electron transport in solids. This can be only achieved by a step-by-step derivation of the analytical formulas. Comparing available experimental data with simulation results is a fundamental step in evaluating the quality of the Monte Carlo codes. Selected applications of the Monte Carlo method to kV electron transport in solids are presented in the second part of this book. The book compares computational simulations and experimental data in order to offer a more global vision. Povo, Italy October 2013

Maurizio Dapor

Acknowledgements

I would like to gratefully thank my beloved children for their immense patience and words of encouragement. I am grateful to my dear parents for their great affection and love. It is also a pleasure to mention many people, friends, and colleagues who, with their enthusiasm, invaluable suggestions, ideas, and competence, considerably contributed to achieve the results presented in this book: I am, in particular, very grateful to Diego Bisero, Lucia Calliari, Giovanni Garberoglio, Simone Taioli, and Paolo Emilio Trevisanutto for their help and sincere friendship. I am also indebted to all the researchers who made possible, with their help, the completion of the project: Gert Aarts, Kerry J. Abrams, Isabel Abril, Martina Azzolini, Nicola Bazzanella, Daniele Binosi, Mauro Borghesi, Eric Bosch, Mauro Ciappa, Michele Crivellari, Pablo de Vera, Sergey Fanchenko, Wolfgang Fichtner, Massimiliano Filippi, Jan Franz, Małgorzata Franz, Rafael GarciaMolina, Stefano Gialanella, Stephan Holzer, Emre Ilgüsatiroglu, Beverley J. Inkson, Gianluca Introzzi, Mark A.E. Jepson, Alexander Koschik, Robert C. Masters, Antonio Miotello, Tommaso Morresi, Francesco Pederiva, Andrea Pedrielli, Nicoletta Plotegher, Maciej Polak, Cornelia Rodenburg, John M. Rodenburg, Manoranjan Sarkar, Paolo Scardi, Giorgina Scarduelli, Siegfried Schmauder, Emanuele Scifoni, Stefano Simonucci, Nicola Stehling, Laura Toniutti, Dionysios Triantafyllopoulos, Chia-Kuang (Frank) Tsung, Jochen Wambach, Quan Wan, Wolfram Weise, and Davide Zari. I wish to express my sincere gratitude to all the students of my course entitled Computational Methods for Transport Phenomena (Department of Physics, University of Trento) for their kind questions, their clever suggestions, and their excellent and illuminating ideas. I would like to thank all my colleagues at the Interdisciplinary Laboratory for Computational Science (LISC) in Trento for the fruitful discussions. The stimulating atmosphere of LISC provided the ideal environment to work on this project. I am grateful to Maria Del Huerto Flammia for assisting me with the proofreading of this book.

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Acknowledgements

I acknowledge the support of the Leverhulme Trust through the Visiting Professorship VP1-2014-011 (Department of Engineering Materials of the University of Sheffield). I wish also to thank the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), the Fondazione Bruno Kessler (FBK), the Trento Institute for Fundamentals Physics and Applications (TIFPA), the Department of Physics of the University of Trento, the Department of Materials Science and Engineering of the University of Sheffield, the Integrated Systems Laboratory of the Swiss Federal Institute of Technology (ETH) of Zurich, the Departament de Física Aplicada of the Universitat d’Alacant, and the Faculty of Applied Physics and Mathematics of the Gda´nsk University of Technology.

Contents

1

Electron Transport in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation: Why Are Electrons Important . . . . . . . . . . . . . . . . . . . 1.2 The Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Monte Carlo Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Electron-Beam Interactions with Solids . . . . . . . . . . . . . . . . . . . . . . 1.5 Electron Energy-Loss Peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Auger Electron Peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Secondary Electron Peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Characterization of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 3 5 7 7 8 9 9

2

Computational Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Numerical Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Numerical Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Trapezoidal Rule, Simpson’s Rule, Bode’s Rule . . . . . . . 2.2.2 Gaussian Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Special Functions of Mathematical Physics . . . . . . . . . . . . . . . . . . 2.4.1 Legendre Polynomials and Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 12 13 15 16 16 17 20 20

Cross-Sections. Basic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Cross-Section and Probability of Scattering . . . . . . . . . . . . . . . . . . 3.2 Stopping Power and Inelastic Mean Free Path . . . . . . . . . . . . . . . . 3.3 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Energy Straggling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 22 23 24 25 26 27

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Contents

Scattering Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Mott Cross-Section Verses Screened Rutherford Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Polarized Electron Beams Elastically Scattered by Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Effect of the Sherman Function on the Differential Elastic Scattering Cross-Section . . . . . . . . . . . . . . . . . . . . 4.1.4 Electron-Molecule Elastic Scattering . . . . . . . . . . . . . . . . 4.2 Quasi-elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Electron-Phonon Interaction . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Stopping: Bethe-Bloch Formula . . . . . . . . . . . . . . . . . . . . . 4.3.2 Stopping: Semi-empirical Formulas . . . . . . . . . . . . . . . . . 4.3.3 Dielectric Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Sum of Drude Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 The Mermin Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Ab-initio Methods for Calculating the Complex Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.7 Exchange Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.8 Polaronic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Surface Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Bulk and Surface Plasmon Losses . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Generating Pseudo-Random Numbers . . . . . . . . . . . . . . . . . . . . . . . 5.2 Testing Pseudo-Random Number Generators . . . . . . . . . . . . . . . . . 5.3 Pseudo-Random Numbers Distributed According to a Given Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Pseudo-Random Numbers Uniformly Distributed in the Interval [a, b] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Pseudo-Random Numbers Distributed According to the Exponential Density of Probability . . . . . . . . . . . . . . . . . . . . 5.6 Pseudo-Random Numbers Distributed According to the Gauss Density of Probability . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 84

86 87 88

Monte Carlo Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Continuous-Slowing-Down Approximation . . . . . . . . . . . . . . 6.1.1 The Step-Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Interface Between Over-Layer and Substrate . . . . . . . . . . 6.1.3 The Polar Scattering Angle . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Direction of the Electron After the Last Deflection . . . . .

89 90 90 90 91 91

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6.1.5

7

8

Electron Position in Three Dimensional Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.6 The Energy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.7 End of the Trajectory and Number of Trajectories . . . . . 6.2 The Energy-Straggling Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 The Step-Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Elastic and Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Energy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Electron-Atom Collisions: Scattering Angle . . . . . . . . . . 6.2.5 Electron–Electron Collisions: Scattering Angle . . . . . . . 6.2.6 Electron-Phonon Collisions: Scattering Angle . . . . . . . . . 6.2.7 Direction of the Electron After the Last Deflection . . . . . 6.2.8 The First Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.9 Transmission Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.10 How Inelastic Scattering Depends on the Distance from the Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.11 End of the Trajectory and Number of Trajectories . . . . . 6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 103 104 104

Electron Beam Interactions with Solid Targets and Thin Films. Basic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Definitions, Symbols, Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Unsupported Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Supported Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Multilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Conservation of the Total Number of Particles . . . . . . . . 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 106 108 110 112 114 114

Backscattering Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Electrons Backscattered from Bulk Targets . . . . . . . . . . . . . . . . . . . 8.1.1 The Backscattering Coefficient: The Everhart Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 The Backscattering Coefficient: The Archard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 The Backscattering Coefficient: The Vicanek and Urbassek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 The Backscattering Coefficient: Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Electrons Backscattered from One Layer Deposited on Semi-infinite Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Carbon Overlayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Gold Overlayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Electrons Backscattered from Two Layers Deposited on Semi-infinite Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92 93 93 93 94 95 95 96 96 98 99 99 99

115 115 116 119 120 122 124 124 126 129

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8.4

A Comparative Study of Electron and Positron Backscattering Coefficients and Depth Distributions . . . . . . . . . . . 133 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 9

Secondary Electron Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Secondary Electron Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Monte Carlo Approaches to the Study of Secondary Electron Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Specific MC Methodologies for SE Studies . . . . . . . . . . . . . . . . . . 9.3.1 Continuous-Slowing-Down Approximation (CSDA Scheme) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Energy-Straggling (ES Scheme) . . . . . . . . . . . . . . . . . . . . 9.4 Secondary Electron Yield of Metals . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Secondary Electron Yield: PMMA and Al2 O3 . . . . . . . . . . . . . . . . 9.5.1 Secondary Electron Emission Yield as a Function of the Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Comparison Between ES Scheme and Experiment . . . . . 9.5.3 Comparison Between CSDA Scheme and Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 CPU Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 140

10 Electron Energy Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Monte Carlo Simulation of the Spectrum . . . . . . . . . . . . . . . . . . . . 10.2 Plasmon Losses and Electron Energy Loss Spectroscopy . . . . . . . 10.2.1 Plasmon Losses in Graphite . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Plasmon Losses in Silicon Dioxide . . . . . . . . . . . . . . . . . . 10.3 Surface Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Energy Losses of Auger Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Elastic Peak Electron Spectroscopy (EPES) . . . . . . . . . . . . . . . . . . 10.6 Spectrum of Secondary Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 Wolff Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.2 Other Formulas Describing the Spectrum of the Secondary Electrons . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.3 Initial Polar and Azimuthal Angle of the Secondary Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.4 Comparison with Theoretical and Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 153 153 154 155 157 160 160 162

140 141 141 142 143 144 144 145 145 146 148 149

166 166 167 170 171

Contents

11 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Linewidth Measurement in Critical Dimension SEM . . . . . . . . . . 11.1.1 Critical Dimension SEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Lateral and Depth Distributions . . . . . . . . . . . . . . . . . . . . . 11.1.3 Linescan of a Silicon Step . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.4 Linescan of PMMA Lines on a Silicon Substrate . . . . . . 11.2 Application to Energy Selective Scanning Electron Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Doping Contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Energy Selective Scanning Electron Microscopy . . . . . . 11.3 Energy Density Radially Deposited Along Ion Tracks . . . . . . . . . 11.3.1 Ion Track Simulation and Bragg Peak . . . . . . . . . . . . . . . . 11.3.2 Damage in the Biomolecules by Ionizations, Excitations, and Dissociative Electron Attachment . . . . . 11.3.3 Simulation of Electron Transport and Further Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.4 Radial Distribution of the Energy Deposited in PMMA by Secondary Electrons Generated by Energetic Proton Beams . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.5 Radial Distribution of the Energy Deposited in Liquid H2 O by Secondary Electrons Generated by Energetic Carbon Ion Beams . . . . . . . . . . . . . . . . . . . . . 11.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxi

173 173 173 174 175 176 176 177 178 179 179 180 180

181

181 183 183

Appendix A: The First Born Approximation and the Rutherford Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Appendix B: Partial Wave Expansion Method . . . . . . . . . . . . . . . . . . . . . . . 195 Appendix C: The Mott Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Appendix D: The Atomic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Appendix E: The Fröhlich Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Appendix F: The Ritchie Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Appendix G: The Chen and Kwei and the Li et al. Theory . . . . . . . . . . . . . 249 Appendix H: The Mermin Theory and the Generalized Oscillator Strength Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Appendix I: The Kramers–Kronig Relations and the Sum Rules . . . . . . 257 Appendix J: From the Electron Energy Loss Spectrum to the Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Chapter 1

Electron Transport in Solids

The Monte Carlo (MC) method is used to evaluate the many physical quantities necessary to study particle beams’ interactions with solid targets. Studies of backscattered and secondary electrons are of great interest for many analytical techniques. A better comprehension of the processes which occur before the emission of backscattered and secondary electrons allows a more comprehensive understanding of surface physics.

1.1 Motivation: Why Are Electrons Important Electrons continuously interact with the matter around us. Plasma processing of materials, electron lithography, electron microscopy and spectroscopies, plasmawall in fusion reactors, the interaction of charged particles with the surfaces of space-crafts, and hadron therapy represent only a few technological examples where electrons are involved and play a role. In fact, we use electron beams for our purposes, either on the front of the production of materials or on that of their characterization. Let us think of the many applications such as the processing of materials with plasma and the local melting of materials for joining large components. We also use electron beams in electron lithography, an important technique for producing microelectronics devices. Let us consider the importance of the beams of electrons in material characterization, performed using techniques such as electron microscopy and electron spectroscopies. Electrons interact with the surfaces of space-crafts. Plasma-wall interaction in fusion reactors also involves electron-matter interaction. Electrons play a role also in ion cancer therapy (hadron therapy), where cascades of secondary electrons are produced. These very low-energy electrons are toxic to human body cells since they produce damage to the biomolecules due to ionizations/excitations and the resulting break of chemical © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1_1

1

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1 Electron Transport in Solids

bonds. Also, secondary electrons which have ultra-low energy—and which, for a long time, were thought to be relatively harmless—are dangerous for biomolecules due to the so-called dissociative electron attachment. And, of course, we want to minimize the effects of irradiation on the healthy tissues near the diseased cells. In all the cases above, modeling the interaction of electrons with matter is very important, as it can be used to provide a solid theoretical interpretation of experimental evidence. This is why an accurate and detailed study of the interaction mechanisms of electrons with matter is paramount.

1.2 The Monte Carlo Method The world is ruled by quantum mechanics. The investigation of the processes of electron-matter interaction requires the use of quantum mechanics-based techniques. And since, typically, the number of particles involved in these processes is huge, it is crucial to use statistical approaches, such as those represented by the Monte Carlo method. This method provides a very accurate description of many of the phenomena that we can observe in nature when electrons interact with materials. The Monte Carlo method is a numerical procedure that uses random numbers, the theory of probability, and statistics to evaluate multiple integrals. Suppose we need to calculate the area of a closed surface. In order to do so, we can surround the curve with a square of a known side. Then we generate a large number of random points inside the square. Whenever a random point falls within the surface, we update a counter. When the number of points is very large, the ratio of those fallen within the surface and the total number of generated points will approach the ratio between the (unknown) area of the surface and the (known) area of the square. It is very important to emphasize that, when the number of dimensions is high (as it is when dealing with all statistical problems) the Monte Carlo method is the best numerical procedure for the calculation of multiple integrals. Complicated problems of physics involving very large numbers of particles can be addressed with the Monte Carlo method: It can realize real numerical simulations of physical processes such as the interaction of an electron beam with a solid. The Monte Carlo method is used, in particular, for evaluating the many physical quantities necessary to the study of the interactions of particle beams with solid targets. By letting the particles carry out an artificial, random walk—taking into account the effect of the single collisions—it is possible to accurately evaluate the diffusion process [1–4].

1.4 Electron-Beam Interactions with Solids

3

1.3 The Monte Carlo Ingredients To work properly, the Monte Carlo method needs a set of input data describing the interaction of the particle beam with the target. Such data specify the kind of materials and, of course, the kind of incident particles. The interactions of the particles impinging on the specimen with the target atoms can be described by the cross-sections that describe different physical phenomena. In fact, various kinds of interaction occur during the passage of the electrons through the material. In particular, we will investigate the elastic electron-atom processes, described by the so-called elastic scattering cross-section. The elastic scattering cross-section can be calculated by using an equation known as the screened Rutherford crosssection formula. This is an analytical expression valid when the energy of the incident electrons is relatively high and the atomic numbers of the target atoms are relatively low, as it was deduced within the so-called first Born approximation. But, since the case of low energy electrons and high atomic numbers is not well described by such a simple formula, in order to simulate the elastic scattering cross-section with a more general formulation—valid for all energies and atomic numbers—a more complex approach is necessary, which is known as the relativistic partial wave expansion method (Mott cross-section [5]). Concerning the inelastic scattering cross-section, we will use analytical formulas, when possible, and the dielectric Ritchie’s theory [6] for dealing with the more general cases. When electron energy becomes relatively small (let us say, smaller than 20–30 eV, depending on the investigated material) another very important mechanism of energy loss (and energy gain) is related to the creation (and annihilation, respectively) of phonons. For describing such a phenomenon, we will introduce the electron-phonon cross-section utilizing the Fröhlich’s theory [7]. In the end, in many cases also trapping phenomena are important and need to be considered in the simulations. They can be due to the polarization of materials (insulators) induced by the passage of very slow electrons through them [8] and/or also to defects in the materials. When dealing with insulating materials, trapping phenomena are mainly due to the so-called polaronic effects, i.e., to the creation of quasi-particles constituted by slow electrons with the polarization field around them [8]. In the case of metals and semiconductors, traps are mainly due to defects in the materials (impurities, structural defects, grain boundaries etc.).

1.4 Electron-Beam Interactions with Solids During their travel in the solid, the incident electrons lose energy and change direction at each collision with the atoms bound in the solid. Because of the large difference between the masses of the electron and the atomic nucleus, nuclear collisions deflect electrons with very small kinetic energy transfers. This process is described by the

4

1 Electron Transport in Solids

differential elastic scattering cross-section (which can be calculated by the so-called relativistic partial wave expansion method, corresponding to the Mott cross-section [5]). The Mott cross-section can be approximated with the screened Rutherford formula: This is possible when the conditions corresponding to the first Born approximation are satisfied, i.e., for high energy of the incident particle and for low atomic number of the target atoms. Additionally, excitation and ejection of atomic electrons, and excitation of plasmons, affect the energy dissipation. These processes only slightly affect the direction of the incident electron in the solid. Plasmon excitations are ruled by the equations for the differential inverse inelastic mean free path, calculated by the use of Ritchie’s dielectric theory [6]. The Fröhlich theory [7] can be used for describing the quasi-elastic electron-phonon interactions in insulating materials. Electron-phonon interactions are considered quasi-elastic for the corresponding energy losses and gains are very small when compared to the plasmon energy losses. When, in insulating materials, electron kinetic energies considerably decrease, trapping phenomena due to the polaronic effects have to be taken into account as well [8]. While for electron kinetic energies higher than 10 keV, MC simulations provide excellent results by just using the Rutherford differential elastic scattering crosssection (elastic scattering) and the Bethe-Bloch stopping power formula1 (inelastic scattering), when the electron energies become much smaller than 5 keV—and this is the case of secondary electron emission—this approach fails [9]. There are many reasons, and the most important ones are related to the three following facts: (i) As the Rutherford formula is a result of the first Born approximation, it is a high energy approximation. (ii) Also the Bethe-Bloch formula is valid only for quite high energies. In particular, the Bethe-Bloch stopping power does not provide the correct predictions when the electron energy E becomes smaller than the mean excitation energy I . It reaches a maximum and then approaches zero as E approaches I /1.166. Below I /1.166, the predicted stopping power becomes negative. The use of semi-empirical approaches can sometimes mitigate the problem. Actually, numerical approaches based on the calculation of the dielectric function—as a function of the energy loss and of the momentum transfer—are necessary to calculate low energy inelastic processes. (iii) The inclusion of the stopping power in the MC code corresponds to the use of the so-called continuous-slowing-down approximation (CSDA). Such a way of describing energy losses completely neglects that actually electrons lose their energy in several inelastic collisions. Sometimes an electron can even lose all its energy in a single collision. In other words, any realistic model of the electron trajectories should avoid the approximation of continuity in describing the electron energy losses. CSDA can be used (and will be used, when possible, in the present work as well) but only in cases where the details of the energy loss mechanisms are not crucial 1

In this book we will use the expression “stopping power” instead of “stopping force” to indicate the energy loss per unit distance of electrons in the solid. Even if consistent with the units, and hence more accurate, the use in the literature of the expression stopping force, as observed by Peter Sigmund [10], is only slowly appearing, after a hundred years of use of the term stopping power.

1.5 Electron Energy-Loss Peaks

5

for the accurate description of the process under investigation. CSDA can be used, for example, for the calculation of the backscattering coefficient. We will see that, in some specific cases, even the calculation of the secondary electron yield can be performed using CSDA. On the contrary, the description of the energy distributions of the emitted electrons (both backscattered and secondary) have to be performed avoiding the approximation of continuity in the energy loss processes and including energy straggling (ES)—i.e., the statistical fluctuations of the energy loss due to the different energy losses suffered by each electron traveling in the solid—in the calculations. A detailed approach able to accurately describe low energy elastic and inelastic scattering and to appropriately take into account the energy straggling is required for the description of secondary electron cascade. The whole cascade of secondary electrons must be followed. Indeed any truncation, or cut-off, underestimates the secondary electron emission yield. Also, as already discussed, for insulating materials the main mechanisms of energy loss cannot be limited to the electron-electron interaction for, when the electron energy becomes very small (lower than 10–20 eV, say), inelastic interactions with quasi-particles are responsible for electron energy losses. In particular, at very low electron energy, trapping phenomena (polaronic effects) and electron-phonon interactions are the main mechanisms of electron energy loss. For electron-phonon interaction, even phonon annihilations and the corresponding energy gains should be taken into account. Actually, the energy gains are often neglected, for their probability of occurrence is very small. Much smaller, in any case, than the probability of phonon creation. Summarizing, incident electrons are scattered and lose energy, due to the interactions with the atoms of the specimen, so the incident electrons’ direction and kinetic energy are changed. It is usual to describe the collision events assuming that they belong to three distinct kinds: Elastic (scattering with atomic nuclei), quasi-elastic (scattering with phonons), and inelastic (scattering with the atomic electrons and trapping due to the polaronic effects).

1.5 Electron Energy-Loss Peaks Electron energy-loss spectroscopy treats the primary process in which the incident electron loses amounts of energy which characterize the target material (see, for example, [6, 11–31]). An electron spectrum represents the number of electrons emerging from the target surface as a function of the energy they have after interaction with the target. The spectrum can be represented as a function of either the electron energy or the electron energy loss. In this second case, the first peak on the left of the spectrum, centered at zero energy-loss, is known as the zero-loss peak. Also known as the elastic peak, it collects all the electrons which were transmitted—in transmission electron energy loss spectroscopy (TEELS)—or backscattered—in reflection electron energy loss spectroscopy (REELS)—without any measurable energy loss. It includes both the electrons which did not suffer any energy loss and those which

6

1 Electron Transport in Solids

were transmitted (TEELS) or backscattered (REELS) after one or more quasi-elastic collisions with phonons (for which the energy transferred is so small that, with conventional spectrometers, it cannot be experimentally resolved). In TEELS, the elastic peak includes also all the electrons which were not scattered at all, namely which were not deflected during their travel inside the target and did not lose energy. Actually, the energy of electrons of the elastic peak is slightly reduced. This is due to the recoil energy transferred to the atoms of the specimen. Elastic peak electron spectroscopy (EPES) is the technique devoted to the analysis of the line-shape of the elastic peak [32, 33]. Since lighter elements show larger energy shifts, EPES can be used to detect hydrogen in polymers and hydrogenated carbon-based materials [34– 41] measuring the energy difference between the position of the carbon elastic peak and that of the hydrogen elastic peak. This difference between the energy positions of the elastic peaks—for incident electron energy in the range 1000–2000 eV—is in the neighborhood of 2–4 eV. In the first 30–40 eV from the elastic peak, a generally quite broad peak collects all the electrons which suffered inelastic interaction with the outer-shell atomic electrons. Typically it includes electrons that suffered energy loss due to inelastic interaction with plasmons (plasmon-losses) and to inter-band and intra-band transitions. If the sample is sufficiently thick (in TEELS) and in the case of bulk targets (in REELS), the probability that an electron, before emerging from the specimen, has suffered more than one inelastic collision with plasmons is not negligible. Such multiple electron-plasmon inelastic collisions are represented in the spectrum by the presence of a set of equidistant peaks (the distance from each other being given by the plasma energy). The relative intensities of these multiple inelastic scattering peaks decrease as the energy loss increases, demonstrating that the probability of suffering one inelastic collision is greater than the probability of suffering two inelastic collisions, which is in turn higher than the probability of suffering three inelastic collisions, and so on. Of course, in transmission EELS, the number of measurable plural scattering peaks is also a function of the thickness of the sample. Plural scattering peaks at multiples of the plasma energy are clearly observable in TEELS—in the energy-loss region between the elastic peak and approximately 100–200 eV (i.e., in the energy spectrum, between 100–200 eV from the elastic peak and the elastic peak)—when the film thickness is greater than the electron inelastic mean free path. On the other hand, when the film thickness is much smaller than the electron inelastic mean free path, a strong elastic peak and only the first plasmon-loss peak can be observed in the energy-loss region below 100–200 eV. For higher energy losses, edges (of relatively low intensity with respect to the plasmon-losses), corresponding to inner-shell atomic electron excitations, can be observed in the spectrum. These edges are followed by slow falls, as the energy-loss increases. The energy position of these steps or, better, sharp rises, corresponds to the ionization threshold. The energy loss of each edge is an approximate measure of the binding energy of the inner-shell energy level involved in the inelastic scattering process. With an energy resolution better than 2 eV, it is possible to observe, in the lowloss peaks and in the ionization threshold edges, detailed features related to the band

1.7 Secondary Electron Peak

7

structure of the target and its crystalline characteristics. For example, in carbon, plasmon peaks can be found at different energies in the spectrum, according to the carbon structure. This is due to the different valence-electron densities of the different allotropic forms of carbon, such as diamond, graphite, C60-fullerite, glassy carbon, and amorphous carbon [26, 27]. For an excellent review about electron energy-loss spectra, see the Egerton book [26].

1.6 Auger Electron Peaks Also, Auger electron peaks can be observed in the spectrum. They are due to the presence of doubly ionized atoms. Auger [42] and Meitner [43] noted the presence of pairs of tracks—originating from the same point—in X-ray irradiated cloud chambers filled with an inert gas. One of them had a variable length which depended on the energy of the incident radiation. The other track had a fixed length. Auger suggested the presence of doubly ionized atoms in the gas. Two years later, Wentzel made the hypothesis of a two-step process. A primary ionization, in the Wentzel interpretation, was followed by a decay process [44]. The incident radiation ionizes the system in the inner shell S. The system can then decay according to two alternative mechanisms. One is radiative: One electron drops out of an outer shell R into the inner shell S and a photon is emitted. The other one is non-radiative: One electron drops out of an outer shell R into the inner shell S, and the excess of energy is used to eject out of the shell R  another electron (the Auger electron). The two processes are competitive. In the electron spectrum, Auger electron peaks—due to the non-radiative process—can be recognized.

1.7 Secondary Electron Peak Secondary electrons—produced by a cascade process—are those electrons extracted from the atoms by inelastic electron-electron collisions. Actually, not all the secondary electrons generated in the solid emerge from the target. In order to emerge from the surface, the secondary electrons generated in the solid must reach the surface and satisfy given angular and energetic conditions. Of course, only the secondary electrons which are able to emerge from the target are included in the spectrum. Their energy distribution presents a pronounced peak in the region of the spectrum below 50 eV. The secondary electron emission yield is conventionally measured by integrating the area of the spectrum from 0 to 50 eV (including, in such a way, also the tail of backscattered electrons whose number, in this energy region, is actually negligible—unless the primary energy is very low as well).

8

1 Electron Transport in Solids

1.8 Characterization of Materials Simulation of the transport of electrons in materials has been demonstrated to be very important for many applications. The determination of electron emission from solids irradiated by a particle beam is of crucial importance, especially in connection with the analytical techniques that utilize electrons to investigate the chemical and compositional properties of solids in the near-surface layers. Electron spectroscopies and microscopies, examining how electrons interact with matter, represent fundamental tools to investigate the electronic and optical properties of matter. Electron spectroscopies and microscopies allow studying the chemical composition, electronic properties, and crystalline structure of materials. According to the energy of the incident electrons, a broad range of spectroscopic techniques can be utilized. For example, low energy electron diffraction (LEED) allows to investigate the crystalline structure of surfaces, Auger electron spectroscopy (AES) permits to analyze the chemical composition of the surfaces of solids, electron energy loss spectroscopy—both in transmission, when the spectrometer is combined with a transmission electron microscope, and in reflection—can be used to characterize materials by comparing the shape of the plasmon-loss peaks and the fine-structure features due to interband and intraband transitions with those of suitable standards, elastic peak electron spectroscopy is a useful tool to detect the presence of hydrogen in carbon-based materials. The study of the properties of a material using electron probes requires knowledge of the physical processes corresponding to the interaction of the electrons with the particular material under investigation. A typical AES peak of an atomic spectrum, for example, has a width in the range from 0.1 to 1.0 eV. In a solid, many energy levels are involved which are very close in energy, so broad peaks are typically observed in AES spectra of solids. Their features also depend on the instrumental resolution. Another important characteristic of the spectra is related to the shift in the energy of the peaks due to the chemical environment: Indeed the core energy levels of an atom are shifted when it is a part of a solid. This property is used to characterize materials, as the shift can be determined theoretically or by comparison with suitable standards. Even the changes in spectral intensities and the appearance of secondary peaks can be used for analyzing unknown materials. Electron spectra are used for self-supported thin film local thickness measurements, multilayer surface thin film thickness evaluation, doping dose determination in semiconductors, radiation damage investigations, and so on. The electron backscattering coefficient can be used for non-destructive evaluation of over-layer film thickness [45, 46], while the study of the energy distribution of the backscattered electrons may be utilized for materials characterization through the study of the shape of plasmon-loss peaks [47, 48]. Secondary electron investigation allows extraction of critical dimensions by modeling the physics of secondary electron image formation [49–51]. It permits to investigate doping contrast in p-n junctions and to evaluate accurate nanometrology for the most advanced CMOS processes [52, 53].

References

9

1.9 Summary Transport Monte Carlo simulation is a very useful mathematical tool for describing many important processes relative to the interaction of electron beams with solid targets. In particular, the backscattered and secondary electron emission from solid materials can be investigated with the use of the Monte Carlo method. Many applications of the Monte Carlo study of backscattered and secondary electrons concern materials analysis and characterization. Among the many applications of MC simulations to analysis and characterization, we can mention non-destructive evaluation of over-layer film thickness [45, 46], materials characterization through the study of the main features of the electron spectra and the shape of plasmon-loss peaks [47], extraction of critical dimensions by modeling the physics of secondary electron image formation [49–51], and doping contrast in p-n junctions for the evaluation of accurate nanometrology for the most advanced CMOS processes [52, 53].

References 1. R. Shimizu, Ding Ze-Jun. Rep. Prog. Phys. 55, 487 (1992) 2. D.C. Joy, Monte Carlo Modeling for Electron Microscopy and Microanalysis (Oxford University Press, 1995) 3. M. Dapor, Electron-Beam Interactions with Solids: Application of the Monte Carlo Method to Electron Scattering Problems (Springer, Berlin, 2003) 4. C.G.H. Walker, L. Frank, I. Müllerová, Scanning 38, 802 (2016) 5. N.F. Mott., Proc. R. Soc. Lond. Ser. 124, 425 (1929) 6. R.H. Ritchie, Phys. Rev. 106, 874 (1957) 7. H. Fröhlich, Adv. Phys. 3, 325 (1954) 8. J.P. Ganachaud, A. Mokrani, Surf. Sci. 334, 329 (1995) 9. M. Dapor, Phys. Rev. B 46, 618 (1992) 10. P. Sigmund, Particle Penetration and Radiation Effects (Springer, Berlin, 2006) 11. R.H. Ritchie, A. Howie, Phil. Mag. 36, 463 (1977) 12. H. Ibach, Electron Spectroscopy for Surface Analysis (Springer, Berlin, Heidelberg, 1977) 13. P.M. Echenique, R.H. Ritchie, N. Barberan, J. Inkson, Phys. Rev. B 23, 6486 (1981) 14. H. Raether, Excitation of Plasmons and Interband Transitions by Electrons (Springer, Berlin, 1982) 15. D.L. Mills, Phys. Rev. B 34, 6099 (1986) 16. D.R. Penn, Phys. Rev. B 35, 482 (1987) 17. J.C. Ashley, J. Electron Spectrosc. Relat. Phenom. 46, 199 (1988) 18. F. Yubero, S. Tougaard Phys. Rev. B 46, 2486 (1992) 19. Y.F. Chen, C.M. Kwei, Surf. Sci. 364, 131 (1996) 20. Y.C. Li, Y.H. Tu, C.M. Kwei, C.J. Tung, Surf. Sci. 589, 67 (2005) 21. A. Cohen-Simonsen, F. Yubero, S. Tougaard, Phys. Rev. B 56, 1612 (1997) 22. Z.-J. Ding, J. Phys.: Condens. Matter 10, 1733 (1988) 23. Z.-J. Ding, R. Shimizu, Phys. Rev. B 61, 14128 (2000) 24. Z.-J. Ding, H.M. Li, Q.R. Pu, Z.M. Zhang, R. Shimizu, Phys. Rev. B 66, 085411 (2002) 25. W.S.M. Werner, W. Smekal, C. Tomastik, H. Störi, Surf. Sci. 486, L461 (2001) 26. R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 3rd edn. (Springer, New York, Dordrecht, Heidelberg, London, 2011)

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1 Electron Transport in Solids

27. R. Garcia-Molina, I. Abril, C.D. Denton, S. Heredia-Avalos, Nucl. Instrum. Methods Phys. Res. B 249, 6 (2006) 28. R.F. Egerton, Rep. Prog. Phys. 72, 016502 (2009) 29. S. Taioli, S. Simonucci, L. Calliari, M. Filippi, M. Dapor, Phys. Rev. B 79, 085432 (2009) 30. S. Taioli, S. Simonucci, M. Dapor, Comput. Sci. Discovery 2, 015002 (2009) 31. S. Taioli, S. Simonucci, L. Calliari, M. Dapor, Phys. Rep. 493, 237 (2010) 32. G. Gergely, Progr. Surf. Sci. 71, 31 (2002) 33. A. Jablonski, Progr. Surf. Sci. 74, 357 (2003) 34. D. Varga, K. Tökési, Z. Berènyi, J. Tóth, L. Kövér, G. Gergely, A. Sulyok, Surf. Interface Anal. 31, 1019 (2001) 35. A. Sulyok, G. Gergely, M. Menyhard, J. Tóth, D. Varga, L. Kövér, Z. Berènyi, B. Lesiak, A. Jablonski, Vacuum 63, 371 (2001) 36. G.T. Orosz, G. Gergely, M. Menyhard, J. Tóth, D. Varga, B. Lesiak, A. Jablonski, Surf. Sci. 566–568, 544 (2004) 37. F. Yubero, V.J. Rico, J.P. Espinós, J. Cotrino, A.R. González-Elipe, Appl. Phys. Lett. 87, 084101 (2005) 38. V.J. Rico, F. Yubero, J.P. Espinós, J. Cotrino, A.R. González-Elipe, D. Garg, S. Henry, Diam. Relat. Mater. 16, 107 (2007) 39. D. Varga, K. Tökési, Z. Berènyi, J. Tóth, L. Kövér, Surf. Interface Anal. 38, 544 (2006) 40. M. Filippi, L. Calliari, Surf. Interface Anal. 40, 1469 (2008) 41. M. Filippi, L. Calliari, C. Verona, G. Verona-Rinati, Surf. Sci. 603, 2082 (2009) 42. P. Auger, P. Ehrenfest, R. Maze, J. Daudin, R.A. Fréon, Rev. Mod. Phys. 11, 288 (1939) 43. L. Meitner, Z. Phys. 17, 54 (1923) 44. G. Wentzel, Z. Phys. 43, 524 (1927) 45. M. Dapor, N. Bazzanella, L. Toniutti, A. Miotello, S. Gialanella, Nucl. Instrum. Methods Phys. Res. B 269, 1672 (2011) 46. M. Dapor, N. Bazzanella, L. Toniutti, A. Miotello, M. Crivellari, S. Gialanella, Surf. Interface Anal. 45, 677 (2013) 47. M. Dapor, L. Calliari, G. Scarduelli, Nucl. Instrum. Methods Phys. Res. B 269, 1675 (2011) 48. M. Dapor, L. Calliari, S. Fanchenko, Surf. Interface Anal. 44, 1110 (2012) 49. M. Dapor, M. Ciappa, W. Fichtner, J. Micro/Nanolith, MEMS MOEMS 9, 023001 (2010) 50. M. Ciappa, A. Koschik, M. Dapor, W. Fichtner, Microelectron. Reliab. 50, 1407 (2010) 51. A. Koschik, M. Ciappa, S. Holzer, M. Dapor, W. Fichtner, Proc. SPIE 7729, 77290X–1 (2010) 52. M. Dapor, M.A.E. Jepson, B.J. Inkson, C. Rodenburg, Microsc. Microanal. 15, 237 (2009) 53. C. Rodenburg, M.A.E. Jepson, E.G.T. Bosch, M. Dapor, Ultramicroscopy 110, 1185 (2010)

Chapter 2

Computational Minimum

This book offers several numerical approaches for the solution of many transport phenomena problems. We need to be able to calculate the relevant differential scattering cross-sections, both elastic and inelastic, and this requires the knowledge of the methods of numerical analysis and computational physics. In this chapter, the basic aspects of computational approaches [1–3] will be briefly summarized. In particular, we will describe the most important and frequently used methods of numerical differentiation, numerical quadrature, and numerical solution of ordinary differential equations. Such methods are crucial for investigating many physics problems such as, for example, the solution of Dirac’s equation in a central field, which is the prerequisite for the computation of the Mott elastic scattering cross-section. A short study of some important special mathematical physics functions (Legendre polynomials, associated Legendre functions, and spherical Bessel functions, both regular and irregular) and recursion formulas for their calculation will be provided at the end of the chapter.

2.1 Numerical Differentiation Let us expand a function f (x) by using a Taylor series. In the neighborhood of x = 0 we can write: x 2  f (0) + · · · . (2.1) f (x) = f (0) + x f  (0) + 2! From this equation, it follows that f (h) = f (0) + h f  (0) +

h 2  f (0) + · · · , 2!

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1_2

(2.2) 11

12

2 Computational Minimum

f (−h) = f (0) − h f  (0) +

h 2  f (0) + · · · , 2!

(2.3)

so that f =

f (h) − f (0) + O(h) , h

(2.4)

f (h) − f (−h) + O(h 2 ) . 2h

(2.5)

and f =

The “two-point" formula expressed by (2.4) is exact if f is a linear function in the interval [0, h]. In other words (2.4) assumes a linear interpolation of the function f through x = 0 and x = h. The “three-point" formula expressed by (2.5) is exact if f is a second-degree polynomial function in the interval [−h, h]. In other words (2.5) assumes a quadratic polynomial interpolation of the function f through x = −h, x = 0, and x = h. Let us note that higher-order derivatives can also be calculated using similar procedures. For example, from (2.2) and (2.3) it is easy to see that f  =

f (h) − 2 f (0) + f (−h) + O(h 2 ) . h2

(2.6)

2.2 Numerical Quadrature 2.2.1 Trapezoidal Rule, Simpson’s Rule, Bode’s Rule Assuming that the function f is approximately linear in the interval [0, h], we obtain the trapezoidal rule:  h f (0) + f (h) h + O(h 3 ) . f (x) d x = (2.7) 2 0 The name “trapezoidal rule" originates from the fact that, if f has positive values, the integral is approximated by the area of a trapezoid. It can also be expressed in a symmetric form about x = 0 as follows (assuming that f is approximately linear in the two intervals [−h, 0] and [0, h]):  h f (−h) + 2 f (0) + f (h) h + O(h 3 ) . f (x) d x = (2.8) 2 −h

2.2 Numerical Quadrature

13

For the sake of increased accuracy, let us consider the following approximation of the Taylor expansion, valid for |x| < h:

f (x) ≈ f (0) +

f (h) − 2 f (0) + f (−h) 2 f (h) − f (−h) x + x . 2h 2h 2

(2.9)

Let us now integrate this equation from −h to h: 

h

−h

f (x) d x = 2h f (0) +

2h 3 f (h) − 2 f (0) + f (−h) + O(h 5 ). 3 2h 2

(2.10)

By rearranging this equation we obtain the so called Simpson’s quadrature rule:  h f (h) + 4 f (0) + f (−h) h + O(h 5 ). f (x) d x = (2.11) 3 −h Considering more terms in the Taylor expansion we obtain higher order formulas, such as the Bode’s quadrature rule: 

x0 +4h

f (x) d x =

(2.12)

x0

2h [7 f (x0 ) + 32 f (x0 + h) + 12 f (x0 + 2h) + 45 + 32 f (x0 + 3h) + 7 f (x0 + 4h)] + O(h 7 ).

=

2.2.2 Gaussian Quadrature Let us consider the integral  S =

1 −1

f (x)d x .

(2.13)

The elementary quadrature formulas described in the previous subsection use the following approach to numerically approximate S: S ≈

N  n=1

cn f (xn ) .

(2.14)

14

2 Computational Minimum

In the last equation xn = 2

n−1 − 1. N −1

(2.15)

The coefficients cn can be determined by solving the following system of N linear equations: 

1

x dx = p

1

N 

cn xnp ,

(2.16)

n=1

where p = 0, 1, · · · , N − 1 . Let us consider, for example, the Simpson’s rule. In this case N = 3, x1 = −1, x2 = 0, x3 = +1, and we have  1 dx = 2 , (2.17) c1 + c2 + c3 = −1

 c1 x1 + c2 x2 + c3 x3 = −c1 + c3 =

x dx = 0 ,

(2.18)

x 2 d x = 2/3 .

(2.19)

−1

 c1 x12 + c2 x22 + c3 x32 = c1 + c3 =

1

1 −1

Hence, c1 = 1/3, c2 = 4/3, c3 = 1/3. In order to increase precision, we can give up the requirement of equally spaced points xn . In this case it is possible to show that, choosing xn as the N zeros of the Legendre polynomial PN (x) [2], cn =

2 (1 −

xn2 )[PN (xn )]2

.

(2.20)

Once the values of xn and cn are known, we can easily compute the integral (2.13). If we are interested in calculating the integral  b f (x) d x , (2.21) Sab = a

we just have to carry out the following change of variable: y = 2

x −a − 1. b−a

(2.22)

Note that the Gaussian quadrature is a very fast algorithm, but it requires that the integrand be a smooth function of x. For rapidly varying integrand functions, Gaus-

2.3 Ordinary Differential Equations

15

sian quadrature can still be used integrating over many subintervals of the integration range [2].

2.3 Ordinary Differential Equations The simplest ordinary differential equation is the following: dy = f (x, y) , dx

(2.23)

where f (x, y) is a known function of x and y. We are interested in finding the function y(x) satisfying (2.23) with the initial condition y(x0 ) = y0 ,

(2.24)

where x0 and y0 are two given real numbers. The Euler’s method is the simplest numerical approach for the solution of this problem. In order to illustrate this method, let us consider the case where the independent variable x ranges in the interval [0, 1]. Let us assume that x0 = 0 and divide the interval [0, 1] in N subintervals of identical length h = 1/N . The Euler’s algorithm is based on the following simple recursion formula, that numerically solves (2.23) by a step-by-step integration: yn+1 = yn + h f (xn , yn ) + O(h 2 ) ,

(2.25)

where yn = y(xn ) and xn = nh . Note that, since the local error is O(h 2 ), we have a global error N O(h 2 ) = (1/ h) O(h 2 ) ≈ O(h) due to the N steps necessary to integrate from x = 0 to x = 1. The accuracy of the Euler’s method is relatively low. A more accurate approach is represented by the Runge-Kutta method. According to the second order Runge-Kutta method, for example, the solution of the ordinary differential equation (2.23) can be approximated as yn+1 = yn + h f (xn + h/2, yn + k/2) + O(h 3 ) ,

(2.26)

k = h f (xn , yn ) .

(2.27)

where

According to the fourth order Runge-Kutta method, we have yn+1 = yn +

1 (k1 + 2 k2 + 2 k3 + k4 ) + O(h 5 ) , 6

(2.28)

16

2 Computational Minimum

where k1 = h f (xn , yn ) , k2 = h f (xn + h/2, yn + k1 /2) , k3 = h f (xn + h/2, yn + k2 /2) , k4 = h f (xn + h, yn + k3 ) .

(2.29)

2.4 Special Functions of Mathematical Physics 2.4.1 Legendre Polynomials and Associated Legendre Functions The Legendre polynomial of degree l (l = 0, 1, 2, . . . , ∞) is defined by the formula Pl (u) =

1 dl 2 (u − 1)l . 2l l! du l

(2.30)

It is a polynomial with l zeros in the range (−1, +1) and with parity (−1)l . The first five Legendre polynomials are P0 = 1 ,

(2.31)

P1 = u ,

(2.32)

P2 =

1 (3u 2 − 1) , 2

(2.33)

P3 =

1 (5u 3 − 3u) , 2

(2.34)

1 (35u 4 − 30u 2 + 3) . 8

(2.35)

P4 =

Let us now introduce the associated Legendre functions: Plm (u) = (1 − u 2 )(1/2)m

dm Pl (u) . du m

(2.36)

The Legendre polynomials are the particular associated Legendre functions corresponding to m = 0: Pl (u) = Pl0 (u) .

(2.37)

2.4 Special Functions of Mathematical Physics

The associated Legendre functions satisfy the orthonormality relations  +1 2 (l + m)! Pkm Plm du = δkl , 2l + 1 (l − m)! −1 and the differential equation   2 m2 d 2 d (1 − u ) 2 − 2u Plm = 0 . + l(l + 1) − du du 1 − u2

17

(2.38)

(2.39)

The following recursion relations are very useful when using the Legendre polynomials: m m (u) + (l + m)Pl−1 (u) = (2l + 1)u Plm (u) , (l − m + 1)Pl+1

(1 − u 2 )

d m m P (u) = (l + m)Pl−1 (u) − lu Plm (u) . du l

(2.40) (2.41)

In order to find the recursion relations corresponding to the Legendre polynomials, we can impose m = 0 in the previous equations and obtain the following: (l + 1)Pl+1 (u) + l Pl−1 (u) = (2l + 1)u Pl (u) , (1 − u 2 )

d Pl (u) = l Pl−1 (u) − lu Pl (u) . du

(2.42) (2.43)

The Legendre polynomials are eigenfunctions of the square of the orbital angular momentum:  2  ∂ 1 ∂2 ∂ 2 2 , (2.44) + L = − + cot θ ∂θ2 ∂θ sin2 θ ∂φ2 L2 Pl (cos θ) = 2 l(l + 1)Pl (cos θ) .

(2.45)

2.4.2 Bessel Functions Let us introduce the Bessel equation of order ν: x2

d2 y dy + (x 2 − ν 2 )y = 0 . +x dx2 dx

(2.46)

The solution of this equation is a linear combination of the Bessel functions J−ν and J+ν . Let us now consider the Schrödinger equation of a particle in a constant potential V0 ,

18

2 Computational Minimum

(∇ 2 + K2 − U0 )Ψ = 0 ,

(2.47)

where ∇ ≡ (∂/∂x, ∂/∂ y, ∂/∂z), K2 = 2m E/2 , and U0 = 2mV0 /2 . In order to proceed, let us expand the wave function in Legendre polynomials: Ψ (r, cos θ) =

∞  l=0

al

yl (r ) Pl (cos θ) . r

(2.48)

Taking into account (2.45), the Schrödinger equation becomes the following:  ∞  2  ∂ l(l + 1) yl (r ) 2 ∂ 2 − Pl (cos θ) = 0 . + + K − U 0 al 2 2 ∂r r ∂r r r l=0

(2.49)

All the coefficients of the expansion must then satisfy the following differential equations:  2  ∂ l(l + 1) yl (r ) 2 ∂ 2 al − =0. (2.50) + + K − U 0 2 2 ∂r r ∂r r r If we define k 2 = K 2 − U0 ,

(2.51)

we can write 

 d2 l(l + 1) 2 yl (r ) = 0 . − + k dr 2 r2

(2.52)

Let us now introduce the variable x ≡ kr , so that the last equation can be rewritten as  2  d l(l + 1) − + 1 yl (x) = 0 . (2.53) dx2 x2 The spherical Bessel function of order l is defined by  π Jl+1/2 (x) . jl (x) = 2x

(2.54)

Since x 1/2 Jl+1/2 is a solution of (2.53), we can conclude that the function kr jl (kr ) is a solution of (2.52). In a similar way, once the spherical Neumann function of order l (also known as the irregular spherical Bessel function of order l) has been defined as

2.4 Special Functions of Mathematical Physics

19

 n l (x) = (−1)

l+1

π J−l−1/2 (x) , 2x

(2.55)

it is possible to show that kr n l (kr ) is a solution of (2.52). The first three regular spherical Bessel functions are j0 =

sin x , x

sin x cos x − , 2 x x   3 3 1 sin x − 2 cos x , − j2 = x3 x x j1 =

(2.56) (2.57)

(2.58)

while the first three spherical Neumann functions are n0 = −

cos x , x

cos x sin x , − 2 x x   3 3 1 cos x − 2 sin x . n2 = − 3 + x x x n1 = −

(2.59) (2.60)

(2.61)

It can be proved that xl , 1 · 3 · ... · (2l + 1) x→0

jl (x) ∼ and that

n l (x) ∼ − x→0

1 · 3 · ... · (2l − 1) . x l+1

(2.62)

(2.63)

The following equation holds: jl (0) = δl0 .

(2.64)

The asymptotic behavior of the Bessel and Neumann functions is described by the following equations: sin(x − lπ/2) , (2.65) jl (x) ∼ x x→∞

20

2 Computational Minimum

n l (x) ∼ − x→∞

cos(x − lπ/2) . x

(2.66)

If we indicate by fl any linear combination of the Bessel and Neumann functions ( fl = a jl + bn l , where a and b are arbitrary coefficients), we have x fl−1 − (2l + 1) fl + x fl+1 = 0 , x fl−1 − (l + 1) fl − x

d fl =0. dx

(2.67) (2.68)

2.5 Summary The basic aspects of the computational methods were summarized. In particular, we described methods of numerical differentiation, rules of numerical quadrature and algorithms for the numerical solution of the ordinary differential equations (Euler method, Runge-Kutta methods). The last part of the chapter was devoted to a short study of some important special functions of the mathematical physics which are very important in the study of the scattering of electrons by central fields. In particular, recursion formulas were provided for the calculation of the Legendre polynomials, the associated Legendre functions, and the spherical Bessel functions, both regular and irregular.

References 1. R.L. Burden, J.D. Faires, Numerical Analysis (PWS, Boston, 1985) 2. S.E. Koonin, D.C. Meredith, Computational Physics (Addison-Wesley, Redwood City, 1990) 3. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C. The Art of Scientific Computing (Cambridge University Press, Cambridge, 1985, 1992, 2007)

Chapter 3

Cross-Sections. Basic Aspects

In electron microscopies and spectroscopies, electrons penetrate a material experiencing many different scattering processes. For a realistic description of the electron emission, it is necessary to know all the scattering mechanisms involved [1, 2]. This chapter is devoted to the introduction of the concepts of cross-section and stopping power. For an excellent review of the topics treated in this chapter, also see [1]. The chosen approach is deliberately elementary since the focus is on the basic aspects of the penetration theory. Specific details will be provided in the next chapters and in the Appendices. From the macroscopic point of view, the cross-section represents the area of a target that can be hit by a projectile, so that it depends on the geometrical properties of both the target and the projectile. Let us consider, for example, a point bullet impinging on a spherical target whose radius is r. The cross-section σ of the target is, in such a case, simply given by σ = πr 2 . In the microscopic world, the concept has to be generalized in order to take into account that the cross-section does not only depend on the projectile and on the target, but also on their relative velocity and on the physical phenomena we are interested in: Examples are represented by the elastic scattering cross-section and the inelastic scattering cross-section of electrons (projectiles) impinging on atoms (targets). The elastic scattering cross-section describes the interactions in which the sum of kinetic energies of the incident particle (the electron) and the target (the atom) does not change. Since the mass of the electron is much smaller than the mass of the atom, the kinetic energy of the electron is, after an elastic collision, practically the same before and after the interaction. The inelastic scattering cross-section, on the other hand, describes the collisions corresponding to an energy transfer from the incident particle (the electron) to the target (the atom). As a consequence, the kinetic energy of the incident electron decreases due to the interaction so the electron slows down. As the cross-section is a function of the kinetic energy of the incident electron, after every inelastic collision the cross-sections (both elastic and inelastic) of the subsequent collision, if any, will be changed accordingly.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1_3

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22

3 Cross-Sections. Basic Aspects

In real experiments, the investigators cannot measure the cross-section for a single electron that hits a single atom. The typical experiment consists, instead, in the collision of a great number of electrons, called the beam, with a medium constituted by a configuration of many atoms and/or molecules (a gas, for example, or an amorphous or crystalline solid). The electrons constituting the beam have, in principle, all the same initial energy (the primary energy) and do not interact with each other but only with the atoms of the medium. Actually, in practical cases, the energies of the electrons constituting the primary beam are distributed around the primary energy which has to be considered their mean energy. Furthermore, the electrons of the beam do not interact only with the target atoms (or molecules) but with each other as well. Neglecting these interactions corresponds to investigating the so-called low current beam approximation [1].

3.1 Cross-Section and Probability of Scattering Let us indicate with σ the cross-section of the physical effect we are interested in describing, and with J the density current, i.e., the number of electrons per unit area and per unit time in the beam. Let us indicate, furthermore, with N the number of targets per unit volume (number density) and with A the area of the target where the beam is spread. Let us assume that the beam spreading is homogeneous. If z is the depth where the collisions occur, then the volume where the electrons interact with the stopping medium is given by zA and, as a consequence, the number of collisions per unit of time can be calculated by NzAJ σ. As the product of A by J is the number of electrons per unit of time, the quantity P = Nzσ

(3.1)

represents the mean number of collisions per electron. In the hypothesis that the target thickness z is very small (thin layers) or the density of the target atoms N is very small (gas targets), so that P  1, P represents the probability that an electron suffers a collision while traveling in the medium. In order to take into account that in the great majority of the experiments the projectile undergoes many collisions, let us associate to the trajectory of each particle a cylindrical volume V = zσ and calculate the probability Pν to hit ν target particles in this volume. If the positions of the particles are not correlated, as in the case, for example, with an ideal gas, such a probability is given by the Poisson distribution Pν =

(Nzσ)ν (NV )ν exp(−NV ) = exp(−Nzσ) , ν! ν!

where ν = 0, 1, 2, 3, ....

(3.2)

3.2 Stopping Power and Inelastic Mean Free Path

23

Let us first consider the single collision problem, ν = 1. The probability of hitting precisely one particle in the volume zσ is given by P1 = P(ν=1) = (Nzσ) exp(−Nzσ) ,

(3.3)

so that, in the limit Nzσ  1, P1 ∼ = P = Nzσ ,

(3.4)

which is the same result deduced above. Also note that, in the same limit, the probability for no collision at all is given by 1 − P = 1 − Nzσ. This is the first order in Nzσ of the well-known Lambert and Beer’s absorption law: P0 = P(ν=0) = exp(−Nzσ) .

(3.5)

Note that, to the first order in Nzσ, the probability for double events is equal to zero. As it is well known, one of the characteristics of the Poisson distribution is that its expected value and variance are identical. In particular, the average value ν and the variance (ν − ν)2 , are given by ν = (ν − ν)2  = Nzσ ,

(3.6)

so that the relative fluctuation goes to zero as the reciprocal of the square root of ν = Nzσ:  (ν − ν)2  1 = √ . (3.7) 2 ν ν

3.2 Stopping Power and Inelastic Mean Free Path Let us now consider the collisions with the stopping medium resulting in a kinetic energy transfer from the projectile to the target atoms and/or molecules constituting the target. Let us assume that the energy transfer Ti (i = 1, 2, ...) is small with respect to the incident particle kinetic energy E. Let us also assume that νi is the number of events corresponding to the energy loss Ti , so that the total energy ΔElost by an incident particle passing through a thin film of thickness Δz is given by i νi Ti . As the mean number of type i collisions, according to (3.6), is given by νi  = N Δzσi , where σi is the energy-loss cross-section, the energy loss is given by  Ti σi . (3.8) ΔE = N Δz i

24

3 Cross-Sections. Basic Aspects

The stopping power is defined as  ΔE Ti σi , = N Δz i and the stopping cross-section S is given by  Ti σi , S =

(3.9)

(3.10)

i

so that ΔE = NS . Δz

(3.11)

If the spectrum of the energy loss is continuous, instead of discrete, the stopping cross-section assumes the form  d σinel dT , (3.12) S = T dT the stopping power is given by ΔE = N Δz

 T

d σinel dT , dT

and the total inelastic scattering cross-section σinel can be calculated by  d σinel dT , σinel = dT

(3.13)

(3.14)

where d σinel /dT is the so-called differential inelastic scattering cross-section. Once the total inelastic scattering cross-section σinel is known, the inelastic mean free path λinel can be calculated by λinel =

1 . N σinel

(3.15)

3.3 Range While the inelastic mean free path is the average distance between two inelastic collisions, the maximum range is the total path length of the projectile. It can be easily estimated—using the simple approach we are describing in this section—in all the cases in which the energy straggling, i.e., the statistical fluctuations in energy loss, can be neglected [3, 4]. Indeed, in this case, the energy of the incident particle

3.4 Energy Straggling

25

is a decreasing function of the depth z calculated from the surface of the target, so that E = E(z). As the stopping cross-section is a function of the incident particle energy, S = S(E), the (3.11) assumes the form of the following differential equation dE = −NS(E) , dz

(3.16)

where the minus sign has been introduced in order to take into account that, as already stated, the projectile energy E(z) is a decreasing function of the depth z. Indicating with E0 the initial energy of the projectiles (the so-called beam primary energy) the maximum range R can be easily obtained by the integration:  0  R dz , (3.17) dz = dE R = dE 0 E0 so that 

E0

R = 0

dE . N S(E)

(3.18)

3.4 Energy Straggling Actually, the range calculated in such a way can be different from the real one because of the statistical fluctuations of the energy loss. The consequences of such a phenomenon, known as energy straggling, can be evaluated following a procedure similar to the one used for introducing the stopping cross-section. Let us firstly consider then, as before, the discrete case and calculate the variance Ω 2 , or mean square fluctuation, in the energy loss ΔE, given by Ω 2 = (ΔE − ΔE)2  .

(3.19)

Since ΔE − ΔE =



(νi − νi ) Ti ,

(3.20)

i

we have Ω2 =

  (νi − νi ) (νj − νj )  Ti Tj , i,j

(3.21)

26

3 Cross-Sections. Basic Aspects

so, due to the statistical independence of the collisions and the properties of the Poisson distribution,1   Ω2 = (νi − νi )2  Ti2 = νi  Ti2 . (3.22) i

i

As a consequence, taking into account that νi  = N Δzσi , the energy straggling can be expressed as  Ti2 σi = N Δz W (3.23) Ω 2 = N Δz i

where we have introduced the straggling parameter defined as  Ti2 σi . W =

(3.24)

i

If the spectrum of the energy loss is continuous, instead of discrete, the straggling parameter assumes the form  d σinel dT . (3.25) W = T2 dT

3.5 Summary In this chapter, the elementary theory of electron penetration in solid targets has been briefly described [1]. We have discussed the fundamental concepts of cross-section, stopping power, maximum range of penetration, and energy straggling. Details of specific applications and calculations of the scattering mechanisms along with the main theoretical approaches, which describe the cross-sections relative to the interaction of the incident electrons with atomic nuclei, atomic electrons, plasmons, and phonons will be the subject of the next chapters.

1

Consider the case i = j. Because of the statistical independence of the collisions, we have  (νi − νi ) (νj − νj )  =  νi −  νi    νj − νj   .

Since  νi −  νi   = 0, all the terms with i = j are 0. As for the terms with i = j, taking into account the properties of the Poisson distribution, we have  (νi − νi )2  =  νi  . .

References

27

References 1. P. Sigmund, Particle Penetration and Radiation Effects (Springer, Berlin, 2006) 2. M. Dapor, Electron-Beam Interactions with Solids: Application of the Monte Carlo Method to Electron Scattering Problems (Springer, Berlin, 2003) 3. M. Dapor, Phys. Rev. B 43, 10118 (1991) 4. M. Dapor, Surf. Sci. 269/270, 753 (1992)

Chapter 4

Scattering Mechanisms

This chapter is devoted to the main mechanisms of scattering (elastic, quasi-elastic, and inelastic) that are relevant to the description of the interaction of electron beams with solid targets. Firstly the elastic scattering cross-section will be described, comparing the screened Rutherford formula to the Mott cross-section [1]. The Mott theory is based on the relativistic partial wave expansion method and the solution of the Dirac equation in a central field. While the quantum-relativistic Mott theory is the best description of the elastic scattering of electrons impinging on atoms, the Rutherford formula represents an approximation of the exact theory, valid only when the ratio between the electron kinetic energy and the atomic potential energy is high and, as a consequence, the scattering is weak (first Born approximation). We will also briefly describe the Fröhlich theory [2], which deals with the quasielastic events occurring when electron energy is very low and the probability of electron-phonon interaction becomes significant. We will discuss energy loss and energy gain due to electron phonon-interactions, and see that electron energy gains can be safely neglected. Electron energy losses are fractions of eV. The Bethe-Bloch stopping power formula [3] and semi-empirical approaches [4, 5] will be presented, along with the limits of these models for the calculation of energy losses. The Ritchie dielectric theory [6] will then be considered, which is used for the accurate calculation of electron energy losses due to electron-plasmon interaction. Polaronic effects will be also mentioned, as they are important mechanisms for trapping very slow electrons in insulating materials [7]. A discussion about the inelastic mean free path will be provided that takes into account all the inelastic scattering mechanisms introduced in this chapter. Lastly, surface phenomena will be described along with numerical calculations of surface and bulk plasmon loss spectra. Many details about the most important theoretical models presented in this chapter can be found in the Appendices. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1_4

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4 Scattering Mechanisms

4.1 Elastic Scattering Electron-atom elastic scattering is the main responsible for the angular deflection of electrons traveling in solid targets. For some reviews about the subject of elastic scattering see, for example, [8–18]. Elastic scattering is not only the cause of the electron deflection: It also accounts for electron energy-loss problems, for it contributes to changing the angular distribution of the inelastically scattered electrons [12, 13]. Since a nucleus is much more massive than an electron, the energy transfer is very small and, typically, negligible in an electron-nucleus collision. The great majority of elastic collisions regard the interaction of the incident electrons with the electrostatic nuclear field in regions that are far from the center of mass of the nucleus where, due to both the inverse square law and the shielding of the nucleus by the atomic electrons, the potential is relatively weak. For this reason, many electrons are elastically scattered through small angles. Conservation of energy and momentum requires small transfers of energy between the electrons and the nuclei that depend on the angle of scattering. Even if the electron energy transfers are a very small fraction of eV, in many circumstances they cannot be neglected. Furthermore, it has to be noted that, despite this general rule, in a few cases significant energy transfers are possible. Indeed, even if electron energy losses are typically very small and often irrelevant in electron-nucleus collisions, for the very rare cases of head-on collisions, where the scattering angle is equal to 180◦ , the energy transfer can be, for the case of light elements, higher than the displacement energy, namely the energy necessary to displace the atom from its lattice position. In these cases, displacement damage and/or atom removal (sputtering) can be observed [12, 13, 19]. The differential elastic scattering cross-section represents the probability per unit solid angle that an electron is elastically scattered by an atom and is given by the square modulus of the complex scattering amplitude f , which is a function of the scattering angle θ, of the incident electron energy E, and of the atomic number Z of the target. The angular distribution, once taken into account that the Coulomb potential is screened by the atomic electrons, can be calculated either by the use of the first Born approximation (screened Rutherford cross-section) (see Appendix A) or, in order to obtain more accurate results—in particular for low-energy electrons—, by solving the Schrödinger equation in a central field (partial wave expansion method, PWEM) (see Appendix B). Typically, for the case of the screened Rutherford formula obtained within the first Born approximation, the screening of atomic electrons is described by the Wentzel formula [20], which corresponds to a Yukawa exponential attenuation of the nuclear potential as a function of the distance from the center of mass of the nucleus. For the case of the more accurate partial wave expansion method, the Dirac-Hartree-FockSlater approach is generally used for calculating the screened nuclear potential.

4.1 Elastic Scattering

31

A further improved approach for obtaining a very accurate calculation of the differential elastic scattering cross-section, valid also for relativistic electrons, is the so-called relativistic partial wave expansion method (RPWEM),—which is based on the solution of the Dirac equation in a central field (Mott cross-section)—where the sum of the squares of the moduli of two complex scattering amplitudes f and g is required for the calculation of the elastic scattering probabilities [1] (see Appendix C). Also in this case, the Dirac-Hartree-Fock-Slater method is utilized to calculate the shielded nuclear potential.

4.1.1 Mott Cross-Section Verses Screened Rutherford Cross-Section The relativistic partial wave expansion method (Mott theory) [1] permits to calculate the differential elastic scattering cross-section for spin-unpolarized electron beams as follows:   dσel dσel = = | f (θ) |2 + | g(θ) |2 , (4.1) dΩ dΩ unpolarized where f (θ) and g(θ) are the scattering amplitudes (direct and spin-flip, respectively). For details about the Mott theory and the calculation of the scattering amplitudes f (θ) and g(θ), see Appendix C and [14, 17, 18]. Also, see [16, 21–23] for several applications. Once the differential elastic scattering cross-section has been calculated, the total elastic scattering cross-section σel and the first transport elastic scattering crosssection σtr can be computed using the following equations:  σel =

dσel dΩ , dΩ

 σtr =

(1 − cos ϑ)

dσel dΩ . dΩ

(4.2)

(4.3)

It can be interesting to investigate the high energy and low atomic number limits of the Mott theory (corresponding to the first Born approximation). Along with the assumption that the atomic potential can be written according to the Wentzel formula [20]:  r Z e2 exp − , (4.4) V (r ) = − r a where r is the distance between the incident electron and the nucleus, Z the target atomic number, e the electron charge, and a approximately represents the screening of the nucleus by the orbital electrons, given by

32

4 Scattering Mechanisms

a =

a0 , Z 1/3

(4.5)

where a0 is the Bohr radius, the first Born approximation permits to write the differential elastic scattering cross-section in an analytical closed form. It is the so-called screened Rutherford cross-section: dσel 1 Z 2 e4 , = dΩ 4E 2 (1 − cos θ + α)2 α =

me4 π 2 Z 2/3 h2 E

(4.6)

(4.7)

In these equations, m is the electron mass, and h is the Planck constant. The screened Rutherford formula has been largely used even if it is unable to describe all the features corresponding to the elastic scattering as a function of the scattering angle that one can observe when incident electron kinetic energies are lower than ∼5–10 keV and the target atomic number is relatively high. In Figs. 4.1, 4.2, 4.3 and 4.4 the differential elastic scattering cross-section dσel /dΩ (DESCS)— calculated with both the Mott and the Rutherford theories—are compared. The presented data concern two different elements (Cu and Au) and two different energies (1000 eV and 3000 eV).1 From the comparison clearly emerges that the Rutherford theory approaches the Mott theory as the atomic number decreases and the primary energy increases. Indeed, the Rutherford formula can be deduced assuming the first Born approximation, which is valid when [24] E 

e2 2 Z . 2a0

(4.8)

In other words, the higher the electron energy—in comparison with the atomic potential—the higher the accuracy of the Rutherford theory (see, in particular, Fig. 4.3). Anyway, the Rutherford formula represents a decreasing function of the scattering angle, so that it should not be surprising that it cannot describe the features that emerge as the electron energy is low and the atomic number is high (see, in particular, Fig. 4.2).

1

Please note that, in Figs. 4.1, 4.2, 4.3, and 4.4, the screening function used for obtaining the Rutherford formula (Wentzel-like) is not the same one we used for the Mott calculation (DiracHartree-Fock-Slater). Thus, the differences observed in the figures are due not only to the more accurate approach represented by the Mott theory but also to the better description of the atomic potential used in the calculation of the Mott cross-section.

4.1 Elastic Scattering

33

1

2

DESCS (Å /sr)

10

0.1

0.01

0

30

60

90

120

150

180

Scattering angle (deg)

Fig. 4.1 Differential elastic scattering cross-section of 1000 eV electrons scattered by Cu as a function of the scattering angle [17]. Solid line: Relativistic partial wave expansion method (Mott theory). Dashed line: Screened Rutherford formula, (4.6)

1

2

DESCS (Å /sr)

10

0.1

0.01

0.001 0

30

60

90

120

150

180

Scattering angle (deg)

Fig. 4.2 Differential elastic scattering cross-section of 1000 eV electrons scattered by Au as a function of the scattering angle [17]. Solid line: Relativistic partial wave expansion method (Mott theory). Dashed line: Screened Rutherford formula, (4.6)

34

4 Scattering Mechanisms

1

2

DESCS (Å /sr)

10

0.1

0.01

0.001 0

30

60

90

120

150

180

Scattering angle (deg)

Fig. 4.3 Differential elastic scattering cross-section of 3000 eV electrons scattered by Cu as a function of the scattering angle [17]. Solid line: Relativistic partial wave expansion method (Mott theory). Dashed line: Screened Rutherford formula, (4.6)

2

DESCS (Å /sr)

10

1

0.1

0.01

0.001 0

30

60

90

120

150

180

Scattering angle (deg)

Fig. 4.4 Differential elastic scattering cross-section of 3000 eV electrons scattered by Au as a function of the scattering angle [17]. Solid line: Relativistic partial wave expansion method (Mott theory). Dashed line: Screened Rutherford formula, (4.6)

4.1 Elastic Scattering

35

In Monte Carlo simulations, when the electron primary energy is higher than 10 keV, Rutherford cross-section is sometimes used—instead of the more accurate Mott cross-section—mainly because it provides a very simple analytical way to calculate both the cumulative probability of elastic scattering into an angular range from 0 to θ, Pel (θ, E), and the elastic mean free path, λel . It is useful to see how Pel (θ, E) and λel can be calculated in a completely analytical way taking advantage of the particular form of the screened Rutherford formula. In the first Born approximation these quantities are in fact given, respectively, by Pel (θ, E) =

(1 + α/2) (1 − cos θ) , 1 + α − cos θ

λel =

(4.9)

α (2 + α) E 2 , N π e4 Z 2

(4.10)

where N is the number of atoms per unit volume in the target. The demonstration of these equations is quite easy. Indeed Pel (θ, E) =  θ  1−cos θ+α e4 Z 2 2π sin ϑ dϑ du πe4 Z 2 = = , 4σel E 2 0 (1 − cos ϑ + α)2 2σel E 2 α u2 where σel



e4 Z 2 = 4E 2

π

2π sin ϑ dϑ πe4 Z 2 = (1 − cos ϑ + α)2 2E 2

0



2+α α

du . u2

Since 

1−cos θ+α

α

du 1 − cos θ = 2 u α(1 − cos θ + α)

and  α

2+α

du 1 , = 2 u α(1 + α/2)

Equations (4.9) and (4.10) immediately follow. Note that from the cumulative probability expressed by (4.9) it follows that the scattering angle can be easily calculated from: cos θ = 1 −

2α Pel (θ, E) . 2 + α − 2Pel (θ, E)

(4.11)

Due to the excellent agreement between experimental data and calculations of the Mott cross-section (see, for a few comparisons, Figs. 4.5, 4.6, and 4.7), the most

36

4 Scattering Mechanisms

10

2

DESCS (Å /sr)

1

0.1

0.01

0.001 0

30

60

90

120

150

180

Scattering angle (deg)

Fig. 4.5 Differential elastic scattering cross-section of 1000 eV electrons scattered by Al as a function of the scattering angle. Solid line: Current calculations (Mott theory) [17]. Circles: Salvat and Mayol calculations (Mott theory) [25]

recent Monte Carlo codes (and also all the calculations presented in this book) use the Mott cross-section to describe the differential elastic scattering cross-section of electrons (see, for example, Figs. 4.8, 4.9, and 4.10) and positrons (see, for example, Fig. 4.11)—and the cumulative probability necessary for sampling the scattering angle. Nevertheless, it is worth stressing that in the case, for example, of the Monte Carlo simulation of the backscattering coefficient, excellent results can also be obtained using the simple screened Rutherford formula, (4.6), provided that the kinetic primary energy of the incident electrons is higher than 10 keV [29].

4.1.2 Polarized Electron Beams Elastically Scattered by Atoms Spin-polarization phenomena are of huge technological and theoretical interest. Today it is possible to experimentally produce polarized ensembles, and the case of spin-polarized beams deserves particular attention for its simplicity, for its potential technological applications, and for its conceptual aspects. An electron beam is a quantum system in a mixed state of spin orientations. A beam of electrons is polarized when the electron spins have a preferential orientation [8, 9]. In other words, in a polarized electron beam, the two populations of spin orientations are different. On the one hand, the spins of electrons produced by thermal emission have arbitrary directions and, on the other hand, today it is experimentally possible to produce electron beams with the two possible spin orientations not equally populated. If all

4.1 Elastic Scattering

37

1

2

DESCS (Å /sr)

10

0.1

0.01

0.001 0

30

60

90

120

150

180

Scattering angle (deg)

Fig. 4.6 Differential elastic scattering cross-section of 1100 eV electrons scattered by Au as a function of the scattering angle. Solid line: Current calculations (Mott theory) [17]. Circles: Reichert experimental data [26] 100

DESCS (Å2/sr)

10

1

0.1

0.01

0.001 0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.7 Differential elastic scattering cross-section of 300 eV electrons scattered by Hg as a function of the scattering angle. Solid line: Current calculations (Mott theory) [18]. Filled circles: Bromberg experimental data [27]. Empty circles: Holtkamp et al. experimental data [28]

38

4 Scattering Mechanisms 100 10

DESCS (Å2/sr)

1 0.1

1 keV 0.01

4 keV

0.001 1E-4

16 keV

1E-5

64 keV

1E-6 1E-7 0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.8 Differential elastic scattering cross-section of 1, 4, 16, and 64 keV electrons scattered by Ar as a function of the scattering angle. Solid line: Current calculations (Mott theory) [18]. Circles: Riley et al. calculations (Mott theory) [10] 100 10

DESCS (Å2/sr)

1

1 keV

0.1 0.01

4 keV

0.001

16 keV 1E-4

64 keV

1E-5 1E-6 0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.9 Differential elastic scattering cross-section of 1, 4, 16, and 64 keV electrons scattered by Kr as a function of the scattering angle. Solid line: Current calculations (Mott theory) [18]. Circles: Riley et al. calculations (Mott theory) [10]

4.1 Elastic Scattering

39

1000 100

DESCS (Å2/sr)

10 1

1 keV

0.1

4 keV

0.01

16 keV

0.001

64 keV

1E-4 1E-5 0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.10 Differential elastic scattering cross-section of 1, 4, 16, and 64 keV electrons scattered by Au as a function of the scattering angle. Solid line: Current calculations (Mott theory) [18]. Circles: Riley et al. calculations (Mott theory) [10] 100 10

DESCS (Å2/sr)

1 0.1

1 keV

0.01

4 keV 0.001

16 keV 1E-4

64 keV 1E-5 0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.11 Differential elastic scattering cross-section of 1, 4, 16, and 64 keV positrons scattered by Au as a function of the scattering angle (Mott theory) [18]

40

4 Scattering Mechanisms

the spins have the same orientation, the beam is said to be totally polarized. A beam of electrons can also be partially polarized when the majority of the spins have the same orientation. When the two possible spin orientations are equally populated, then the beam is said to be unpolarized [8, 9]. Within the density-matrix formalism a set of observable quantities can be defined [8, 9]. These observables—the polarization parameters S, T , and U , which depend on the scattering angle and on the incident electron kinetic energy—describe the elastic scattering process, providing a way for calculating the differential elastic scattering cross-section for spin-polarized electron beams. In fact, while for an unpolarized electron beam the differential elastic scattering cross-section can be determined without knowing the S, T , and U spin-polarization parameters, for the case of polarized electron beams the cross-section depends on the S function (also known as the Sherman function, or the left-right asymmetry function) and on the state of polarization of the beam. Furthermore, it is possible to demonstrate that, after an elastic collision, an initially unpolarized electron beam acquires a polarization (whose magnitude is equal to the Sherman function S). In general, for an initially polarized beam, the final polarization after an elastic scattering collision is a function of all the S, T , and U spin-polarization parameters. Once the scattering amplitudes are known, it is possible to calculate the differential elastic scattering cross-section as dσel ˆ , = [| f (θ)|2 + |g(θ)|2 ][1 + S(θ) Pi · n] dΩ where nˆ =

(4.12)

ki × k f |ki × k f |

(4.13)

and ki and k f are, respectively, the initial and final momenta of the electron. In other words, nˆ is the unit vector normal to the scattering plane (see Fig. 4.12). y

z

θ

n ˆ

ki kf

x

cos φ

y

− sin φ

cos φ φ

sin φ x

φ

Fig. 4.12 Unit vector nˆ = (ki × k f )/|ki × k f | perpendicular to the scattering plane. The z-axis is oriented along the direction of electron incidence. ki is the electron momentum before scattering, k f is the electron momentum after scattering, θ is the scattering angle, and φ is the azimuthal angle

4.1 Elastic Scattering

41

Let us indicate with s the spin operator for the spin 1/2 particles. Its components are the 2 × 2 Pauli matrices σ = (σx , σ y , σz ) divided by two. Concerning Pi , it represents the initial polarization vector. Please note that the polarization vector P is the mean value of σ calculated over the spin functions: P = σ .

(4.14)

Regarding S(θ) it is a real function known as Sherman’s asymmetry function. It is given by f (θ)g ∗ (θ) − f ∗ (θ)g(θ) . (4.15) S(θ) = i | f (θ)|2 + |g(θ)|2 Since an unpolarized electron beam is composed of equal numbers of particles polarized parallel and antiparallel to a given direction (for example the incidence direction), averaging over the initial spin orientations we obtain (4.1) 

dσel dΩ

 = | f (θ)|2 + |g(θ)|2

(4.16)

unpolarized

and hence we can write that, in general, dσel = dΩ



dσel dΩ



+ i [ f (θ)g ∗ (θ) − f ∗ (θ)g(θ)] Pi · nˆ .

(4.17)

unpolarized

An interesting result concerning initially unpolarized electron beams (Pi = 0) is that, after scattering, the final polarization P f is a function of the scattering angle θ and is given by [8, 9] (4.18) P f = S(θ)nˆ . In other words, an electron beam initially not polarized, i.e., composed of equal numbers of particles polarized parallel and antiparallel to the incidence direction, becomes polarized due to the scattering. The magnitude of the polarization is Sherman’s asymmetry function (sometimes called, for this reason, polarization function) and the direction is normal to the scattering plane. The experimental evaluation of the asymmetry function is typically performed through the so-called double scattering experiments [8, 9]. Let be k f1 and k f2 , respectively, the final momenta after the first and the second scattering. We have two scattering planes and the unit vectors normal to the two planes are

42

4 Scattering Mechanisms

nˆ 1 =

ki × k f1 , |ki × k f1 |

(4.19)

nˆ 2 =

k f1 × k f2 , |k f1 × k f2 |

(4.20)

where, as before, ki is the initial momentum. The differential elastic scattering crosssection for the second scattering, if the beam is initially unpolarized, is given by dσel2 = [| f (θ2 )|2 + |g(θ2 )|2 ][1 + S(θ1 )S(θ2 )nˆ 1 · nˆ 2 ] , dΩ2

(4.21)

where θ1 and θ2 are, respectively, the first and the second scattering angle. So, the cross-section depends on the scattering angles, the Sherman functions and the angle between the scattering planes. Considering only the scatterings occurring in the same plane, we have nˆ 1 · nˆ 2 = ±1. Let us define  ηl ≡  ηr ≡

dσel2 dΩ2 dσel2 dΩ2

 ,

(4.22)

.

(4.23)

left

 right

In practice, when the two scatterings occur in the same plane, ηl and ηr are the differential elastic scattering cross-sections for the second scattering and correspond to nˆ 1 · nˆ 2 = +1 and nˆ 1 · nˆ 2 = −1, respectively. It is easy to see that

≡

ηl − ηr = S(θ1 )S(θ2 ) . ηl + ηr

(4.24)

¯ Once |S(θ)| ¯ is Consequently, a measure of for θ1 = θ2 = θ¯ allows to obtain S 2 (θ). ¯ known for a given angle θ, a second experiment is performed varying θ1 and keeping ¯ Since |S(θ)| ¯ is known from the first experiment, by utilizing the constant θ2 = θ. equation defining it is now possible to determine |S(θ1 )| for different angles θ1 . The sign of S(θ1 ) can be found by measuring the sign of S in the scattering of ¯ using (4.12). polarized electrons (whose polarization direction is known) through θ, Note that, in the general case for which the initial polarization is not zero, i.e., for 0 ≤ |Pi | ≤ 1, it is possible to show that [8, 9] Pf =

ˆ n] ˆ + U (θ)nˆ × Pi [Pi · nˆ + S(θ)]nˆ + T (θ)[Pi − (Pi · n) , ˆ 1 + Pi · nS(θ)

(4.25)

4.1 Elastic Scattering

43

where

| f (θ)|2 − |g(θ)|2 (dσel /dΩ)unpolarized

(4.26)

f (θ)g ∗ (θ) + f ∗ (θ)g(θ) . (dσel /dΩ)unpolarized

(4.27)

T (θ) = and U (θ) = Also note that

S 2 (θ) + T 2 (θ) + U 2 (θ) = 1 .

(4.28)

The experimental determination of T (θ) and U (θ) is performed by triple scattering experiments [8, 9]. The functions S and T are represented in Figs. 4.13 and 4.14, respectively, for electrons with energy from 0 to 10 eV elastically scattered by Xe atoms [30]. Figure 4.15 compares the calculated Sherman function [30] with the Berger and Kessler experimental data [31] for 100 eV electrons elastically scattered by Xe atoms. Figures 4.16 and 4.17 compares the calculations [32] with the Kessler et al. [33] experimental data for, respectively, 900 eV and 1200 eV electrons elastically scattered by Xe atoms. Figures 4.18 and 4.19 compares the calculations [32] with the Steidl et al. experimental data and with the Berger and Kessler experimental data for, respectively, 150 eV [31] and 1200 eV [34] electrons elastically scattered by Hg atoms.

Fig. 4.13 Sherman asymmetry function of electrons elastically scattered by Xe atoms [30]

44

4 Scattering Mechanisms

Fig. 4.14 T function of electrons elastically scattered by Xe atoms [30] 0.5

S

0.0

-0.5

0

30

60

90

120

150

180

Scattering Angle (deg) Fig. 4.15 Sherman asymmetry function of 100 eV electrons elastically scattered by Xe atoms [30]. Symbols represent the Berger and Kessler experimental data [31]

4.1 Elastic Scattering

45

0.4

S

0.2

0.0

-0.2

0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.16 Sherman asymmetry function of 900 eV electrons elastically scattered by Xe atoms [32]. Symbols represent the Kessler et al. experimental data [33]

S

0.2

0.0

-0.2 0

30

60

90

120

150

180

Scattering angle (deg)

Fig. 4.17 Sherman asymmetry function of 1200 eV electrons elastically scattered by Xe atoms [32]. Symbols represent the Kessler et al. experimental data [33]

46

4 Scattering Mechanisms

0.5

S

0.0

-0.5

-1.0 0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.18 Sherman asymmetry function of 150 eV electrons elastically scattered by Hg atoms [32]. Symbols represent the Berger and Kessler experimental data [31]

S

0.5

0.0

-0.5 0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.19 Sherman asymmetry function of 700 eV electrons elastically scattered by Hg atoms [32]. Symbols represent the Steidl et al. experimental data [34]

4.1 Elastic Scattering

47

4.1.3 Effect of the Sherman Function on the Differential Elastic Scattering Cross-Section We know that the differential elastic scattering cross-section as a function of the scattering angle θ and of the azimuthal angle φ is given by: dσel ˆ (θ, φ) = (| f (θ)|2 + |g(θ)|2 )[1 + S(θ) P · n(φ)] , dΩ

(4.29)

where the z axis is oriented along the direction of electron incidence, P is the spinˆ polarization of the beam, n(φ) is the unit vector perpendicular to the scattering plane (see Fig. 4.12), ⎛ ⎞ − sin φ ˆ n(φ) = ⎝ cos φ ⎠ , (4.30) 0 and S(θ) is the Sherman function. Please note that (4.29) is valid both for fully polarized beams, so that |P| = 1 (pure spin states), and for any degree of polarization, i.e., 0 ≤ |P| ≤ 1 (a mixture of spin states). If the spin-polarization P is zero, the Sherman function does not affect the differential elastic scattering cross-section. When, on the other hand, P is not zero, changes are observed in the differential elastic scattering cross-section. Since (4.31) P · nˆ = − Px sin φ + Py cos φ , no asymmetry can be observed when Px = Py = 0 and Pz = 0 (longitudinal spinpolarization), i.e., when spin is parallel or antiparallel to the direction of the incident electron beam. On the other hand, asymmetry can be observed when the spin polarization possesses transverse components (i.e., Px = 0 and/or Py = 0). If the primary electron beam is fully spin-polarized in the transverse direction, e.g., when Px = 1, Py = 0, Pz = 0, it is not axially symmetric with respect to the direction of incidence. In such a case, the elastic scattering cross-section depends on the azimuthal angle φ, and the maximum asymmetry is reached when the scattering plane is orthogonal to P, i.e., when P · nˆ = ± 1. This asymmetry in the differential elastic scattering cross-section represents a way to measure the spin-polarization of electron beams. The spin functions S(θ), T (θ), and U (θ) for 1000 eV electrons impinging on uranium atoms are represented in Fig. 4.20. Let us now investigate the effect of the Sherman asymmetry function S(θ) on the differential elastic scattering cross-section [35]. In Fig. 4.21 we compare the differential elastic scattering cross-section of an unpolarized 1000 eV electron beam with the differential elastic scattering cross-section

48

4 Scattering Mechanisms 1.0

S, T, U

0.5

0.0

-0.5

-1.0 0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.20 S(θ) (black line), T (θ) (red line), and U (θ) (green line) functions for 1000 eV electrons impinging on uranium atoms

of a spin-polarized 1000 eV electron beam (transverse spin-polarization) impinging on uranium atoms. The following spin-polarization conditions were considered in the calculation: Px = 1 , Py = 0 , Pz = 0, P · nˆ = − 1. At θ = 104.2◦ we can observe a minimum in the differential elastic scattering cross-section of the unpolarized beam. This minimum is shifted to a smaller polar angle (θ = 102.7◦ ) for the spin-polarized beam. Also, the peak intensity changes. While it was 8.0 × 10−4 Å/sr for the unpolarized beam, it is 2.7 × 10−7 Å/sr for the spin-polarized beam. Since the contribution of the Sherman function is particularly strong in the θ range from ≈ 102◦ to ≈ 105◦ (see Fig. 4.20), the effect of the transverse spin-polarization is particularly evident in this angular range.

4.1.4 Electron-Molecule Elastic Scattering When the target is a molecule and the spin-polarization is null, then  dσel = exp(iq · rmn ) [ f m (θ) f n∗ (θ) + gm (θ)gn∗ (θ)] , dΩ m,n

(4.32)

4.1 Elastic Scattering

49

1000 100

DESCS (Å2/sr)

10 1 0.1 0.01 0.001 1E-4 1E-5 1E-6 1E-7 0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.21 Differential elastic scattering cross-section of 1000 eV electrons impinging on uranium atoms. Black line: Unpolarized beam. Red line: Polarized beam: Px = 1 , Py = 0 , Pz = 0 , φ = 90◦ , P · nˆ = − 1. Circles: Unpolarized beam (Riley et al. [10])

where q is the momentum transfer, rmn = rm − rn , and rm is the position vector of the m th atom in the molecule [15]. If the molecules in the target are randomly oriented, we are allowed to average out all the orientations, so that the last equation can be simplified as follows [15]2 :  sin qrmn dσel = [ f m (θ) f n∗ (θ) + gm (θ)gn∗ (θ)] , dΩ qr mn m,n

(4.33)

where the modulus of the momentum transfer is given by: q = 2 K sin(θ/2) . In this last equation, K is the relativistic wave number of the projectile.

2

Please note that, if we indicate with ϑ the angle between q and rmn , we have π 0

.

2π sin ϑ eiqrmn cos ϑ dϑ π = 0 2π sin ϑ dϑ

1

−1

eiqrmn t dt sin qrmn = . 1 qrmn dt −1

(4.34)

50

4 Scattering Mechanisms 100

10

DESCS (Å2/sr)

Fig. 4.22 Differential elastic scattering cross-section of 40 eV electrons scattered by water molecules as a function of the scattering angle. Circles: Cho et al. experimental data [36]

1

0.1

0.01 0

30

60

90

120

150

180

Scattering angle (deg)

Let us consider, for example, the simple case of a triatomic molecule, such as H2 O. The differential elastic scattering cross-section of electrons impinging on randomly oriented water molecules is given by 

+2

dσel dΩ



 = 2 H2 O

dσel dΩ



 +

H

dσel dΩ

 + O

sin qrOH [ f H (θ) f O∗ (θ) + f O (θ) f H∗ (θ) + gH (θ)gO∗ (θ) + gO (θ)gH∗ (θ)] + qrOH sin qrHH [| f H (θ)|2 + |gH (θ)|2 ] . +2 qrHH

Note that the the subscript H indicates the hydrogen atoms, the subscript O the oxygen atom, rOH is the average distance between the nuclei of the oxygen and the hydrogen atoms, and rHH is the average distance between the nuclei of the hydrogen atoms in the water molecule. In Figs. 4.22, 4.23, 4.24, and 4.25, calculations of the differential elastic scattering cross-section of electrons scattered by water molecules are compared with the Cho et al. [36], with the Johnston and Newell [37], and with the Katase et al. [38] experimental data. Calculations of the total elastic scattering cross-section of electrons scattered by water molecules are compared with the Katase et al. experimental data [38] and with the Itikawa and Mason [39] recommended data in Fig. 4.26.

4.1 Elastic Scattering

51

100

DESCS (Å2/sr)

10

1

0.1

0.01 0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.23 Differential elastic scattering cross-section of 50 eV electrons scattered by water molecules as a function of the scattering angle. Circles: Cho et al. experimental data [36]. Triangles: Johnston and Newell experimental data [37]

DESCS (Å2/sr)

10

1

0.1

0.01

0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.24 Differential elastic scattering cross-section of 500 eV electrons scattered by water molecules as a function of the scattering angle. Circles: Katase et al. experimental data [38]

52

4 Scattering Mechanisms

DESCS (Å2/sr)

10

1

0.1

0.01

0.001 0

30

60

90

120

150

180

Scattering angle (deg) Fig. 4.25 Differential elastic scattering cross-section of 700 eV electrons scattered by water molecules as a function of the scattering angle. Circles: Katase et al. experimental data [38]

ESCS (Å2)

10

1

0.1 10

100

1000

Energy (eV) Fig. 4.26 Total elastic scattering cross-section of electrons scattered by water molecules as a function of the electron energy. Circles: Katase et al. experimental data [38]. Triangles: Itikawa and Mason recommended data [39]

4.2 Quasi-elastic Scattering

53

4.2 Quasi-elastic Scattering Due to thermal excitations, atoms in crystalline structures vibrate around their equilibrium lattice sites. These vibrations are known as phonons. A mechanism of energy loss (and energy gain as well) is represented by the interaction of the electrons with the optical modes of the lattice vibrations. These transfers of small amounts of energy among electrons and lattice vibrations are due to quasi-elastic processes known as phonon creation (electron energy-loss) and phonon annihilation (electron energygain) [2, 40]. Phonon energies do not exceed kB TD , where kB is the Boltzmann constant and TD is the Debye temperature. As typically kB TD is not greater than 0.2 eV (C: kB TD = 0.19 eV, Cu: kB TD = 0.03 eV, Fe: kB TD = 0.04 eV), the energy losses and gains due to electron-phonon interaction are usually smaller than 0.2 eV, so that they are generally not resolved by conventional spectrometers [12]. These mechanisms of electron energy loss—and, with much smaller probability, energy gain—are particularly relevant when the electron energy is low (a few eV) [7].

4.2.1 Electron-Phonon Interaction According to Fröhlich [2] and Llacer and Garwin [40], the inverse mean free path for electron energy loss due to phonon creation can be written as λ−1 phonon

√ 

1 + 1 − ω/E 1 ε0 − ε∞ ω n(T ) + 1 , ln = √ a 0 ε0 ε∞ E 2 1 − 1 − ω/E

(4.35)

where E is the energy of the incident electron, Wph = ω the electron energy loss (in the order of 0.01 eV – 0.1 eV), ε0 the static dielectric constant, ε∞ the high frequency dielectric constant, a0 the Bohr radius and n(T ) =

1 eω/kB T

−1

(4.36)

the occupation number. Notice that a similar equation can be written to describe electron energy gain (corresponding to phonon annihilation). The occurrence probability of phonon annihilation is much lower than that of phonon creation. Electron energy gain can thus be safely neglected for many practical purposes. For further details about electron-phonon interaction and Fröhlich theory [2, 40] see Appendix E.

54

4 Scattering Mechanisms

4.3 Inelastic Scattering Let us consider now the inelastic scattering due to the interaction of the incident electrons with the atomic electrons located around the nucleus (both the core and the valence electrons). For an excellent review of this subject, see [12]. If the incident electron energy is high enough, it can excite an inner-shell electron that can make a transition from its ground state to one of the unoccupied electron states above the Fermi level. Due to energy conservation, the incident electron loses an amount of energy equal to the difference between the state above the Fermi level occupied by the excited atomic electron and its ground state. While the atom is left in an ionized state. The following de-excitation of the target atom generates excess energy that can be released in one of two competitive ways: Either by the generation of an X-ray photon (Energy dispersive spectroscopy, EDS, is based on this process) or by the emission of another electron: This is the phenomenon on which Auger electron spectroscopy, AES, is based. Outer-shell inelastic scattering can occur according to two alternative processes. In the first one, an outer-shell electron can suffer a single-electron excitation. A typical example is constituted by inter-band and intra-band transitions. If the atomic electron excited in such a way is able to reach the surface with an energy higher than the potential barrier between the vacuum level and the minimum of the conduction band, it can emerge from the solid as a secondary electron, the energy needed for this transition being provided by the fast incident electron. De-excitation can occur through the emission of electromagnetic radiation in the visible region—corresponding to the phenomenon known as cathode-luminescence—or through radiation-less processes generating heat. Outer-shell electrons can also be excited in collective states corresponding to the oscillation of the valence electrons denoted as plasma resonance. It is generally described as the creation of quasi-particles known as plasmons, with energies—characteristic of the material—that range, typically, in the interval from 5 to 30 eV. Plasmon decay generates secondary electrons and/or produces heat.

4.3.1 Stopping: Bethe-Bloch Formula The Bethe-Bloch formula allows calculating electron stopping power once the target atomic number and the incident electron kinetic energy are known. It can be used when electron energy is relatively high. In order to deduce the Bethe-Bloch formula, we will use a very simple approach. Let us assume that a free electron is at rest. At a distance r from it, another electron is traveling along the z direction. If θ represents the angle between r and z, v represents the velocity of the incident electron, and the instant of impact is t = 0 then (see Fig. 4.27) − vt = r cos θ .

(4.37)

4.3 Inelastic Scattering

55

Fig. 4.27 At a distance r from a free electron at rest, another electron is traveling along the z direction

t

Defining the impact parameter b by b ≡ r sin θ ,

(4.38)

it is possible to see that b is the distance from the trajectory of the incident electron to the target: (4.39) b2 + v 2 t 2 = r 2 . F indicates the force of repulsion between the two electrons, where Fz =

e2 cos θ r2

(4.40)

Fx =

e2 sin θ , r2

(4.41)

and

and pz and px are the components of the momentum transferred to the target electron. As vt (4.42) cot θ = − , b differentiating with respect to θ, we can obtain

Therefore,

b dt = . dθ v sin2 θ

(4.43)

1 dt 1 = . 2 r dθ bv

(4.44)

The x component of p is thus given by  px =

∞ −∞

Fx dt =

e2 bv

 0

π

sin θ dθ =

2e2 , bv

(4.45)

56

4 Scattering Mechanisms

while the z component of p is zero. The energy T transferred to the electron at rest is e4 p2 (4.46) T = x = 2 , 2m b E where E = mv 2 /2 is the kinetic energy of the incident electron. The number of electrons in the volume 2πb db dz is given by 2πb db dz N Z , where N is the number of atoms per unit of volume in the target and Z the target atomic number. Thus, the energy lost by the incident electron per unit length in the target (i.e., the stopping power) may be calculated from  dσinel dE = NZ T dT = dz dT   2πe4 N Z bmax db = N Z T 2πb db = , E b bmin

−

(4.47)

where bmin and bmax are, respectively, the minimum and the maximum impact parameter, and    d(πb2 )   dT = 2π b db . (4.48) dσinel =  dT  The atomic electrons may be considered as a set of oscillators with frequencies νi and amplitudes f i . As a consequence, we can rewrite (4.48) as  bmax i dE 2πe4 N  db − = . fi dz E b bmin i

(4.49)

The sum of the amplitudes of the oscillators has to be equal to the atomic number Z : 

fi = Z .

(4.50)

i

If hνi indicates the binding energy of the ith atomic electron, then the maximum impact parameter, on the one hand, is of the order of i ∼ v/νi . bmax

(4.51)

The minimum impact parameter, on the other hand, is of the order of bmin ∼ h/mv ,

(4.52)

4.3 Inelastic Scattering

57

so that −

  k mv 2 2πe4 N  dE = f i ln , dz E hνi i

(4.53)

where k is a constant whose value depends on the exact limits of integration. Let us define the mean excitation energy as IZ ≡



(hνi ) fi ,

(4.54)

i

so that the stopping power can be written as   2πe4 N Z 2k E dE = ln , − dz E I

(4.55)

The indistinguishability of the scattered electrons from the ejected ones requires further consideration. In order to take into account both the electrons emerging from the inelastic collision, the stopping power can be calculated as follows:  dσinel dE = NZ T dT = dz dT

 E/2   E dσinel dσinel = NZ dT + dT T (E − T ) dT dT Tmin E/2 −

Since b2 =

e4 , TE

(4.56)

(4.57)

the differential inelastic scattering cross-section can be written as   4  db2  dσinel   = πe 1 = π dT dT  E T2

(4.58)

and therefore 

dσinel πe4 dT = [1 − ln(2)] = dT E E/2  2πe4 e 2πe4 ln = ln(1.166) , = E 2 E E

(E − T )

(4.59)

where e is the base for natural logarithms. Selecting Tmin = I /2, so that 

E/2

T Tmin

dσinel 2πe4 dT = ln dT E



E I

 ,

(4.60)

58

we finally obtain

4 Scattering Mechanisms

  2πe4 N Z 1.166E dE = ln . − dz E I

(4.61)

This is the Bethe-Bloch formula for the stopping power including electron-exchange effects [3]. Note that accurate tables of the mean excitation potential I as a function of the atomic number Z are available [41]. The Bethe-Bloch formula is valid for relatively high energies. It is considered quite accurate for electron energies larger than the K-shell binding energies of the target atoms. Note that, on the one hand, when relativistic velocities are considered, relativistic corrections have to be included in the Bethe-Bloch theoretical formulation. On the other hand, when E becomes smaller than I /1.166, the stopping power predicted by the Bethe-Bloch formula becomes negative. This is an unrealistic consequence of the application of the Bethe-Bloch formula to very slow electrons. Therefore, the low-energy stopping power requires a different approach (see the dielectric approach below).

4.3.2 Stopping: Semi-empirical Formulas The stopping power can also be described with semi-empirical expressions, such as the following: K e N Z 8/9 dE = (4.62) − dz E 2/3 proposed in 1972 by Kanaya and Okayama (with K e = 360 eV5/3 Å2 ) [5]. The latter formula allows us to analytically evaluate the maximum range of penetration R as a function of the primary energy E 0 :  R=

0 E0

5/3

3E 0 dE = ∝ E 01.67 . d E/dz 5K e N Z 8/9

(4.63)

A similar empirical formula for the evaluation of the maximum range of penetration of electrons in solid targets was proposed for the first time in 1954 by Lane and Zaffarano [4] who found that their range-energy experimental data (obtained by investigating electron transmission in the energy range 0–40 keV by thin plastic and metal films) can be described by the following simple equation: E 0 = 22.2R 0.6 ,

(4.64)

4.3 Inelastic Scattering

59

where E 0 was expressed in keV and R in mg/cm2 . As a consequence, the Kanaya and Okayama formula is consistent with the Lane and Zaffarano experimental observations, which are described as well by the relationship R ∝ E 01.67 .

(4.65)

4.3.3 Dielectric Theory In order to obtain an accurate description of the electron energy loss processes, of the stopping power, and of the inelastic mean free path, valid even when electron energy is low, it is necessary to consider the response of the ensemble of conduction electrons to the electromagnetic field generated by the electrons passing through the solid: This response is described by a complex dielectric function. In Appendix F the Ritchie dielectric theory [6, 42] is described. It demonstrates, in particular, that the energy loss function, f (k, ω), necessary to calculate both the stopping power and the inelastic mean free path, is the reciprocal of the imaginary part of the dielectric function 

1 . (4.66) f (k, ω) = Im ε(k, ω) In (4.66), k represents the momentum transferred and ω the electron energy loss. Once the energy loss function has been obtained, the differential inverse inelastic mean free path can be calculated as [43] dλ−1 1 inel = dω π E a0 where  k± =



k+ k−

dk f (k, ω) , k

 √ 2 m E ± 2 m (E − ω) ,

(4.67)

(4.68)

E is the electron energy, m is the electron mass, and a0 is the Bohr radius. The limits of integration, expressed by (4.68), come from conservation laws (see section 6.2.5). In order to calculate the dielectric function, and hence the energy loss-function, let us consider the electric displacement D [22, 23]. If P is the polarization density of the material and E the electric field, then P = χε E ,

(4.69)

ε−1 4π

(4.70)

where χε =

60

4 Scattering Mechanisms

and D = E + 4 π P = (1 + 4 πχε ) E = ε E .

(4.71)

If n is the density of the outer-shell electrons, i.e., the number of outer-shell electrons per unit volume in the solid, and ξ the electron displacement due to the electric field, then P = enξ, (4.72) so that |E| =

4πenξ . ε−1

(4.73)

Let us consider the classical model of an electron elastically bound, with elastic constant k0 = m ω02 and subject to a frictional damping effect, due to collisions, described by a damping constant Γ0 . We have indicated here with m the electron mass and with ω0 the natural frequency. The electron displacement satisfies the equation [44] (4.74) m ξ¨ + β0 ξ˙ + k0 ξ = e E where β0 = m Γ0 . Assuming that ξ = ξ0 exp(iωt) ,

(4.75)

a straightforward calculation allows concluding that ε(0, ω) = 1 −

ωp2 ω 2 − ω02 − iΓ0 ω

,

(4.76)

where ωp is the plasma frequency, given by ωp2 =

4 π n e2 . m

(4.77)

Please note that, for the case of a free electron (ω0 = 0), using (4.75) the dielectric function assumes the following form ε(0, ω) = 1 −

ωp2 ω 2 − iΓ0 ω

.

(4.78)

On the other hand, if we describe ξ as ξ = ξ0 exp(−iωt) ,

(4.79)

4.3 Inelastic Scattering

61

then we obtain ε(0, ω) = 1 −

ωp2 ω 2 + iΓ0 ω

,

(4.80)

Indicating with ε1 and ε2 the real and imaginary parts of the dielectric function, respectively, we see that, using (4.78), ε2 is negative and given by ε2 (0, ω) = −

ω 2p Γ0 , ω ω 2 + Γ02

(4.81)

while, using (4.80), ε2 is positive and given by ω 2p Γ0 ω ω 2 + Γ02

ε2 (0, ω) =

(4.82)

To make the energy loss function positive using (4.79), the energy loss function is defined as the imaginary part of the opposite of reciprocal of the dielectric function:

f (0, ω) = Im −

1 ε(0, ω)

 =

ε2 (0, ω) . ε21 (0, ω) + ε22 (0, ω)

(4.83)

On the other hand, using (4.75), the energy loss function is positive if it is defined as the imaginary part of the reciprocal of the dielectric function:

f (0, ω) = Im

1 ε(0, ω)

 =

− ε2 (0, ω) , ε21 (0, ω) + ε22 (0, ω)

(4.84)

It is then clear that the sign used in the definition of the energy loss function is just a matter of the choice of the function describing the time dependence of the electron displacement. In this book, we will use (4.75), so the energy loss function is defined as the imaginary part of the reciprocal of the dielectric function.3 There exists a relationship between the dielectric function in the optical limit, i.e., for k → 0, and the coefficients of refraction and extinction. We have

3

ε(0, ω) = ε1 (0, ω) + iε2 (0, ω) ,

(4.85)

ε1 (0, ω) = Re[ε(0, ω)] = ν 2 − κ2 ,

(4.86) (4.87)

where ε2 (0, ω) = Im[ε(0, ω)] = − 2νκ , ν is the refraction index, and κ is the extinction coefficient. Therefore,

 1 2νκ Im . = ε(0, ω) (ν 2 + κ2 )2

(4.88)

The calculations of the refraction index ν and of the extinction coefficient κ can be performed by using the following equations [45]:

62

4 Scattering Mechanisms

Let us now consider a superimposition of free and bound oscillators. In such a case the dielectric function can be written as: ε(0, ω) = 1 −

f 0 ωp2 ω 2 − iΓ0 ω

−

J 

f j ωp2

j=1

ω 2 − ω 2j − iΓ j ω

,

(4.93)

where Γ j are positive frictional damping coefficients and f j are the fractions of the electrons bound with energies ω j ( j = 1, 2, . . . , J ). Please note that ω0 = 0, i.e., f 0 represents the fraction of free electrons with frictional damping coefficient Γ0 . The extension of the dielectric function from the optical limit (corresponding to k = 0) to k > 0 is obtained including, in the previous formula, an energy ωk related to the dispersion relation, so that ε(k, ω) = 1 −

f 0 ωp2 ω 2 − ωk2 − iΓ0 ω

−

J 

f j ωp2

j=1

ω 2 − ω 2j − ωk2 − iΓ j ω

,

(4.94)

In the determination of the dispersion relation, one has to take into account a constraint, known as the Bethe ridge. According to the Bethe ridge, as k → ∞, ωk should approach 2 k 2 /2m. Of course, an obvious way to obtain this result (the simplest one, actually) is to assume that [43, 46], 2 k 2 . 2m

(4.95)

 e2 λ2 N x p f1 p , 2 2πmc p

(4.89)

ωk =

ν = 1−

 e2 λ2 N x p f2 p . 2πmc2 p

κ =

(4.90)

In these equations, c is the speed of light, N is the number of molecules per unit volume (each having x p atoms), and λ is the photon wavelength. f 1 p and f 2 p are, respectively, the real and the imaginary components of the atomic scattering factor of the p th atom. The calculation of the real and imaginary components of the atomic scattering factor of each atom can be performed using the following equations [45]: f1 p = Z +

A mc2 2π 2 ce2 N A



∞ 0

2 μ( ) d , E 2p − 2

(4.91)

mc2 A E p μ(E p ) . (4.92) 4πce2 N A In these equations E p is the incident photon energy, Z the atomic number, A the atomic weight, N A the Avogadro number and μ the photoabsorption cross-section (cm2 /g). f2 p =

4.3 Inelastic Scattering

63

This law is valid at high momentum transfer. Another way of satisfying the constraint represented by the Bethe ridge is to use, according to Ritchie [6] and to Richie and Howie [42], the following equation: 2 ωk2 =

3 2 v 2F k 2 12 E F 2 k 2 4 k 4 = + + 2 5 4m 5 2m



2 k 2 2m

2 ,

(4.96)

where v F and E F represent, respectively, the Fermi velocity and the Fermi energy. This last equation is actually the most general dispersion law that can be derived within the Random Phase Approximation (RPA) of the 3D electron gas to extend the optical data to finite momentum transfer for bulk plasmons. For intermediate k, in fact, one cannot neglect the quadratic term in the dispersion law for bulk plasmons in (4.96). Once the dielectric function is known, the loss function Im[1/ε(k, ω)] is given by  1 ε2 (k, ω) , = − 2 Im ε(k, ω) ε1 (k, ω) + ε22 (k, ω)

(4.97)

ε(k, ω) = ε1 (k, ω) + iε2 (k, ω) ,

(4.98)

where

The calculation of the energy loss function can be also performed through the direct use of experimental optical data. In Figs. 4.28 and 4.29, the optical energy loss function of polymethyl methacrylate (PMMA) and silicon dioxide (SiO2 ) are represented, respectively. A quadratic extension into the energy- and the momentum-transfer plane of the energy loss function allows the extension of the dielectric function from the optical limit to k > 0 [50–52]. Penn [50] and Ashley [51, 52] calculated the energy loss function by using optical data and by extending as discussed above the dielectric function from the optical limit to k > 0. According to Ashley [51, 52], the inverse inelastic mean free path λ−1 inel of electrons penetrating solid targets can be calculated by: λ−1 inel (E)

me2 = 2π2 E



Wmax

0

   w 1 L dw , Im ε(0, w) E

(4.99)

where E is the incident electron energy and Wmax = E/2 (as usual, we have indicated with e the electron charge and with  the Planck constant h divided by 2π). According to Ashley, in the dielectric function ε(k, w), the momentum transfer k was set to 0 and the ε dependence on k was factorized through the function L(w/E). Ashley [51] demonstrated that a good approximation of the function L(x) is given by: L(x) = (1 − x) ln

33 4 7 − x + x 3/2 − x 2 . x 4 32

(4.100)

64

4 Scattering Mechanisms 10 1

Energy loss function

0.1 0.01 0.001 1E-4 1E-5 1E-6 1E-7 1E-8 1E-9 1

10

100

1000

10000

Energy transfer (eV)

Fig. 4.28 Optical energy loss function for electrons in polymethyl methacrylate (PMMA). For energies lower than 72 eV we used the optical data of Ritsko et al. [47]. For higher energies the calculation of the optical loss function was performed by using the Henke et al. atomic photoabsorption data [45, 48] 1

Energy loss function

0.1

0.01

0.001

1E-4

1E-5

1E-6 10

100

1000

Energy transfer (eV)

Fig. 4.29 Optical energy loss function for electrons in silicon dioxide (SiO2 ). For energies lower than 33.6 eV we used the optical data of Buechner [49]. For higher energies the calculation of the optical loss function was performed by using the Henke et al. atomic photo-absorption data [45, 48]

4.3 Inelastic Scattering

65

The calculation of the stopping power, −d E/dz, can be performed by using the following equation [51]: me2 dE = − dz π2 E



Wmax 0

   w 1 S w dw , Im ε(0, w) E

(4.101)

where S(x) = ln

x 4 1 1.166 3 x 2 4 31 2 − x − ln + x 3/2 − ln − x . x 4 4 x 2 16 x 48

(4.102)

The inelastic mean free path and the stopping power for positrons may be calculated in a similar way [52]:    w 1 Lp dw , ε(0, w) E

(4.103)

      Wmax 1 w me2 dE w dw , = Im − Sp 2 dz p 2π E 0 ε(0, w) E

(4.104)

(λ−1 inel )p =

where

me2 2π2 E



Wmax

Im 0

 √ 1 − 2x L p (x) = ln √ 1 − x/2 − 1 − 2x 

1 − x/2 +



and Sp (x) = ln

1−x + 1−x −

√ √

1 − 2x 1 − 2x

(4.105)

 .

(4.106)

In Figs. 4.30 and 4.31 the stopping powers of electrons in polymethyl methacrylate (PMMA) and in silicon dioxide (SiO2 ) are, respectively, shown. In Figs. 4.32 and 4.33 the inelastic mean free paths of electrons in PMMA and in SiO2 are also, respectively, represented. The current calculations, obtained using the Ashley theory that we just described, are compared with results of other authors. Once ε(0, w) is known, the electron differential inverse inelastic mean free path dλ−1 inel (w, E)/dw can be calculated, according to the Ashley’s approach, by using the following equation:

   me2 1 w dλ−1 inel (w, E) = Im L . dw 2π2 E ε(0, w) E

(4.107)

66

4 Scattering Mechanisms 4

-dE/dz (eV/Å)

3

2

1

0

10

100

1000

10000

Energy (eV)

Fig. 4.30 Stopping power of electrons in PMMA. The solid line represents the current calculation, obtained according to the Ashley recipe [51]. The dashed line provides the Ashley original results [51]. Dotted line describes the Tan et al. computational results [53] Fig. 4.31 Stopping power of electrons in SiO2 . The solid line represents the current calculation, obtained according to the Ashley recipe [51]. The dashed line provides the Ashley and Anderson data [54]

4

-dE/dz (eV/Å)

3

2

1

0

100

1000

10000

Energy (eV)

4.3.4 Sum of Drude Functions Ritchie and Howie [42] have proposed to calculate the energy loss function as a linear superimposition of Drude functions, as follows:

Im

1 ε(k, ω)

 =

 j

A j Γ j ω , [w 2j (k) − 2 ω 2 ]2 + 2 ω 2 Γ j2

(4.108)

4.3 Inelastic Scattering

67

IMFP (Å)

1000

100

10

10

100

1000

10000

Energy (eV)

Fig. 4.32 Inelastic mean free path of electrons in PMMA due to electron-electron interaction. The solid line represents the current calculation, based on the Ashely model [51]. The dashed line describes the original Ashley results [51]

IMFP (Å)

100

10

10

100

1000

10000

Energy (eV)

Fig. 4.33 Inelastic mean free path of electrons in SiO2 due to electron-electron interaction. The solid line represents the current calculation, based on the Ashely model [51]. The dashed line describes the Ashley and Anderson data [54]. The dotted line provides the Tanuma, Powell and Penn computational results [55]

68

4 Scattering Mechanisms

Table 4.1 Parameters calculated by Garcia-Molina et al. [59] to fit the contribution to the optical energy loss function of the outer electrons for five selected carbon allotropes (amorphous carbon, glassy carbon, C60-fullerite, graphite, and diamond) Target j wj Γj Aj (eV) (eV) (eV2 ) Amorphous carbon Glassy carbon

C60-fullerite

Graphite

Diamond

1 2 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3

6.26 25.71 2.31 5.99 19.86 23.67 38.09 6.45 14.97 24.49 28.57 40.82 2.58 6.99 21.77 28.03 38.09 22.86 29.93 34.77

5.71 13.33 4.22 2.99 6.45 12.38 54.42 2.45 6.26 13.06 12.24 27.21 1.36 1.77 8.16 6.80 68.03 2.72 13.61 11.43

9.25 468.65 0.96 6.31 77.70 221.87 110.99 6.37 16.52 175.13 141.21 141.47 0.18 7.38 73.93 466.69 103.30 22.21 140.64 843.85

where, according to (4.96),  w j (k) = ω j (k) =

w 2j +

12E F 2 k 2 + 5 2m



2 k 2 2m

2 .

(4.109)

In these equations E F is the Fermi energy and w j , Γ j , A j are the excitation energies, the damping constants, and the oscillator strengths, respectively [56]. Garcia-Molina et al. [59] provided, for example, the values of these parameters calculated from the fit of experimental optical data for five allotropic forms of carbon (amorphous carbon, glassy carbon, C60-fullerite, graphite, and diamond). In Table 4.1 the GarciaMolina et al. parameters have been reported, for the reader’s convenience. The optical energy loss functions so calculated for these five carbon allotropes are represented in Fig. 4.34. In Fig. 4.35 the differential inverse inelastic mean free path for the considered allotropes of carbon, calculated by [see (4.66) and (4.67)]

4.3 Inelastic Scattering

69

3

Energy loss function

Amorphous carbon Glassy carbon C60-fullerite Graphite Diamond

2

1

0 0

20

40

60

80

Energy transfer (eV)

Fig. 4.34 Optical energy loss function of selected allotropic forms of carbon (amorphous carbon, glassy carbon, C60-fullerite, graphite, and diamond) as a function of the excitation energy, calculated by sum of Drude-Lorentz functions, according to Ritchie and Howie [42], using the Garcia-Molina et al. parameters [59] Amorphous carbon Glassy carbon C60-fullerite Graphite Diamond

0.006

0.004

DIIMFP (eV

-1

-1

)

0.005

0.003

0.002

0.001

0.000 0

20

40

60

80

Energy transfer (eV)

Fig. 4.35 Differential inverse inelastic mean free path for electrons impinging upon several allotropic forms of carbon (amorphous carbon, glassy carbon, C60-fullerite, graphite, and diamond) as a function of the electron energy loss, calculated using Drude-Lorentz functions. The incident electron energy is E=250 eV

70

4 Scattering Mechanisms

1 dλ−1 inel = dω πa0 E



k+ k−

 1 dk Im , k ε(k, ω)

(4.110)

where k− and k+ are given by (4.68), is represented as a function of the energy loss w = ω for incident electron energy E=250 eV. The comparison with the shapes of the curves representing the optical energy loss function in Fig. 4.34 clearly shows that the peaks broaden and become more asymmetric (see the tail on their right side). The electron inelastic mean free path is the reciprocal of λ−1 inel (E)

 =

Wmax Wmin

dλ−1 inel dw , dw

(4.111)

where, according to [56], Wmin is set to zero for conductors and to the energy of the band gap for semiconductors and insulating materials. Wmax represents the minimum between E − E F and (E + Wmin )/2: 

Wmax

E + Wmin = min E − E F , 2

 .

(4.112)

Please note that the calculation of Wmax requires further consideration. On the one hand, according to Emfietzoglou et al. [56], the factor 1/2 in Wmax is due to the indistinguishability of the electrons (with the convention to consider the incident electron as the most energetic one after the collision). On the other hand, according to Bourke and Chantler [58], when the dominant cause of inelastic losses is represented by plasmon excitations, the integration limit should always be: (4.113) Wmax = E − E F . Being a plasmon a distinct entity, it is distinguishable from an electron. Thus, when only plasmon losses are involved in the calculations, (4.113) should be used for evaluating Wmax instead of (4.112). Please note that also the following equation has been suggested   E + Wmin , Wmax = min E − E Pauli , 2

(4.114)

where, for metals, E Pauli = E F . Garcia-Molina et al. proposed the value E Pauli = 4 eV for several biomaterials [57] and have set Wmin to the energy gap for the outershell electron excitations and to the inner-shell threshold energy for the inner-shell electron excitations [57].

4.3 Inelastic Scattering

71

4.3.5 The Mermin Theory A more accurate approach to the calculation of the ELF is based on the use of the Mermin functions [60] instead of the Drude-Lorentz functions in the sum. It is known as the “Mermin energy loss function-generalized oscillator strength” method (MELF-GOS method) and was proposed by Abril et al. [61]. Details of the MELFGOS method can be found in Appendix H. According to Abril et al. [61], a linear combination of Mermin-type energy loss functions, one per oscillator, allows the calculation of the ELF for any given material. The procedure is very similar to the previous one, apart from the use of Mermin functions instead of Drude-Lorenz functions. Note that the dispersion law is included in the Mermin theory, and it is not necessary, as in the case of the Drude-Lorentz approach, to introduce an approximate expression of it to extend the ELF beyond the optical domain. The Mermin differential inverse inelastic mean free path of electrons in PMMA is represented in Fig. 4.36, for kinetic energies of the incident electrons in the range from 50 to 1000 eV. The inverse of the integral of every curve presented in Fig. 4.36 provides, for each kinetic energy E, the inelastic mean free path. According to de la Cruz and Yubero [62], the values of the inelastic mean free path calculated using the Tanuma, Powell, and Penn (TPP) empirical predictive formula [63] are systematically higher than the corresponding values calculated within the Mermin theory. For PMMA, according to our calculation, when E = 100 eV the

E = 50 eV E = 100 eV E = 200 eV E = 300 eV E = 400 eV E = 500 eV E = 1000 eV

0.006

-1

-1

DIIMFP (eV Å )

0.008

0.004

0.002

0.000 0

20

40

60

80

100

Energy transfer (eV)

Fig. 4.36 Mermin differential inverse inelastic mean free path of electrons in PMMA as a function of the energy loss for selected values of the incident electron kinetic energy E in the range 50-1000eV [65]. The calculations are based on the MELF-GOS method [61] (see Appendix H for details)

72

4 Scattering Mechanisms

Mermin IMFP is equal to 6.3 Å while when E = 1000 eV it is equal to 27.6 Å. According to TPP, the inelastic mean free path of PMMA is equal to 7.9 Å for E = 100 eV and to 33.7 Å for E = 1000 eV [63]. Also approaches based on the Drude-Lorentz theory provide values of the inelastic mean free path higher than those obtained using the Mermin theory. The inelastic mean free path of PMMA calculated according to the Drude-Lorentz theory is equal to 10.1 Å for E = 100 eV and to 33.5 Å for E = 1000 eV [64, 65]. Since the Mermin theory is more accurate, it is preferable, although both the predictive TPP formula and the Drude-Lorentz theory provide results that are compatible with the available experimental data.

4.3.6 Ab-initio Methods for Calculating the Complex Dielectric Function The complex dielectric function, and hence the energy loss function, can also be calculated by using first principles, i.e., by using ab-initio methods. Here we give just a short description of these methods and show an application. It is clear that since ab-initio methods provide very accurate results, they should be preferred even if they are not always easy to be implemented. Please note that, by using firstprinciples calculations, we can determine the energy loss function including the momentum transfer dispersion. The computational cost is the same as the zeromomentum transfer. Today, user-friendly codes performing such calculations can be obtained free of charge (or from the authors upon request). For example, the ELK website is available at: http://elk.sourceforge.net/. ELK program implements an all-electron Full-Potential Linearized Augmented-Plane-Wave approach. ELK is not affected by any pseudo-potential approximation for the ion-electron Coulomb interaction description. The microscopic dielectric function of materials depends on the bare Coulomb potential and on the full density-response function χ(k, ω). The full density-response function can be obtained by solving a Dyson-like equation: χ−1 (k, ω) = χ−1 0 (k, ω) − vC (k) − f xc (k, ω) .

(4.115)

Here we have indicated with χ0 (k, ω) the non-interacting density-response function and with vC (k) the bare Coulomb potential. If ρ is the ground state density calculated with the density-functional theory (DFT) and vxc [ρ] the exchange-correlation functional which can be obtained, for example, by using the local density approximation (LDA), then   d vxc [ρ] (4.116) f xc (r, t) = dρ ρ=ρ(r,t)

4.3 Inelastic Scattering

73

is the time-dependent density-functional theory (TDDFT) kernel. Let us consider a periodic crystal. In this case, the microscopic dielectric function is a matrix, i.e., [67]

−1 G,G (k, ω) = δG,G + vG (k)χG,G (k, ω)

(4.117)

where G and G are reciprocal lattice vectors, k is the transferred momentum vector in the first Brillouin zone, and vG (k) =

4π . |k + G|2

(4.118)

The macroscopic dielectric function ε(k, ω) is given by  −1 ε(k, ω) = −1 G=0,G =0 (k, ω)

(4.119)

In Fig. 4.37 we present the energy loss function of Au in the optical limit (k → 0) calculated using the described ab-initio method and compared with the Werner et al. experimental data [66]. Note that collective excitations (plasmons) occur when the real part of the dielectric function ε1 is zero and the imaginary part ε2 of the dielectric function approaches zero, as, in this case, the reciprocal of the imaginary part of the dielectric function presents a resonance. Indeed, if ε1 = 0, then Im

 1 1 = − . ε ε2

(4.120)

From an analysis of the real and imaginary parts of the dielectric function, Gurtubay et al. [67] concluded that the energy of the plasmon peak of Au (expected at ∼9 eV in correspondence to the density of the free electrons) is lowered at ∼5–6 eV as, at these energies, the real part of the dielectric function is zero. Gurtubay et al. observe that, on the other hand, the imaginary part of the dielectric function, at ∼5–6 eV, is different from zero and is not small due to inter-band transitions damping the freeelectron collective excitations. Also, at ∼23–24 eV and ∼31–32 eV, two peaks can be observed, in Fig. 4.37, with the real part of the dielectric function close to zero, indicating resonances in the energy loss function related to collective excitations. So, according to Gurtubay et al. [67], these peaks are due to a combination of band structure effects and d-like collective excitations. The peak at ∼16 eV and those with energies higher than ∼37 eV, in Fig. 4.37, are weakly dispersing with increasing momentum transfer, indicating their inter-band transition nature [68].

74

4 Scattering Mechanisms

Eneegy loss function

1.0

0.5

0.0 0

20

40

60

80

Energy transfer (eV) Fig. 4.37 Energy loss function of Au in the optical limit (k → 0) calculated using the ab-initio method (red line) compared with the Werner et al. experimental data (black line) [66]

4.3.7 Exchange Effects Exchange effects in the electron-electron interaction are due to the indistinguishability of scattered and ejected electrons. In order to take into account the exchange effects, the differential inverse inelastic mean free path have to be calculated as 1 dλ−1 inel = dω πa0 E



k+

k−



dk 1 [1 + f ex (k)] Im , k ε(k, ω)

(4.121)

where, according to the Born-Ochkur approximation [69–71],  f ex (k) =

k mv

4

 −

k mv

2 .

(4.122)

In Equation (4.122) m is the electron mass and v is the electron velocity. Exchange effects were discussed by de Vera et al. in [71]. The Mermin inelastic mean free path of electrons in PMMA, calculated including the Born-Ochkur approximation in order to take into account the exchange effects (black line) is compared, in Fig. 4.38, with the Mermin inelastic mean free path of electrons in PMMA calculated without exchange effects (gray line) [71]. The calculations are based on the MELF-GOS method [61]. The Born-Ochkur approximation is based on the premise that the exchange scattering is calculated by a scaling of the direct scattering, so the need for a phase factor

4.3 Inelastic Scattering

75

40 30

IMFP (Å)

20

10

10

100

1000

Energy (eV) Fig. 4.38 Mermin inelastic mean free path of electrons in PMMA [71] calculated according to the MELF-GOS method [61] (see Appendix H for details). Black line: With exchange. Red line: Without exchange. Courtesy of P. de Vera, I. Abril, R. Garcia-Molina

is removed and all the theories can be based on the direct scattering amplitude. Equation (4.121), with exchange described by the Born-Ochkur approximation, (4.122), according to Bourke [72] requires corrections. In particular, proper implementation of the relative phase of the final-state excitations is proposed. The Bourke model improves the calculation of exchange interference by providing a more physical description of the phenomenon and of the scattering processes.

4.3.8 Polaronic Effects A low-energy electron moving in an insulating material induces a polarization field that has a stabilizing effect on the moving electron. This phenomenon can be described as the generation of a quasi-particle called polaron. The polaron has a relevant effective mass and mainly consists of an electron (or a hole created in the valence band) with its polarization cloud around it. According to Ganachaud and Mokrani [7], the polaronic effects can be described assuming that the inverse inelastic mean free path governing the phenomenon—that is proportional to the probability for a low-energy electron to be trapped in the ionic lattice—is given by −γ E λ−1 pol = C e

(4.123)

76

4 Scattering Mechanisms

where C and γ are constants depending on the dielectric material. Thus the lower the electron energy, the higher the probability for an electron to lose its energy and create a polaron. This approach implicitly assumes that, once a polaron has been generated, the residual kinetic energy of the electron is negligible. Furthermore, it is assumed that the electron stays trapped in the interaction site. This is quite a rough approximation, as trapped electrons—due to phonon induced processes—can actually hop from one trapping site to another. Anyway, it is often sufficiently a good approximation for Monte Carlo simulation purposes, so it will be used in this book when we will deal with secondary electron emission from insulating materials.

4.4 Surface Phenomena 4.4.1 Bulk and Surface Plasmon Losses The plasma frequency ωp is given, in the Drude free electron theory, by (4.77) and represents the frequency of the volume collective excitations, which correspond to the propagation in the solid of bulk plasmons with energy E p = ωp .

(4.124)

In the electron energy loss spectra, it is thus expected to observe a bulk plasmon peak whose maximum is located at an energy E p [given by (4.124)]. Also, features related to surface plasmon excitations appear in spectra acquired either in reflection mode from bulk targets or in transmission mode from very thin samples or small particles [73]. Indeed, in the proximity of the surface, due to Maxwell’s equation boundary conditions, surface excitations modes (surface plasmons) take place with a resonance frequency slightly lower than the bulk resonance frequency. A rough evaluation of the energy of the surface plasmons can be performed— for a free electron metal—through the following very simple considerations [12]. In general, similarly to the volume plasmons propagating inside the solid, in the presence of an interface between two different materials—which we indicate here with a and b—, longitudinal waves travel as well along the interface. From continuity considerations, it follows that [12] ε a + εb = 0 ,

(4.125)

where we have indicated with εa the dielectric function on side a and with εb the dielectric function on side b of the interface. Let us now consider the particular case of a vacuum/metal interface and ignore, for the sake of simplicity, the damping, so that Γ ≈ 0. Then, if a represents the vacuum, we have εa = 1 ,

(4.126)

4.4 Surface Phenomena

77

and εb ≈ 1 −

ωp2 ωs2

,

(4.127)

where we have indicated with ωs the frequency of the longitudinal waves of charge density traveling along the surface. Then we obtain, from (4.125) 2−

ωp2 ωs2

= 0.

As a consequence, the surface plasmon energy E s = ωs , i.e., the surface plasmon peak position in the energy loss spectrum, is expected to be found at an energy Ep Es = √ . 2

(4.128)

Theoretical, experimental, and computational methods have been developed that allow estimating the surface excitation phenomena. Several examples can be find in [74–82]. Chen and Kwei theory. Chen and Kwei [78] used the dielectric theory to show that the differential inverse inelastic mean free path for electrons emerging from a solid surface can be split up into two terms. The first one is the differential inverse inelastic mean free path in an infinite medium. The second one is the so-called surface term which is related to a surface layer extending on both sides of the vacuum-solid interface. As a consequence, electrons can interact inelastically with the solid even if outside, if they are close enough to the surface. Spectra of electrons originating near the surface are therefore influenced by these surface effects. The original version of the Chen and Kwei theory concerned only outgoing projectiles [78]. It was generalized by Li et al. [80] for incoming projectiles. See Appendix G for details. So the theory predicts different trends for the inverse inelastic mean free path (IIMFP) for incoming and outgoing electrons when electrons are close to the surface. In particular, the inverse inelastic mean free path of the incoming electrons is found to slightly oscillate around the mean value, i.e., the bulk inverse inelastic mean free path. This phenomenon is attributed to the behavior of the electrons passing through the surface. Using the Chen and Kwei theory, one can calculate the dependence on z of the inverse inelastic mean free path, for any given electron kinetic energy. In Figs. 4.39 and 4.40 the inverse inelastic mean free path of Al and Si, respectively, are presented as a function of the electron’s energy and depth (both outside and inside the solid). The Chen and Kwei theory [78] and its generalization by Li et al. [80] have been used for simulating the surface and bulk plasmon loss peaks in Al and Si [81]. The energy loss spectrum can be calculated under the assumption that experimental spectra arise from electrons undergoing a single large angle elastic scattering

78

4 Scattering Mechanisms 0.10

incoming outgoing

IIMFP (

-1

)

500eV

0.05

1000eV 2000eV

0.00 -60

-40

-20

0

20

40

60

z( )

Fig. 4.39 Inverse inelastic mean free path (IIMFP) of electrons in Al as a function of the distance from the surface (in the solid and in the vacuum) for several kinetic energies of both incoming and outgoing electrons 0.10

incoming outgoing

IIMFP (

-1

)

500eV

0.05

1000eV 2000eV

0.00 -60

-40

-20

0

20

40

60

z( )

Fig. 4.40 Inverse inelastic mean free path (IIMFP) of electrons in Si as a function of the distance from the surface (in the solid and in the vacuum) for several kinetic energies of both incoming and outgoing electrons

4.4 Surface Phenomena

79

0.20

Intensity (arb. units)

0.15

0.10

0.05

0.00 0

10

20

Energy loss (eV)

Fig. 4.41 Comparison between the experimental (black line) and theoretical (red line) electron energy loss spectra for 1000 eV electrons impinging upon Al [81]. Calculated and experimental spectra were normalized to a common height of the bulk plasmon peak after linear background subtraction. Experimental data: Courtesy of Lucia Calliari and Massimiliano Filippi

Intensity (arb. units)

0.10

0.05

0.00 0

10

20

Energy loss (eV)

Fig. 4.42 Comparison between the experimental (black line) and theoretical (red line) electron energy loss spectra for 1000 eV electrons impinging upon Si [81]. Calculated and experimental spectra were normalized to a common height of the bulk plasmon peak after linear background subtraction. Experimental data: Courtesy of Lucia Calliari and Massimiliano Filippi

80

4 Scattering Mechanisms

event (so-called V-type trajectories [83]). In Figs. 4.41 and 4.42, the current calculation based on the combination of the Chen and Kwei and Li et al. theories [78, 80] with a single V-type trajectory modeling for Al and Si, are compared to experimental data [81]. Calculated and experimental spectra are normalized to a common height of the bulk plasmon peak. Dispersion law. Due to the presence of a quasi 2D slab on the top of a semiinfinite bulk, a further term has to be added in the momentum transfer dispersion law expressed by (4.96) [84, 85]. According to Kyriakou et al. [84], in order to properly take into account surface effects, the more general dispersion law is the following one:   2 2 2 12 E F 2 k 2  k 6E F 2 k 2 2 2 + + . (4.129)  ωk = ω p 5 2m 5 2m 2m

4.5 Summary In this chapter, elastic and inelastic scattering cross-sections were described. They are the main ingredients of the Monte Carlo simulation. In particular, the elastic scattering collisions can be calculated by the Mott crosssection, the electron-plasmon inelastic scattering events by the Ritchie dielectric theory, and electron-phonon energy losses by the Fröhlich theory. Polaronic effects can be computed according to Ganachaud and Mokrani. Chen and Kwei theory and its generalization due to Li et al. were also described. These theories allow dealing with the surface phenomena, particularly important for the investigation of reflection electron energy loss spectroscopy when the incident electron energy is smaller than 2–3 keV.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

N.F. Mott, Proc. R. Soc. Lond. Ser. 124, 425 (1929) H. Fröhlich, Adv. Phys. 3, 325 (1954) H.A. Bethe, Ann. Phys. Leipzig 5, 325 (1930) R.O. Lane, D.J. Zaffarano, Phys. Rev. 94, 960 (1954) K. Kanaya, S. Okayama, J. Phys. D Appl. Phys. 5, 43 (1972) R.H. Ritchie, Phys. Rev. 106, 874 (1957) J.P. Ganachaud, A. Mokrani, Surf. Sci. 334, 329 (1995) J. Kessler, Polarized Electrons (Springer, Berlin, 1985) P.G. Burke, C.J. Joachain, Theory of Electron-Atom Collisions (Plenum Press, New York, 1995) M.E. Riley, J. MacCallum, F. Biggs, At. Data Nucl. Data Tables 15, 443 (1975) P. Sigmund, Particle Penetration and Radiation Effects (Springer, Berlin, 2006) R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 3rd edn. (Springer, New York, Dordrecht, Heidelberg, London, 2011) 13. R.F. Egerton, Rep. Prog. Phys. 72, 016502 (2009) 14. A. Jablonski, F. Salvat, C.J. Powell, J. Phys. Chem. Data 33, 409 (2004)

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60. N.D. Mermin, Phys. Rev. B 1, 2362 (1970) 61. I. Abril, R. Garcia-Molina, C.D. Denton, F.J. Pérez-Pérez, N.R. Arista, Phys. Rev. A 58, 357 (1998) 62. W. de la Cruz, F. Yubero, Surf. Interface Anal. 39, 460 (2007) 63. S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interface Anal. 21, 165 (1994) 64. M. Dapor, Nucl. Instrum. Methods Phys. Res. B 352, 190 (2015) 65. M. Dapor, Front. Mater. 2, 27 (2015) 66. W.S.M. Werner, K. Glantschnig, C. Ambrosch-Draxl, J. Phys. Chem. Ref. Data 38, 1013 (2009) 67. I.G. Gurtubay, J.M. Pitarke, I. Campillo, A. Rubio, Comput. Mater. Sci. 22, 123 (2001) 68. S. Taioli, M. Dapor, F. Dimiccoli, M. Fabi, V. Ferroni, C. Grimani, M. Villani, W.J. Weber, Class. Quantum Grav. 40, 075001 (2023) 69. V.I. Ochkur, Soviet Phys. J.E.T.P. 18, 503 (1964) 70. J.M. Fernández-Varea, R. Mayol, D. Liljequist, F. Salvat, J. Phys.: Condens. Matter 5, 3593 (1993) 71. P. de Vera, I. Abril, R. Garcia-Molina, J. Appl. Phys. 109, 094901 (2011) 72. J.D. Bourke, Phys. Rev. B 100, 184311 (2019) 73. C.J. Powell, J.B. Swann, Phys. Rev. 115, 869 (1959) 74. G. Chiarello, E. Colavita, M. De Crescenzi, S. Nannarone, Phys. Rev. B 29, 4878 (1984) 75. Y. Ohno, Phys. Rev. B 39, 8209 (1989-II) 76. F. Yubero, J.M. Sanz, E. Elizalde, L. Galán, Surf. Sci. 237, 173 (1990) 77. C.J. Tung, Y.F. Chen, C.M. Kwei, T.L. Chou, Phys. Rev. B 49, 16684 (1994-I) 78. Y.F. Chen, C.M. Kwei, Surf. Sci. 364, 131 (1996) 79. W.S.M. Werner, W. Smekal, C. Tomastik, H. Stoöri, Surf. Sci. 486, L461 (2001) 80. Y.C. Li, Y.H. Tu, C.M. Kwei, C.J. Tung, Surf. Sci. 589, 67 (2005) 81. M. Dapor, L. Calliari, S. Fanchenko, Surf. Interface Anal. 44, 1110 (2012) 82. M. Dapor, Front. Mater. 9, 1068196 (2022) 83. A. Jablonski, C.J. Powell, Surf. Sci. 551, 106 (2004) 84. I. Kyriakou, D. Emfietzoglou, R. Garcia-Molina, I. Abril, K. Kostarelos, J. Appl. Phys. 110, 054304 (2011) 85. M. Azzolini, O.Y. Ridzel, P.S. Kaplya, V. Afanas’ev, N.M. Pugno, S. Taioli, M. Dapor, Comput. Mat. Sci. 173, 109420 (2020)

Chapter 5

Random Numbers

As Monte Carlo is a statistical method, the accuracy of its results depends on the number of simulated electron trajectories and on the pseudo-random number generator used to perform the simulations. We will briefly summarize how pseudo-random numbers can be generated. We will describe, as well, how to calculate selected random number distributions that are particularly relevant for the Monte Carlo method purposes [1]. We are firstly interested in a generator of pseudo-random numbers uniformly distributed in the range [0, 1]. Once it is given, we will describe the way to generate pseudo-random numbers uniformly distributed in a given interval, pseudo-random numbers distributed according to the exponential density of probability, and pseudorandom numbers distributed according to the Gauss density of probability [2].

5.1 Generating Pseudo-Random Numbers The algorithm most frequently used for the generation of pseudo-random numbers uniformly distributed in a given interval provides the entire sequence from a “seed” number: Starting with an initial number, known as the seed, the subsequent random numbers are calculated using an equation that permits to obtain each random number from the previous one. Every number of the sequence is computable knowing the value of the last calculated random number [2]. Let us suppose that μn is the n th pseudo-random number in the sequence. Then the next random number μn+1 is given by μn+1 = (aμn + b) mod m

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1_5

(5.1)

83

84

5 Random Numbers

where a, b and m are three integer numbers. Choosing the values of the three “magic” numbers a, b, and m in a proper way, sequences of random numbers corresponding to the maximum period (which is equal to m) are obtained. In such a way, for every initial seed μ0 , all the integer numbers from 0 to m − 1 will be in the sequence. For b = 0, (5.1) is known as a linear congruential generator while, if b = 0, it is known as a multiplicative linear congruential generator [2]. Several proposals were provided for the three “magic” numbers a, b, and m. Statistical tests have been used to establish the values of the three numbers a, b, and m in order to appropriately approximate a sequence of integer random numbers uniformly distributed in the interval from 0 to m − 1. A simple proposal is the so-called “minimal standard” which corresponds to a = 75 = 16807, b = 0, m = 231 − 1 [3, 4]. Pseudo-random number generators used today in programming languages do not present the weaknesses of the minimal standard and should be preferred. For a description of many very accurate pseudo-random number generators, see [2].

5.2 Testing Pseudo-Random Number Generators A classical (and very simple) test to check the quality and the uniformity of a pseudorandom number generator consists in simulating π. Let us generate a statistically significant number of pairs of random numbers in the range [−1, 1]. If the distribution of the generated pseudo-random numbers approaches a perfectly uniform distribution of random numbers, then the fraction of generated points that lie within the unit circle (i.e., the number of points in the circle divided by the total number of generated points) should approach π/4.

5.3 Pseudo-Random Numbers Distributed According to a Given Probability Density Let us indicate with ξ a random variable defined in the range [a, b] distributed according to a given probability density p(s). If μ represents a random variable uniformly distributed in the range [0,1], then the values of ξ can be obtained by the use of the equation:  ξ p(s) ds = μ . (5.2) a

5.5 Pseudo-Random Numbers Distributed According …

85

5.4 Pseudo-Random Numbers Uniformly Distributed in the Interval [a, b] Starting from a distribution μ uniformly distributed in the range [0, 1], we can use (5.2) to obtain a uniform distribution η in the interval [a, b]. The distribution η corresponds to the probability density: pη (s) =

1 . b−a

(5.3)

η satisfies the equation:  μ=

η

 pη (s) ds =

a

η

a

ds . b−a

(5.4)

As a consequence, η = a + μ(b − a) .

(5.5)

The expected value of the distribution is given by:  η =

b

 s pη (s) ds =

a

a

b

a+b s ds = . b−a 2

(5.6)

5.5 Pseudo-Random Numbers Distributed According to the Exponential Density of Probability Starting from a distribution μ uniformly distributed in the range [0, 1], we can use (5.2) to obtain the exponential distribution as well. It is a very important distribution for the Monte Carlo simulations, as the stochastic process for multiple scattering follows an exponential-type law. The exponential distribution is defined by the following probability density: pχ (s) =

 s 1 exp − , λ λ

(5.7)

where λ is a constant. A random variable χ distributed according to the exponential law, and defined in the interval [0, ∞), is given by the solution of the equation: 

χ

μ= 0

 s 1 ds , exp − λ λ

(5.8)

86

5 Random Numbers

where μ is, as usual, a random variable uniformly distributed in the range [0, 1]. Then χ = −λ ln(1 − μ) . (5.9) Since the distribution of 1 − μ is equal to that of μ, we also have: χ = −λ ln(μ) .

(5.10)

The constant λ is the expected value of χ: 

∞

χ =

s pχ (s) ds =

0

1 λ



∞ 0

 s ds = λ . s exp − λ

(5.11)

5.6 Pseudo-Random Numbers Distributed According to the Gauss Density of Probability The sequences of random numbers distributed with Gaussian density can be calculated by using the Box-Muller method [2]. Let us indicate with μ1 and μ2 two sequences of random numbers uniformly distributed in the interval [0, 1]. Let us consider the transformation: γ1 = γ2 =



−2 ln μ1 cos 2πμ2 ,



−2 ln μ1 sin 2πμ2 .

(5.12) (5.13)

Algebraic manipulations permit to calculate μ1 and μ2 ,   1 μ1 = exp − (γ12 + γ22 ) , 2 μ2 =

γ2 1 arctan . 2π γ1

(5.14)

(5.15)

Let us now consider the Jacobian determinant J of the random variables μ1 and μ2 with respect to the random variables γ1 and γ2 . Since   1 ∂μ1 = −γ1 exp − (γ12 + γ22 ) , ∂γ1 2

(5.16)

  ∂μ1 1 = −γ2 exp − (γ12 + γ22 ) , ∂γ2 2

(5.17)

5.7 Summary

87

and

γ2 ∂μ2 1 = − , 2 ∂γ1 2π γ1 + γ22

(5.18)

∂μ2 γ1 1 = , ∂γ2 2π γ12 + γ22

(5.19)

the Jacobian determinant J is given by J =

∂μ2 ∂μ1 ∂μ1 ∂μ2 − = −g(γ1 ) g(γ2 ) , ∂γ1 ∂γ2 ∂γ1 ∂γ2

where g(γ) =

exp(−γ 2 /2) √ 2π

(5.20)

(5.21)

The relationship between the joint probability distribution p(γ1 , γ2 ) dγ1 dγ2 (of the random variables γ1 and γ2 ) and the joint probability distribution p(μ1 , μ2 ) dμ1 dμ2 (of the random variables μ1 and μ2 ) is given by: p(γ1 , γ2 ) dγ1 dγ2 = p(μ1 , μ2 ) J dμ1 dμ2 = = − p(μ1 , μ2 ) g(γ1 ) g(γ2 ) dμ1 dμ2 .

(5.22)

The Jacobian determinant is the product of two Gaussian functions, given by (5.21), one of γ1 alone, and the other one of γ2 alone. Thus the two random variables γ1 and γ2 are independently distributed according to the Gaussian density.

5.7 Summary We have described a very simple algorithm used for the generation of pseudo-random numbers uniformly distributed in a given interval. It provides the whole sequence from a seed number. Starting with a given initial number, the algorithm computes the subsequent pseudo-random numbers according to a simple rule. Knowing the value of the last calculated pseudo-random number, any other number in the sequence can be easily computed. Pseudo-random number generators used today in programming languages do not present the weaknesses of the minimal standard. Once provided a generator of pseudo-random numbers uniformly distributed on the range [0, 1], sequences of pseudo-random numbers distributed according to given densities of probability can be obtained by the use of specific algorithms. Several examples, useful for the purposes of transport Monte Carlo, were provided in this chapter.

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5 Random Numbers

References 1. M. Dapor, Surf. Sci. 600, 4728 (2006) 2. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C. The Art of Scientific Computing (Cambridge University Press, Cambridge, 1985, 1992, 2007) 3. P.A. Lewis, A.S. Goodman, J.M. Miller, IBM Syst. J 8, 136 (1969) 4. S.K. Park, K.W. Miller, Commun. ACM 31, 1192 (1988)

Chapter 6

Monte Carlo Strategies

Monte Carlo is one of the most powerful theoretical methods for evaluating the physical quantities related to the interaction of electrons with a solid target. A Monte Carlo simulation can be considered as an idealized experiment. It is based on several fundamental ingredients. It is necessary to have a good knowledge of them—in particular of the energy loss and angular deflection phenomena—to obtain a good simulation. All the cross-sections and mean free paths have to be previously accurately calculated: They are then used in the Monte Carlo code in order to obtain the macroscopic characteristics of the interaction processes by simulating a large number of single particle trajectories and then averaging them. Due to the recent evolution in computer calculation capability, we are now able to obtain statistically significant results in very short calculation times. Two main strategies can be utilized in order to simulate electron transport in solid targets. The first one, the so-called continuous-slowing-down approximation, is very simple and assumes that electrons continuously lose energy as they travel inside the solid, changing direction when elastic collisions occur. It is frequently used—for it is a very fast procedure—when the description of the statistical fluctuations of the energy loss due to the different energy losses suffered by each electron of the penetrating beam and of the shower of the secondary electron are not crucial for simulating the desired quantities. This is the case, for example, for the calculation of the backscattering coefficient or the depth distribution of the absorbed electrons. If, instead, accurate descriptions of all the inelastic events which occur along the electron path—i.e., of the statistical fluctuations of the energy loss—are needed to study the investigated phenomena, a second strategy is required. It is a strategy where energy straggling is properly taken into account simulating all the single energy losses occurring along the electron trajectory (together with the elastic events in order to take into account the changes of direction). This is the case, for example, for the calculation of the energy distribution of the electrons emitted by the surface of the solid target. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1_6

89

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6 Monte Carlo Strategies

In this chapter, both these strategies will be briefly described, while the discussion of specific features and details will be found in the chapters devoted to the applications. For both descriptions, we will adopt spherical coordinates (r, θ, φ) and assume that the surface of the target is in the plane z = 0. When simulating electron spectra, streams of electrons irradiate the solid target with an incident angle θ0 with respect to the normal to the surface, i.e., the z-axis.

6.1 The Continuous-Slowing-Down Approximation Let us first describe the Monte Carlo method based on the continuous-slowing-down approximation. It requires the use of the stopping power—for calculating the energy losses along the electron trajectories—while the electron angular deflections are ruled by the Mott cross-section.

6.1.1 The Step-Length The stochastic process for multiple scattering is assumed to follow an exponentialtype law. The step-length Δs is then given by Δs = −λel ln(μ1 ) ,

(6.1)

where μ1 is a random number uniformly distributed in the range [0, 1] and λel is the elastic mean free path: 1 . (6.2) λel = N σel Here we have indicated with N the number of atoms per unit volume in the solid and with σel the total elastic scattering cross-section, given by  σel (E) =

dσel dΩ = dΩ

 0

π

dσel 2 π sin ϑ dϑ . dΩ

(6.3)

6.1.2 Interface Between Over-Layer and Substrate For surface films, the interface between the overlayer and the substrate must be properly taken into account. The change in the scattering probabilities per unit length in passing from the film to the substrate and vice versa, have to be considered so that (6.1) has to be accordingly modified. Let us denote with p1 and p2 the scattering

6.1 The Continuous-Slowing-Down Approximation

91

probabilities per unit length for the two materials, where p1 refers to the material in which the last elastic collision occurred and p2 to the other material and let us indicate with d the distance along the scattering direction between the initial scattering and the interface. According to Horiguchi et al. [1] and Messina et al. [2], if μ1 is a random number uniformly distributed in the range [0, 1], the step-length Δs is given by, ⎧  ⎨ 1 [− ln(1 − μ1 )], 0 ≤ μ1 < 1 − exp(− p1 d); p1   Δs = 1 ⎩d + [− ln(1 − μ1 ) − p1 d], 1 − exp(− p1 d) ≤ μ1 ≤ 1. p2

(6.4)

6.1.3 The Polar Scattering Angle The polar scattering angle θ resulting from an elastic collision is calculated assuming that the probability of elastic scattering into an angular range from 0 to θ, Pel (θ, E) =

2π σel

 0

θ

dσel sin ϑ dϑ , dΩ

(6.5)

is a random number μ2 uniformly distributed in the range [0, 1]: μ2 = Pel (θ, E) .

(6.6)

In other words, the sampling of the elastic scattering is performed looking for the angle of scattering corresponding to a random number uniformly distributed in the range [0, 1] (see Fig. 6.1). The angle of scattering, for any given electron energy, is calculated looking for the upper limit of integration in (6.5), once imposed that μ2 be equal to Pel (θ, E) [ (6.6)].

6.1.4 Direction of the Electron After the Last Deflection The azimuthal angle φ can assume any value selected by a random number μ3 uniformly distributed in the range [0, 2π]. Both the θ and φ angles refer to the last direction before the impact. The direction  θz in which the electron is moving after the last deflection, relative to the z direction, is given by [3–5]  (6.7) cos θz = cos θz cos θ − sin θz sin θ cos φ . In the last equation, θz is the angle relative to the z direction before the impact. The step of trajectory along the z direction, Δz, is then obtained by

92

6 Monte Carlo Strategies

Pel

1.0

0.5

0.0 0

30

60

90

120

150

180

Scattering angle (deg)

Fig. 6.1 Sampling of the elastic scattering angle for electrons in silicon. Pel is the cumulative probability for elastic scattering into an angular range from 0 to θ calculated numerically by solving the Dirac equation in a central field, according to the relativistic partial wave expansion method (Mott cross-section). Solid line: E = 500 eV; Dashed line: E = 1000 eV; Dotted line: E = 2000 eV



Δz = Δs cos θz .

(6.8)



The new angle θz is the incident angle θz corresponding to the next path length.

6.1.5 Electron Position in Three Dimensional Cartesian Coordinates In order to describe, at each scattering point, the electron position in three dimensional Cartesian coordinates (x, y, z), let us indicate with Δsn the length of nth step. Thus ⎧ ⎨xn+1 = xn + Δsn sin θn cos φn yn+1 = yn + Δsn sin θn sin φn ⎩ z n+1 = z n + Δsn cos θn

(6.9)

where the following transformations ⎧ ⎨cos θn = cos θn−1 cos θ − sin θn−1 sin θ cos φ sin(φn − φn−1 ) = sin θ sin φ/ sin θn ⎩ cos(φn − φn−1 ) = (cos θ − cos θn−1 cos θn )/(sin θn−1 sin θn )

(6.10)

6.2 The Energy-Straggling Strategy

93

relate the angles in the coordinate system of the target with the angles in the coordinate systems moving with the electron [5].

6.1.6 The Energy Loss The basic idea of the continuous-slowing-down approximation, is to assume that electrons lose energy with continuity while they are traveling in the solid: In order to calculate the energy loss along the various segments of the electron trajectory, use is made of the stopping power. Monte Carlo codes typically approximate the energy loss ΔE along the segment of trajectory Δs by the following equation ΔE = (d E/ds)Δs ,

(6.11)

where −d E/ds is the electron stopping power. With this approach, statistical fluctuations of the energy losses are completely neglected. As a consequence this kind of Monte Carlo strategy should be avoided when detailed information about energy loss mechanisms are required (for example when we are interested in the energy distribution of the emitted electrons).

6.1.7 End of the Trajectory and Number of Trajectories Each electron is followed until its energy becomes lower than a given value or until it emerges from the target surface. The selection of the cut-off value of the energy depends on the particular problem one is investigating. For the calculation of the backscattering coefficient, for example, the electrons are followed until their energy becomes smaller than 50 eV. It should be noted that even the number of trajectories is a crucial quantity to obtain statistically significant results and to improve the signal to noise ratio. In this work the typical number of trajectories, used for the presented simulations based on the continuous-slowing-down approximation strategy, ranges from 105 to 106 , depending on the particular problem investigated.

6.2 The Energy-Straggling Strategy Let us now describe the Monte Carlo method based on the energy-straggling strategy [5, 6]. It requires a detailed knowledge of all the energy loss phenomena (due to electron-electron interactions [7] and, for the case of very slow electrons in insulating materials, also to electron-phonon interactions [8] and polaronic effects [9]).

94

6 Monte Carlo Strategies

The electron angular deflections are due to electron-atom collisions (Mott theory), electron-electron collisions, and, for low energy electrons in insulators, electronphonon collisions (Fröhlich theory [8]).

6.2.1 The Step-Length The Monte Carlo method based on the energy-straggling strategy requires an approach different from that based on the continuous-slowing-down approximation. Also in this case the stochastic process for multiple scattering is assumed to follow an exponential-type law. The step-length Δs is thus given by Δs = −λ ln(μ1 ) ,

(6.12)

where μ1 is, as for the previous case, a random number uniformly distributed in the range [0, 1]. Now λ is no longer the elastic mean free path. It is instead given by λ=

1 , N (σel + σin )

(6.13)

where σel is the total elastic scattering cross-sections and σin is the total inelastic scattering cross-section. Note that the total scattering cross-section σel + σin includes all the scattering phenomena that are relevant for the specific material involved in the calculation. In the specific case of metals, σel is the elastic scattering cross-section due to the elastic interaction of the incident electron with the screened atomic nucleus while σin is the inelastic scattering cross-section due to the interaction of the incident electron with the atomic electrons located around the nucleus (valence and core electrons). When the target is an insulator, on the other hand, also trapping phenomena (polaronic effects) and electron-phonon quasi-elastic scattering cross-section have to be added. Concerning the well-known problem related to the excessively high values of the Mott elastic scattering cross-section (and the corresponding unreasonably small values of the elastic mean free paths) which are observed at very low energy, Ganachaud and Mokrani [9] maintain that the elastic scattering cross-section has to be multiplied by a cut-off function Rc (E) such that, when the electron energy becomes very low (and the electron-phonon scattering cross-section values become high) the elastic scattering cross-section will become negligible. The cut-off function proposed by these authors is the following: Rc (E) = tanh[αc (E/E g )] ,

(6.14)

where E is the electron energy, E g is the energy gap, and αc is a dimensionless parameter. The effect of multiplying the Mott elastic scattering cross-section by Rc (E) is to reduce the elastic effects at low energy, when electron-phonon interactions

6.2 The Energy-Straggling Strategy

95

become important, and to recover the behavior of the Mott cross-section at high energies when the electron-phonon cross-section becomes negligible.

6.2.2 Elastic and Inelastic Scattering The probability of inelastic scattering, pin , is given by pin =

σin λ = , σin + σel λin

(6.15)

while that of elastic scattering is given by pel = 1 − pin .

(6.16)

Before each collision, a random number μ2 uniformly distributed in the range [0, 1] is generated and compared with the probability of inelastic scattering pin . If the random number μ2 is less than or equal to pin , then the collision will be inelastic, otherwise it will be elastic.

6.2.3 Energy Loss In each electron-electron inelastic collision the function Pinel (W, E) that provides the fraction of electrons losing energies less than or equal to W [10] (see Fig. 6.2 where the function Pinel (W, E) is represented for 1000 eV electrons impinging on Si) is calculated. The energy loss W is obtained by generating a random number μ3 uniformly distributed in the range [0, 1], and imposing that μ3 is equal to Pinel (W, E): μ3 = Pinel (W, E) =

1 σinel

 0

W

dσinel dw . dw

(6.17)

In the case of ionization, a secondary electron is generated whose energy is equal to the energy lost by the incident electron, W , minus the binding energy, B. In the case of excitation (W ≤ B), the incident electron energy loss does not create any secondary electron. If an electron-lattice interaction occurs, with the creation of a phonon, the energy lost by the electron is equal to the energy of the created phonon, Wph . Lastly, we adopted the approximation below for the cases in which a polaron is generated: The electron ends its travel in the solid, as it is trapped where the interaction has taken place.

96

6 Monte Carlo Strategies

Pinel

1.0

0.5

0.0 0

10

20

30

40

50

W (eV)

Fig. 6.2 Sampling of the energy loss for electrons in silicon. Pinel is the cumulative probability for inelastic collisions of electrons in Si (calculated according to the dielectric Ritchie theory) causing energy losses less than or equal to W . The cumulative probability is here represented, as a function of the energy loss W, for E = 1000 eV

6.2.4 Electron-Atom Collisions: Scattering Angle As for electron-atom collisions (Mott theory), the polar scattering angle θ is calculated by generating a random number μ4 , uniformly distributed in the range [0, 1], representing the probability of elastic scattering into an angular range from 0 to θ: μ4 = Pel (θ, E) =

1 σel

 0

θ

dσel 2π sin ϑ dϑ . dΩ

(6.18)

6.2.5 Electron–Electron Collisions: Scattering Angle Let us consider the collision between two electrons when one of them (the secondary electron) is initially at rest. Let us indicate with p and E the initial momentum and energy, respectively, of the incident electron, with p and E  the momentum and energy, respectively, of the incident electron after the collision, and with q = p − p and W = E − E  the momentum and energy, respectively, of the secondary electron. Let us indicate with θ and θs , respectively, the polar scattering angle of the incident electron and of the secondary electron. Useful relationships between these quantities can be provided using the so-called classical binary-collision model,

6.2 The Energy-Straggling Strategy

97

which is sufficiently accurate for many practical purposes. Due to the conservation of momentum and energy, (6.19) sin θs = cos θ , where the polar scattering angle θ depends on the energy loss W = E − E  = ΔE according to the equation ΔE W = = sin2 θ . (6.20) E E Let us first demonstrate (6.19). To do that, we have to prove that the momentum of each of the two electrons in the final state is perpendicular to that of the other one. Let us then introduce the angle β between p and q. From momentum conservation,

we obtain

p = p + q ,

(6.21)

p 2 = p 2 + q 2 + 2 p  q cos β .

(6.22)

From conservation of energy,

it follows that

E = E  + ΔE ,

(6.23)

p 2 = p 2 + q 2 .

(6.24)

The comparison between (6.22) and (6.24) allows us to conclude, as mentioned before, that π (6.25) β = , 2 which is equivalent to (6.19). Let us now examine the consequences of the conservation laws on the dependence of the scattering angle θ—the angle between the initial momentum p and the final momentum p of the incident electron—on the electron energy loss. As

we obtain

p − p = q ,

(6.26)

q 2 = p 2 + p 2 − 2 pp  cos θ .

(6.27)

Equation (6.27) has two important consequences. The first one is that, in the final state, the absolute value of momentum q of the electron initially at rest, q, can assume only values belonging to the finite interval [q− ,q+ ] where q± =

 √ 2m E ± 2m(E − ΔE) ,

(6.28)

98

6 Monte Carlo Strategies

Fig. 6.3 Collision between two electrons

p

p |p| cos θ θ

θs

|p| sin θs q

as one can immediately see assuming that, in (6.27), θ = 0 (corresponding to q− ) and θ = π (corresponding to q+ ). The second consequence of (6.27) is that it offers the possibility, when conservation of energy is considered as well, to obtain the relationship between the scattering angle and the energy loss of the incident electron represented by (6.20). Indeed, according to (6.24), q 2 = p 2 − p 2 , so that, comparing this result with that expressed by (6.27), we obtain p 2 E , (6.29) cos2 θ = 2 = p E which is equivalent to (6.20). This equation can also be easily deduced from Fig. 6.3. Note that the angular pattern of inelastic collisions can also be calculated by using the dielectric theory. Within the framework of the dielectric theory, the scattering angles are found to belong to the interval [0, θmax ], where [11] θmax =

ΔE . E

(6.30)

6.2.6 Electron-Phonon Collisions: Scattering Angle In the case of electron-phonon collision, the corresponding polar scattering angle can be calculated according to the Llacer and Garwin theory [12]. Details can be found in Appendix E. Indicating with μ5 a new random number uniformly distributed in the range [0, 1], the polar scattering angle corresponding to an electron-phonon collision can be calculated as E + E cos θ = √ (1 − B μ5 ) + B μ5 , (6.31) 2 E E where B =

E + E + 2 E + E − 2

√ √

E E E E

.

(6.32)

6.2 The Energy-Straggling Strategy

99

6.2.7 Direction of the Electron After the Last Deflection Once the polar scattering angle has been calculated, the azimuthal angle is obtained by generating a random number μ6 uniformly distributed in the range [0, 2π]. The  direction θz in which the electron is moving after the last deflection, relative to the z direction, is calculated by (6.7). However, (6.9) and (6.10) have to be used in order to describe, at each scattering point, the particle position in three-dimensional Cartesian coordinates.

6.2.8 The First Step According to [5], the electrons perform the first step without any scattering at the interface between the vacuum and the sample. In other words, the energy losses and angular deflections occur at the last point of each step length.

6.2.9 Transmission Coefficient When dealing with very slow electrons, another important question to be considered is their capability to emerge from the surface of the solid [13]. In fact, the condition for an electron to emerge from the surface of a solid is not always satisfied. The interface with the vacuum represents a potential barrier, and not all the electrons that reach the surface can go beyond it. When the electrons reaching the surface cannot emerge, they are reflected back into the material. This problem is particularly important when investigating secondary electron emission, as secondary electrons typically have very low energy so they often cannot satisfy the condition to emerge. When a very slow electron with energy E reaches the target surface, it can emerge from the surface only if this condition is satisfied E cos2 θ = χ ,

(6.33)

where θ is the angle of emergence with respect to the normal to the surface, measured inside the specimen, and χ is the so-called electron affinity, i.e., the potential barrier represented by the difference between the vacuum level and the bottom of the conduction band. Its value depends on the investigated material. For example, the electron affinity of un-doped silicon is 4.05 eV [14].1 In order to study the transmission coefficient of slow electrons through the potential barrier χ, let us consider two regions along the z direction, inside and outside 1

Please note that, when dealing with metals, the same quantity, i.e., the potential barrier between the vacuum level and the bottom of the conduction band, is typically called work function.

100

6 Monte Carlo Strategies

the solid, respectively. Let us further assume that the potential barrier χ is located at z = 0. The first region, inside the solid, corresponds to the following solution of the Schrödinger equation: ψ1 = A1 exp(i k1 z) + B1 exp(−i k1 z) ,

(6.34)

while the solution in the vacuum is given by: ψ2 = A2 exp(i k2 z) .

(6.35)

In these equations, A1 , B1 , and A2 are three constants while k1 and k2 are, respectively, the electron wave-numbers in the solid and in the vacuum. They are given by k1 = k2 =

2m E cos θ , 2

(6.36)

2 m (E − χ) cos ϑ . 2

(6.37)

Here θ and ϑ represent the angles of the emergence of the electrons—with respect to the normal to the surface—measured, respectively, inside and outside the material. As the following conditions of continuity have to be satisfied ψ1 (0) = ψ2 (0) ,

(6.38)

ψ1 (0) = ψ2 (0) ,

(6.39)

A1 + B1 = A2 ,

(6.40)

(A1 − B1 ) k1 = A2 k2 ,

(6.41)

we have

and

so that the transmission coefficient T can be easily calculated to be

2

B1

4 k1 k2 . T = 1 −



= A1 (k1 + k2 )2

(6.42)

Taking into account the definition of electron wave-numbers given above, we obtain  4 (1 − χ/E) cos2 ϑ/ cos2 θ T =  2  1 + (1 − χ/E) cos2 ϑ/ cos2 θ

(6.43)

6.2 The Energy-Straggling Strategy

101

Due to the conservation of the momentum parallel to the surface E sin2 θ = (E − χ) sin2 ϑ . As a consequence cos2 θ =

(6.44)

(E − χ) cos2 ϑ + χ , E

(6.45)

E cos2 θ − χ . E −χ

(6.46)

cos2 ϑ =

In conclusion, the transmission coefficient T is given, as a function of ϑ, by  4 1 − χ/[(E − χ) cos2 ϑ + χ] T = 2 ,  1 + 1 − χ/[(E − χ) cos2 ϑ + χ]

(6.47)

and, as a function of θ,  4 1 − χ/(E cos2 θ) T =  2 .  1 + 1 − χ/(E cos2 θ)

6.2.9.1

(6.48)

Transmission Coefficient and the Monte Carlo Method

The transmission coefficient is an important quantity for the Monte Carlo description of low energy electrons emerging from the surface of a solid: The code generates a random number, μ7 , uniformly distributed in the range [0, 1] and enables the electron to be emitted into the vacuum if the condition μ7 < T

(6.49)

is satisfied. Those electrons which, once reached the surface, cannot satisfy the condition to emerge, are reflected back into the bulk of the specimen without energy loss and can contribute to the generation of further secondary electrons.

6.2.10 How Inelastic Scattering Depends on the Distance from the Surface In order to describe the surface plasmon loss peaks that can be observed in reflection electron energy loss spectroscopy when primary electron energy is smaller than ∼1000 eV, it is necessary to take into account that the inelastic scattering depends

102

6 Monte Carlo Strategies

Pinel

1.0

0.5

z = 0.0 Å z = 0.5 Å z = 1.0 Å z = 2.0 Å z = 5.0 Å z = 10.0 Å 0.0 0

10

20

30

40

50

W (eV)

Fig. 6.4 Sampling of the energy loss for electrons in silicon. Pinel is the cumulative probability for inelastic collisions of electrons in Si (calculated according to the Chen and Kwei theory [15, 16]) causing energy losses less than or equal to W. The cumulative probability is represented as a function of the energy loss for a few selected distances from the surface in the case of incoming electrons inside the solid. Similar curves can be calculated for incoming electrons outside the solid, outgoing electrons inside the solid, and outgoing electrons outside the solid. E0 = 1000 eV

on the distance from the surface (in the solid and in vacuum) and on the angle of surface crossing (see Figs. 4.39 and 4.40). The consequence is that, in the Monte Carlo simulation, the sampling of the energy loss previously discussed (see Fig. 6.2) is no longer sufficient. If we wish to describe the surface phenomena, cumulative probabilities for inelastic collisions of electrons have to be calculated not only as a function of energy loss W but also as a function of the distance from the surface. Furthermore, since in the Chen and Kwei and Li et al. [15, 16] theory also the vacuum (close to the surface) contributes to inelastic scattering, cumulative probabilities have to be calculated in the vacuum as well [17, 18]. In Fig. 6.4 the cumulative probability for inelastic collisions of electrons in Si is represented as a function of the energy loss for a few selected distances from the surface—in the case of incoming electrons inside the solid. It should be noted that, as z approaches ∞, the curves shown in Fig. 6.4 approach the “bulk” curve in Fig. 6.2. In order to include the information that the electron inelastic mean free path explicitly depends on z in the Monte Carlo code, the sampling procedure for the electron length between the successive scattering events also has to be accordingly modified. Let us assume that the distribution function, pχ (s), has the form [19]: pχ (s) =

   s ds  1 , exp −  λ(s) 0 λ(s )

(6.50)

6.2 The Energy-Straggling Strategy

103

For the sake of comparison, see (5.7), valid when λ does not depend on the depth. Since this equation is difficult to solve, according to Ding and Shimizu [19] the following procedure can be followed to calculate z. The first step is to put z = z i , where z i is the z component of the electron current position. If λmin represents the minimum value of the mean free path (the inelastic one when the particle is in a vacuum and, when it is in the material, calculated taking also into account the elastic mean free path), the second step involves the generation of two independent random numbers, μ8 and μ9 , uniformly distributed in the range [0, 1]. A new value of z is calculated by the equation: z = z i − cos θ λmin ln μ8 .

(6.51)

If μ9 ≤ λmin /λ, then the new value of z is accepted. Otherwise we put z = z i (i.e., z is now considered as the new z component of the initial position), two new random numbers μ8 and μ9 are generated, and a new value of z is calculated according to (6.51). Theoretical calculations and Monte Carlo results concerning surface and bulk plasmon loss peaks show a very good agreement with the available experimental data [19–29]. The agreement, in particular, between Monte Carlo simulated and experimental data, is very good even on an absolute scale [19, 27, 28].

6.2.11 End of the Trajectory and Number of Trajectories As for continuous-slowing-down approximation, described in the previous section, each electron is followed until its energy becomes lower than a given agreed threshold or until it emerges from the target surface. If, for example, we are studying the plasmon losses, the electrons can be followed until their energy becomes smaller than E 0 − 150 eV, as typically all the plasmon losses can be found in the energy ranges from E 0 − 150 eV to E 0 (where we have indicated with E 0 the primary energy expressed in eV). If, instead, we are facing the problem of simulating secondary electron energy distribution, the electrons must be followed until they reach a very small minimum energy. The use of trajectory methods when electron energies are very low is, of course, questionable. The problem was discussed by Lijequist [30] who demonstrated that, if short-range order is neglected, then trajectory simulations provide a good approximation down to 15 eV (less than 5% error). Short-range order produces diffraction effects under single scattering conditions. If multiple inelastic scattering is included, depending on the frequency of the inelastic scattering, trajectory simulation was demonstrated by Liljequist to be a good approximation, for electrons in water, even at energies as low as a few eV [31]. The number of trajectories is also a very important parameter. In this work, the typical number of trajectories, using the energy-straggling strategy for simulating spectra of energy distributions, ranges from 107 to 109 .

104

6 Monte Carlo Strategies

6.3 Summary This chapter briefly described the Monte Carlo method for the study of electron transport in solid targets. Its main features and characteristics were summarized, considering in particular two different strategies: One based on the so-called continuousslowing-down approximation, the other one on a scheme that takes into account the energy straggling, i.e., the statistical fluctuations of energy losses. We considered trapping phenomena and electron-atom, electron-electron, electron-plasmon, electron-phonon interactions with all the related effects, both in terms of energy losses and scattering angles.

References 1. S. Horiguchi, M. Suzuki, T. Kobayashi, H. Yoshino, Y. Sakakibara, Appl. Phys. Lett. 39, 512 (1981) 2. G. Messina, A. Paoletti, S. Santangelo, A. Tucciarone, La Rivista del Nuovo Cimento 15, 1 (1992) 3. J.F. Perkins, Phys. Rev. 126, 1781 (1962) 4. M. Dapor, Phys. Rev. B 46, 618 (1992) 5. R. Shimizu, Z.J. Ding, Rep. Prog. Phys. 55, 487 (1992) 6. H. Chen, Y. Zou, S. Mao, M.S.S. Khan, K. Tókési, Z.J. Ding, Sci. Rep. 12, 18201 (2022) 7. R.H. Ritchie, Phys. Rev. 106, 874 (1957) 8. H. Fröhlich, Adv. Phys. 3, 325 (1954) 9. J.P. Ganachaud, A. Mokrani, Surf. Sci. 334, 329 (1995) 10. H. Bichsel, Nucl. Instrum. Methods Phys. Res. B 52, 136 (1990) 11. L. Calliari, M. Dapor, G. Garberoglio, S. Fanchenko Surf. Interface Anal. 46, 340 (2014) 12. J. Llacer, E.L. Garwin, J. Appl. Phys. 40, 2766 (1969) 13. M. Dapor, Nucl. Instrum. Methods Phys. Res. B 267, 3055 (2009) 14. P. Kazemian, Progress towards Quantitive Dopant Profiling with the Scanning Electron Microscope (Doctorate Dissertation, University of Cambridge, 2006) 15. Y.F. Chen, C.M. Kwei, Surf. Sci. 364, 131 (1996) 16. Y.C. Li, Y.H. Tu, C.M. Kwei, C.J. Tung, Surf. Sci. 589, 67 (2005) 17. A. Jablonski, C.J. Powell, Surf. Sci. 551, 106 (2004) 18. M. Dapor, L. Calliari, S. Fanchenko, Surf. Interface Anal. 44, 1110 (2012) 19. Z.-J. Ding, R. Shimizu, Phys. Rev. B 61, 14128 (2000) 20. M. Novák, Surf. Sci. 602, 1458 (2008) 21. M. Novák, J. Phys. D: Appl. Phys. 42, 225306 (2009) 22. H. Jin, H. Yoshikawa, H. Iwai, S. Tanuma, S. Tougaard, e-J. Surf. Sci. Nanotech. 7, 199 (2009) 23. H. Jin, H. Shinotsuka, H. Yoshikawa, H. Iwai, S. Tanuma, S. Tougaard, J. Appl. Phys. 107, 083709 (2010) 24. I. Kyriakou, D. Emfietzoglou, R. Garcia-Molina, I. Abril, K. Kostarelos, J. Appl. Phys. 110, 054304 (2011) 25. B. Da, S.F. Mao, Y. Sun, Z.J. Ding, e-J. Surf. Sci. Nanotech. 10, 441 (2012) 26. B. Da, Y. Sun, S.F. Mao, Z.M. Zhang, H. Jin, H. Yoshikawa, S. Tanuma, Z.J. Ding, J. Appl. Phys. 113, 214303 (2013) 27. F. Salvat-Pujol, Secondary-electron emission from solids: Coincidence experiments and dielectric formalism (Doctorate Dissertation, Technischen Universität Wien, 2012) 28. F. Salvat-Pujol, W.S.M. Werner, Surf. Interface Anal. 45, 873 (2013) 29. T. Tang, Z.M. Zhang, B. Da, J.B. Gong, K. Goto, Z.J. Ding, Phys. B 423, 64 (2013) 30. D. Liljequist, Rad. Phys. Chem 77, 835 (2008) 31. D. Liljequist, J. Electron. Spectrosc. Relat. Phenom. 189, 5 (2013)

Chapter 7

Electron Beam Interactions with Solid Targets and Thin Films. Basic Aspects

Before describing the use of the Monte Carlo method to simulate the processes of electron-matter interaction, a few elementary concepts and definitions concerning electron beams impinging on solid targets and thin films will be provided. The model presented in this chapter was firstly proposed by Schmidt [1] and then developed by other authors [2–10]. It is known as the multiple reflection method.

7.1 Definitions, Symbols, Properties Let us consider an electron beam impinging on a solid target. If R represents the maximum penetration range, the target is a thin film if its thickness s is smaller than R. If s is greater than R, the target is a bulk target. The fraction of particles transmitted in such a case is zero, while the fraction of particles backscattered is equal to the backscattering coefficient r 1 (which is a function of the target atomic number and the electron primary energy). Let us now consider an electron beam impinging on a thin film, i.e., s ≤ R. We introduce, in such a case, the three quantities η A , η B and ηT representing the fraction of particles absorbed, backscattered, and transmitted, respectively. Each of them lies in the range [0, 1]. Since the total number of particles has to be conserved,2 then η A + η B + ηT = 1 for any given thickness. Note that if the film is deposited on a substrate, the fractions mentioned above are different from those corresponding to unsupported thin films. The difference is due to the backscattering from the substrate. In order to distinguish supported and In order to simplify the notation, we will use the symbol r instead of η in this chapter only to indicate the backscattering coefficient. 2 In this chapter we will not consider the generation of secondary electrons. 1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1_7

105

106

7 Electron Beam Interactions with Solid Targets and Thin Films. Basic Aspects

unsupported thin films, we will use the symbols ζ A , ζ B and ζT for the absorbed, backscattered, and transmitted fractions, respectively, for supported thin films. Note that each fraction ζ lies in the range [0, 1]. Conservation of the number of particles for the fractions ζ reads ζ A + ζ B + ζT = 1. Due to the backscattering from the substrate, ζ B > η B and ζT < ηT for any given film thickness. The fractions η and ζ depend on ξ(s), a function related to the scattering processes satisfying the following conditions3 : lim ξ(s) = 0 ,

(7.1)

lim ξ(s) = +∞ .

(7.2)

lim η A (ξ) = 0 ,

(7.3)

lim η B (ξ) = 0 ,

(7.4)

lim ηT (ξ) = 1 ,

(7.5)

lim η A (ξ) = 1 − r ,

(7.6)

lim η B (ξ) = r ,

(7.7)

lim ηT (ξ) = 0 .

(7.8)

s→0

s→R

Let us observe that ξ→0

ξ→0

ξ→0

ξ→∞

ξ→∞

ξ→∞

7.2 Unsupported Thin Films Let us indicate by Δξ an increment of ξ, i.e., Δξ = ξ(s + Δs) − ξ(s) ,

(7.9)

and let us calculate the fraction of particles absorbed by an unsupported thin film of thickness s + Δs. This is given by the fraction absorbed by ξ plus the fraction transmitted through ξ and absorbed by Δξ, plus the fraction transmitted through 3

Note that the simplest function satisfying these conditions is the following: ξ(s) ∝

s , R−s

where the constant of proportionality depends on the nature of the target.

7.2 Unsupported Thin Films

107

ξ, backscattered by Δξ and absorbed by ξ, and so on, in a sequence of infinite reflections: η A [ξ(s + Δs)] = η A (ξ + Δξ) = ∞  [η B (ξ)η B (Δξ)]n + = η A (ξ) + ηT (ξ)η B (Δξ)η A (ξ) n=0

+ ηT (ξ)η A (Δξ)

∞ 

[η B (ξ)η B (Δξ)]n =

n=0

η A (ξ)ηT (ξ)η B (Δξ) + ηT (ξ)η A (Δξ) . = η A (ξ) + 1 − η B (ξ)η B (Δξ)

(7.10)

Proceeding in a similar way, we obtain: η B (ξ + Δξ) = η B (ξ) + ηT (ξ + Δξ) =

ηT2 (ξ)η B (Δξ) , 1 − η B (ξ)η B (Δξ)

ηT (ξ)ηT (Δξ) . 1 − η B (ξ)η B (Δξ)

(7.11)

(7.12)

For ξ → ∞, η A (ξ) → 1 − r , η B (ξ) → r , and ηT (ξ) → 0. So, as a consequence, for Δξ → ∞, η A (ξ + Δξ) → 1 − r , η A (Δξ) → 1 − r , and η B (Δξ) → r . Let us now define 1 + r2 (7.13) μ= 2r and ν=

1 − r2 . 2r

(7.14)

Using (7.10) for Δξ → ∞, we obtain ηT2 = η 2B − 2μη B + 1 .

(7.15)

Let us now introduce the derivative β of η B calculated for ξ = 0: 

 η B (ξ) dη B (ξ) = |ξ=0 . β ≡ lim ξ→0 ξ dξ

(7.16)

From (7.11), we obtain the following equation: η B (Δξ) ηT2 (ξ) η B (ξ + Δξ) − η B (ξ) = . Δξ Δξ 1 − η B (ξ)η B (Δξ)

(7.17)

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7 Electron Beam Interactions with Solid Targets and Thin Films. Basic Aspects

Therefore

dη B (ξ) = βηT2 (ξ) = β[η 2B (ξ) − 2μη B (ξ) + 1] . dξ

(7.18)

This equation is equivalent to dη B dη B − = 2νβ dξ . η B − (1/r ) η B − r

(7.19)

Keeping in mind that η B (0) = 0, the integration of this equation gives η B (ξ) = r

1 − exp(−2νβξ) . 1 − r 2 exp(−2νβξ)

(7.20)

Since ηT2 = η 2B − 2μη B + 1 and η A = 1 − η B − ηT , we can conclude that 1 − r2 exp(−νβξ) 1 − r 2 exp(−2νβξ)

(7.21)

r exp(−2νβξ) − (1 + r ) exp(−νβξ) + 1 . 1 − r 2 exp(−2νβξ)

(7.22)

ηT (ξ) = and η A (ξ) = (1 − r )

Equations (7.20), (7.21) and (7.22) were deduced by Schmidt [1].

7.3 Supported Thin Films Let us now consider an electron beam impinging on a film of a given material x of thickness s which has been deposited on a substrate composed of a material y (different from material x). Rx indicates the maximum penetration range in a bulk target composed of x, and R y the maximum penetration range in a bulk target composed of y. Let us assume that the thickness of the substrate is greater than R y , thus the substrate is a bulk target for the primary energy considered. The maximum penetration range is a combination of Rx and R y whose value depends on the thickness of the film. It approaches R y for s → 0 and Rx for s → Rx . Indicating by r x the backscattering coefficient of x and by r y the backscattering coefficient of y, the following conditions have to be satisfied: lim ζ A (ξ) = 0 ,

(7.23)

lim ζ B (ξ) = r y ,

(7.24)

ξ→0

ξ→0

7.3 Supported Thin Films

109

lim ζT (ξ) = 1 − r y ,

(7.25)

lim ζ A (ξ) = 1 − r x ,

(7.26)

lim ζ B (ξ) = r x ,

(7.27)

lim ζT (ξ) = 0 .

(7.28)

ξ→0

ξ→∞

ξ→∞

ξ→∞

Let us keep in mind that ζ A is the fraction of electrons absorbed in the supported thin film, ζ B the fraction of electrons backscattered, and ζT the fraction of electrons transmitted across the interface between the film x and substrate y. To calculate ζ A , let us observe that the fraction η A is absorbed by the film, and then the fraction ηT r y comes back across the interface so that the fraction ηT r y η A is absorbed. From the infinite sum of contributions to the absorbed fraction ζ A , we obtain ζ A = η A + ηT r y η A + ηT r y η B r y η A + · · ·   ∞  n = η A 1 + ηT r y (η B r y )  = ηA 1 +

n=0

ηT r y 1 − ηB ry

 .

(7.29)

In a similar way, we obtain the following equations: ζ B = η B + ηT2 r y

∞  (η B r y )n = η B +

ηT2 r y , 1 − ηB ry

n=0

ζT = ηT

∞ ∞   ηT (1 − r y ) (η B r y )n − ηT r y (η B r y )n = . 1 − ηB ry n=0 n=0

(7.30)

(7.31)

If the film material is the same as the substrate material we can avoid the subscripts x and y in the equations above, so we can write  ζA = ηA 1 + ζB = ηB +

ηT r 1 − ηB r

 ,

(7.32)

ηT2 r =r , 1 − ηB r

(7.33)

ζT = ηT

1−r . 1 − ηB r

(7.34)

110

7 Electron Beam Interactions with Solid Targets and Thin Films. Basic Aspects

Upon substituting the Schmidt equations (7.20), (7.21) and (7.22) into the previous equations, we obtain [7]: ζ A = (1 − r )[1 − exp(−νβξ)] ,

(7.35)

ζB = r ,

(7.36)

ζT = (1 − r ) exp(−νβξ) .

(7.37)

Let us observe that the derivative of ζ A gives the implantation profile or the depth distribution, of the trapped electrons [7]: dξ dζ A = νβ(1 − r ) exp(−νβξ) . ds ds

(7.38)

Note that the relationship (7.15) between ηT and η B for unsupported thin films of the material x, taking into account the definition of μ (7.13), can also be rewritten as ηT =

(r x − η B )(1 − r x η B ) . rx

(7.39)

Substituting (7.39) in (7.30), we can conclude that the fraction of backscattered particles for a thin film of material x supported on a substrate of material y [8, 9] can be calculated by using the equation (Fig. 7.1): ζ B = η B + (r x − η B )

r y 1 − rx η B . rx 1 − r y η B

(7.40)

7.4 Multilayers Let us now consider a system composed of N layers. We assume that the materials and thicknesses of each layer are different from those of the other layers. Let ξn represents the extended thickness of the nth layer and let us indicate with f A (ξ1 , ..., ξ N ), f B (ξ1 , ..., ξ N ), and f T (ξ1 , ..., ξ N ), respectively, the fractions of particles absorbed, backscattered, and transmitted. Below, we will indicate with f A (ξ1 ), f B (ξ1 ), and f T (ξ1 ), respectively, our fractions for the first unsupported layer. The fractions of absorbed, backscattered and transmitted particles for the case in which the first layer has been removed will be indicated by f A (ξ2 , ..., ξ N ), f B (ξ2 , ..., ξ N ), and f T (ξ2 , ..., ξ N ). Let us now count the number of particles absorbed in our multilayer. We need to add the fraction of particles absorbed in the first layer to the fraction of particles transmitted through the first layer and then absorbed in the remaining system, plus the fraction that, once transmitted through the first layer and backscattered by the

7.4 Multilayers

111

Electron backscattering ratio

0.5

0.4

0.3

0.2

0.1

0

2000

4000

6000

Film thickness (Å) Fig. 7.1 Electron backscattering ratio of surface films of Ag and Au deposited on bulk substrates of Al as a function of the film thickness. Primary electron energy: 20 keV. Symbols: Niedrig’s experimental data [11] (circles: Ag/Al, squares: Au/Al). Solid lines: (7.40) (black line: Ag/Al, red line: Au/Al)

remaining system, is absorbed by the first layer and so on. The following infinite sum provides the searched result: f A (ξ1 , ..., ξ N ) = f A (ξ1 ) + f T (ξ1 ) f A (ξ2 , . . . , ξ N ) + + f T (ξ1 ) f B (ξ2 , . . . , ξ N ) f A (ξ1 ) + + f T (ξ1 ) f B (ξ2 , . . . , ξ N ) f B (ξ1 ) f A (ξ2 , ..., ξ N ) + + f T (ξ1 ) f B (ξ2 , . . . , ξ N ) f B (ξ1 ) f B (ξ2 , . . . , ξ N ) f A (ξ1 ) + . . . = ∞  [ f B (ξ1 ) f B (ξ2 , . . . , ξ N )]n + = f A (ξ1 ) + f T (ξ1 ) f A (ξ2 , . . . , ξ N ) n=0 ∞  [ f B (ξ1 ) f B (ξ2 , . . . , ξ N )]n . + f T (ξ1 ) f B (ξ2 , . . . , ξ N ) f A (ξ1 ) n=0

As a consequence we can write f A (ξ1 , . . . , ξ N ) = f A (ξ1 ) + f T (ξ1 )

f A (ξ2 , . . . , ξ N ) + f A (ξ1 ) f B (ξ2 , . . . , ξ N ) 1 − f B (ξ1 ) f B (ξ2 , . . . , ξ N ) (7.41)

112

7 Electron Beam Interactions with Solid Targets and Thin Films. Basic Aspects

Proceeding in a similar way we find that: f B (ξ1 , . . . , ξ N ) = f B (ξ1 ) + and f T (ξ1 , . . . , ξ N ) =

f T2 (ξ1 ) f B (ξ2 , . . . , ξ N ) 1 − f B (ξ1 ) f B (ξ2 , . . . , ξ N )

f T (ξ1 ) f T (ξ2 , . . . , ξ N ) . 1 − f B (ξ1 ) f B (ξ2 , . . . , ξ N )

(7.42)

(7.43)

Equations (7.41), (7.42), and (7.43) with N = 2, ξ1 = ξ, ξ2 → ∞, f A (ξ2 ) = f A (∞) = 1 − r , f B (ξ2 ) = f B (∞) = r , f A (ξ) = η A (ξ), f B (ξ) = η B (ξ), f T (ξ) = ηT (ξ), give: f A (ξ1 , ξ2 ) = f A (ξ, ∞) = f A (ξ) + f T (ξ)

(1 − r ) + r f A (ξ) = 1 − r f B (ξ)

(1 − r ) + r η A (ξ) = η A (ξ) + ηT (ξ) = 1 − r η B (ξ)   ηT (ξ) r 1−r + ηT (ξ) = = η A (ξ) 1 + 1 − r η B (ξ) 1 − r η B (ξ) = ζ A (ξ) + ζT (ξ),

(7.44)

so that the fraction of particles absorbed in the target is given by the sum of the fraction of particles absorbed in the supported film plus the fraction of particles transmitted through the layer (and absorbed in the substrate). Similarly, we obtain f B (ξ1 , ξ2 ) = f B (ξ, ∞) = f B (ξ) + = η B (ξ) +

f T2 (ξ) r = 1 − r f B (ξ)

ηT2 (ξ) r = ζ B (ξ), 1 − r η B (ξ)

(7.45)

and, since f T (ξ2 ) = f T (∞) = 0, f T (ξ1 , ξ2 ) = f T (ξ, ∞) = 0.

(7.46)

7.4.1 Conservation of the Total Number of Particles Equations (7.41), (7.42), and (7.43) provide the fractions of particles respectively absorbed, backscattered and transmitted when a particle beam impinges on a multilayer. To conserve the total number of particles, the sum of these fractions must be equal to 1. This is the case, indeed, and it can be demonstrated by induction. Let us assume that this properties holds for N = 1 and N = n − 1 and show that, in such a

7.4 Multilayers

113

case, it holds also for N = n, n being an integer > 1 and N representing the number of layers. In order to keep it simple, let us write f X (ξ1 ) = f X 1 ,

(7.47)

f X (ξ2 , ..., ξn ) = f X n−1

(7.48)

f X (ξ1 , ξ2 , ..., ξn ) = f X n

(7.49)

and where X = A, B, T . Assuming that f A1 + f B1 + f T1 = 1

(7.50)

f An−1 + f Bn−1 + f Tn−1 = 1 ,

(7.51)

f An + f Bn + f Tn = 1 .

(7.52)

and

we wish to show that

This can be simply done by using (7.41), (7.42), and (7.43) which, with the new notation become, with N = n, f An = f A1 + f T1

f Bn = f B1 + and f Tn =

f A1 f Bn−1 + f An−1 , 1 − f B1 f Bn−1 f T21 f Bn−1 1 − f B1 f Bn−1

f T1 f Tn−1 1 − f B1 f Bn−1

(7.53)

(7.54)

(7.55)

Thus we have f An + f Bn + f Tn = (1 − f B1 f Bn−1 )−1 ( f A1 − f A1 f B1 f Bn−1 + f B1 − f B21 f Bn−1 + + f A1 f T1 f Bn−1 + f An−1 f T1 + f T21 f Bn−1 + f T1 f Tn−1 ) = = (1 − f B1 f Bn−1 )−1 (1 − f B1 f Bn−1 ) = 1 ,

(7.56)

where we used (7.50) and (7.51) to substitute f A1 with 1 − f B1 − f T1 and f An−1 with 1 − f Bn−1 − f Tn−1 .

114

7 Electron Beam Interactions with Solid Targets and Thin Films. Basic Aspects

7.5 Summary The multiple reflection method of Schmidt [1], which allows evaluating the fractions of electrons absorbed, backscattered, and transmitted when an electron beam impinges on an unsupported thin film, was described and briefly discussed. The method was then extended to the study of the interaction of particle beams with supported thin films and multilayers [7–9].

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

H.W. Schmidt, Ann. Phys. (Leipzig) 23, 671 (1907) V.E. Cosslet, R.N. Thomas, Br. J. Appl. Phys. 15, (883) 1964 V.E. Cosslet, R.N. Thomas, Br. J. Appl. Phys. 16, (779) 1965 V. Lantto, J. Phys. D: Appl. Phys. 7, 703 (1974) V. Lantto, J. Phys. D: Appl. Phys. 9, 1647 (1976) D. Liljequist, J. Phys. D: Appl. Phys. 10, 1363 (1977) M. Dapor, Phys. Rev. B 43, (1991) 10118 Erratum: Phys. Rev. B 44, 9784 (1991-I) M. Dapor, Phys. Rev. B 48, 3003 (1993) M. Dapor, Eur. Phys. J. AP 18, 162 (2002) M. Dapor, Electron-Beam Interactions with Solids: Application of the Monte Carlo Method to Electron Scattering Problems (Springer, Berlin, 2003) 11. H. Niedrig, J. Appl. Phys. 53, R15 (1982)

Chapter 8

Backscattering Coefficient

The backscattering coefficient is defined as the fraction of electrons of the primary beam emerging from the surface of an electron-irradiated target. Secondary electrons, generated in the solid by a cascade process of extraction of the atomic electrons, are not included in the definition of the backscattering coefficient. The energy cut-off is typically 50 eV. In other words, in a typical experiment aimed at measuring the fraction of backscattered electrons, investigators consider as backscattered all the electrons emerging from the surface of the target with energies higher than the cutoff energy (50 eV), while all the electrons emerging with energies lower than this conventional cut-off are considered as secondary. Of course, secondary electrons with energy higher than any predefined cut-off energy and backscattered electrons with energy lower than such a cut-off also exist. If the primary energy of the incident electron beam is not too low (higher than ≈200–300 eV), the introduction of the 50 eV energy cut-off is generally considered as a good approximation, and it will be therefore adopted in this chapter. This choice is particularly useful, as we are interested in comparing the Monte Carlo results to experimental data that can be found in the literature, where the 50 eV energy cut-off approximation has been widely used.

8.1 Electrons Backscattered from Bulk Targets When an electron beam impinges upon a solid target, some electrons of the primary beam are backscattered and re-emerge from the surface. We already know that the backscattering coefficient is defined as the fraction of the electrons of the incident beam which emerge from the surface with energy higher than 50 eV. This definition is very convenient and useful from the experimental point of view. It is also quite accurate, as the fraction of secondary electrons (i.e., the electrons extracted from the atoms of the target and able to reach the surface and emerge) with energy higher

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1_8

115

116

8 Backscattering Coefficient

than 50 eV is negligible for any practical purpose, as is the fraction of backscattered electrons emerging with energy lower than 50 eV. The backscattering coefficient of electrons in bulk targets has been studied both experimentally and theoretically. Many data, both experimental and theoretical, are available for energies higher than 5–10 keV [1, 2]. The case of energies lower than 5 keV has been investigated too, but not many experimental data are available. Furthermore, there are very few data concerning the case in which the electron energy approaches 0 [3, 4].

8.1.1 The Backscattering Coefficient: The Everhart Model In 1960 Everhart [5] showed that the backscattering coefficient for normal incidence, electron energy greater than ∼10 keV, and target atomic number lower than ∼45, can be calculated as: a − 1 + 0.5a η = , (8.1) a+1 where a = 0.045Z and Z is the target atomic number. The Everhart deduction is based on the Thomson–Whiddington energy loss relation [6] E 02 − E 2 = cz ,

(8.2)

where E 0 is the primary energy, E is the most probable electron beam energy at depth z in the target, and the value of c depends on the target.1 Everhart observed that, when the electron energy is smaller than the K-shell ionization energy, the mean excitation energy decreases as the electron energy increases. The Thomson– Whiddington energy loss relation can be obtained from the Bethe-Bloch formula by assuming that the mean excitation energy I linearly decreases as E decreases. It is easy to see that the Bethe-Bloch formula in such a case becomes −

k dE = , dz E

(8.3)

E2 E2 E dE = 0 − . k 2k 2k

(8.4)

where k is a constant. Thus  z=−

E E0

Note that the value of a obtained from the Thomson–Whiddington energy loss relation is ≈0.012 Z . Everhart proposed to use instead a = 0.045Z in order to obtain reasonable agreement with the available experimental data.

1

8.1 Electrons Backscattered from Bulk Targets

117

Fig. 8.1 A beam of electrons impinging on a surface of area S

ν S dz

ϑ=π−θ θ

With c ≡ 2k, this is the Thomson–Whiddington formula. In particular, we have E 02 = c R

(8.5)

E 2 = c(R − z) .

(8.6)

so that Using this formula and introducing the angle ϑ defined as ϑ = π −θ,

(8.7)

where θ is the scattering angle (see Fig. 8.1), the Rutherford cross-section for a pure Coulomb field (i.e., without screening) becomes dσel =

Z 2 e4 1 2π sin ϑ dϑ . 4c R − z (1 + cos ϑ)2

(8.8)

The fractional range is defined as u=

z . R

(8.9)

If N is the number of atoms per unit of volume in the target, then d N = N S dz gives the number of atoms in the volume S dz. The incremental number of electrons at a fractional depth u deflected through ϑ is given by dν(ϑ, u) = d N or dν(ϑ, u) = 2a

ν(u) dσel S

ν(u) du sin ϑ dϑ , 1 − u (1 + cos ϑ)2

where du = dz/R and a=

π Z 2 e4 N . 4c

(8.10)

(8.11)

(8.12)

118

8 Backscattering Coefficient

On the plane at the fractional depth u, the number ν(u) is given by 

u

ν(u) = ν0 −



0

π/2



and therefore

dν(ϑ, u  ) ,

(8.13)

0

u

ν(u) = ν0 − a 0

ν(u  )du  . 1 − u

(8.14)

This integral equation can be easily solved. We have ν(u) = ν0 (1 − u)a .

(8.15)

sin ϑ dϑ dν(ϑ, u) = 2a(1 − u)a−1 du . ν0 (1 + cos ϑ)2

(8.16)

As a consequence

The total distance traveled by an electron reflected at an angle ϑ at depth z into the target is (see Fig. 8.1) s = z + z sec ϑ . (8.17) If ϑ0 is the maximum escape angle for a given z, then we have u(1 + sec ϑ0 ) = 1 .

(8.18)

The backscattering coefficient η, i.e., the fraction of particles of the primary beam emerging from the surface, is given by:  η =

dν(ϑ, u) = ν0



1/2



ϑ0

2a(1 − u)a−1 du

0

0

sin ϑ dϑ . (1 + cos ϑ)2

(8.19)

The integration over ϑ gives 

ϑ0

0

1 sin ϑ dϑ 1 1 = − = −u 2 (1 + cos ϑ) 1 + cos ϑ0 2 2

(8.20)

and, hence, 

1/2

η = 2a 0

 (1 − u)a−1

 1 a − 1 + (0.5)a − u du = . 2 a+1

(8.21)

8.1 Electrons Backscattered from Bulk Targets

119

8.1.2 The Backscattering Coefficient: The Archard Model According to Archard [7], electrons move equally in all directions from the depth of complete diffusion, D, so that their paths equal the maximum penetration range, R. The backscattered electrons are, as a consequence, the electrons contained in the conical sector projecting onto the target surface. Once defined the angle ϑ so that (see Fig. 8.2) cos ϑ =

D , R − D

(8.22)

we have η =

1 4π



ϑ

2π sin θ dθ =

0

  1 D 1 (1 − cos ϑ) = 1− . 2 2 R − D

(8.23)

The complete diffusion, D, is proportional, according to Archard, to the maximum range of penetration, R. If Z is the target atomic number, then [7]: D ≈

40 R . 7Z

(8.24)

As a consequence, the backscattering coefficient is given by η =

7Z − 80 . 14 Z − 80

(8.25)

Fig. 8.2 Complete diffusion, D, of electrons penetrating in a solid target

D ϑ

R−D

120

8 Backscattering Coefficient

Backscattering coefficient

0.6

0.4

0.2

0.0 0

10

20

30

40

50

60

70

80

90

Atomic number Fig. 8.3 Backscattering coefficient calculated according to the Everhart’s formula [5] (black line), (8.1), and according to the Archard’s formula [7] (red line), (8.25), compared with the Neubert and Rogaschewski experimental data [8] (circles: 20 keV, squares: 60 keV)

Archard’s formula fails for small values of the atomic number (Z ≤ 11) as, in this case, D becomes larger than R/2. For Z higher than 50, it roughly agrees with the available experimental data, predicting backscattering coefficient values lower than those predicted by Everhart’s formula (see Fig. 8.3). Please note that Tomlin [9] proposed to calculate D as 8R , (8.26) D ≈ Z +8 so that η =

Z −8 . 2Z

(8.27)

With this definition of D, the formula describing η fails for Z ≤ 8. For an extended and detailed discussion about single scattering and diffusion models, see Niedrig’s review [10].

8.1.3 The Backscattering Coefficient: The Vicanek and Urbassek Model Another way for analytically calculating the backscattering coefficient, valid for a wide range of the energies of the incident particles, was proposed by Vicanek and

8.1 Electrons Backscattered from Bulk Targets

121

Urbassek [11]. It will help to introduce, according to the Vicanek and Urbassek approach, the quantity (8.28) m = N R σtr , i.e., the so-called mean number of wide-angle collisions, where R is the maximum range of penetration, N is the number of target atoms per unit volume, and σtr is the transport cross-section, given by 

π

σtr = 2π

(1 − cos ϑ)

0

dσel sin ϑdϑ . dΩ

(8.29)

Note that, actually, due to the energy lost by the particles inside the solid target, (8.28) gives only an approximate evaluation of the number of wide-angle collisions suffered by the electron before slowing down to rest [11]. Vicanek and Urbassek have shown that the backscattering coefficient η for light ions may be evaluated by the following simple function of the mean number of wide-angle collisions [11]: η = 

1 1+

a1 √μm0

+

μ2 a2 m0

μ3

μ4

0 + a3 m 3/2 + a 4 m02

,

(8.30)

where a1 ≈ 3.39, a2 ≈ 8.59, a3 ≈ 4.16, a4 ≈ 135.9, μ0 = cos θ0 , and θ0 is the polar angle of incidence of the beam. The Vicanek and Urbassek formula has been shown to provide results in reasonable agreement with the available experimental data for light ions [11], electrons, and positrons [12, 13]. In Tables 8.1 and 8.2 we show the backscattering coefficient of positron beams impinging on Al, Cu, Ag, and Au—calculated according to (8.30) [13]—compared with the Coleman et al. experimental data [12].

Table 8.1 Backscattering coefficients of 1000 eV positrons impinging on Al, Cu, Ag, and Au [13]. Comparison between the results obtained using the Vicanek and Urbassek formula [11], (8.30), and the Coleman et al. experimental data [12] Atomic number Equation (8.30) [11] Experimental data [12] 13 29 47 79

0.086 0.110 0.100 0.111

0.069 0.135 0.106 0.123

122

8 Backscattering Coefficient

Table 8.2 Backscattering coefficients of 3000 eV positrons impinging on Al, Cu, Ag, and Au [13]. Comparison between the results obtained using the Vicanek and Urbassek formula [11], (8.30), and the Coleman et al. experimental data [12] Atomic number Equation (8.30) [11] Experimental data [12] 13 29 47 79

0.107 0.159 0.156 0.186

0.086 0.177 0.168 0.186

8.1.4 The Backscattering Coefficient: Monte Carlo Simulation In Fig. 8.4 we show the trend of the backscattering coefficient as a function of the electron beam primary energy for electrons impinging upon bulk targets of C and Al as calculated by the Monte Carlo method. For the simulations presented here, the continuous-slowing-down approximation was adopted, and we calculated the stopping power by using Ashley’s recipe [14] (within the Ritchie dielectric theoretical scheme [15]). As everywhere in this book, the elastic scattering cross-section was

Backscattering coefficient

0.3

0.2

0.1

0.0 0

2000

4000

6000

8000

10000

E0 (eV)

Fig. 8.4 Monte Carlo simulation of the trend of the backscattering coefficient η as a function of the electron beam primary energy E 0 for electrons impinging upon bulk targets of C (empty circles) and Al (filled circles). The stopping power was taken from Ashley [14] (dielectric theory). The elastic scattering cross-section was calculated using the relativistic partial wave expansion method (Mott theory) [16]. Boxes: Bishop experimental data for C [17]. Diamonds: Bishop experimental data for Al [17]. Triangles: Hunger and Kükler experimental data for C [18]

8.1 Electrons Backscattered from Bulk Targets

123

Table 8.3 Backscattering coefficient of Si as a function of the electron primary kinetic energy. The elastic scattering cross-section was calculated using the Mott theory [16]. Continuous-slowing-down approximation was used. The stopping power was taken from Tanuma et al. [23] (dielectric theory). Comparison between the present Monte Carlo simulated results and the available experimental data (taken from Joy’s database [22]) Energy (eV) Monte Carlo Bronstein and Fraiman Reimer and Tolkamp [19] [20] 1000 2000 3000 4000 5000

0.224 0.185 0.171 0.169 0.162

0.228 0.204 0.192 0.189 –

0.235 – 0.212 – 0.206

Table 8.4 Backscattering coefficient of Cu as a function of the electron primary kinetic energy. The elastic scattering cross-section was calculated using the Mott theory [16]. Continuous-slowing-down approximation was used. The stopping power was taken from Tanuma et al. [23] (dielectric theory). Comparison between the present Monte Carlo simulated results and the available experimental data (taken from Joy’s database [22]) Energy (eV) Monte Carlo Bronstein and Koshikawa and Reimerand Fraiman [19] Shimizu [21] Tolkamp [20] 1000 2000 3000 4000 5000

0.401 0.346 0.329 0.317 0.314

0.381 0.379 0.361 0.340 –

0.430 0.406 0.406 – 0.398

– – 0.311 – 0.311

calculated using the relativistic partial wave expansion method (Mott theory) [16]. The experimental data of Bishop [17] and Hunger and Kükler [18] are also presented in Fig. 8.4 for evaluating the accuracy of the Monte Carlo simulation. In the case of both the examined elements, the backscattering coefficient increases as the energy decreases towards 1000 eV. In Tables 8.3, 8.4, and 8.5 the Monte Carlo simulated data (for the backscattering coefficient of Si, Cu, and Au, respectively) are compared with the available experimental data [19–21] (taken from Joy’s database [22]). The Monte Carlo results were obtained using the stopping power taken from Tanuma et al. [23] (within the Ritchie dielectric theory [15]) for describing the inelastic processes, and the Mott theory [16] for describing the elastic scattering.

124

8 Backscattering Coefficient

Table 8.5 Backscattering coefficient of Au as a function of the electron primary kinetic energy. The elastic scattering cross-section was calculated using the Mott theory [16]. Continuous-slowing-down approximation was used; the stopping power was taken from Tanuma et al. [23] (dielectric theory). Comparison between the present Monte Carlo simulated results and the available experimental data (taken from Joy’s database [22]) Energy (eV) Monte Carlo Bronstein and Reimer and Böngeler et al. Fraiman [19] Tolkamp [20] [24] 1000 2000 3000 4000 5000

0.441 0.456 0.452 0.449 0.446

0.419 0.450 0.464 0.461 –

– 0.415 – 0.448

– 0.373 0.414 0.443 0.459

From these tables, one can observe that the backscattering coefficient is a decreasing function of the primary energy with the exception of gold, which presents an increasing trend as the energy increases in the range 1000–2000 eV. It is worth noting that the issue of the behavior of the backscattering coefficient at low primary energy is quite controversial [25, 26].

8.2 Electrons Backscattered from One Layer Deposited on Semi-infinite Substrates It is well known that over-layer films affect the electron backscattering coefficient of bulk targets. The experimental data available in the literature for the backscattering coefficient are rather sparse and, sometimes, difficulties arise in their interpretation due to the lack of knowledge of the thickness, uniformity, and nature of the surface layers. In particular, a quantitative treatment of the effect of surface films deposited on bulk targets and a systematic comparison with experimental data are not available. The main ingredient of the present approach is the evaluation of the backscattered electron emission coefficient—that results from the interplay between average atomic number and interaction volume—as compared to the actual thickness of the overlayer [27–31].

8.2.1 Carbon Overlayers Let’s start with the study of the low energy backscattering coefficient for the special case of layers of carbon deposited on aluminum [27]. Carbon films are deposited on various substrates (polymers, polyester fabrics, polyester yarns, metal alloys) both for experimental and technological motivations.

8.2 Electrons Backscattered from One Layer Deposited on Semi-infinite Substrates

125

Backscattering coefficient

0.3

0.2

0.1

0.0 0

2000

4000

6000

8000

10000

E0 (eV)

Fig. 8.5 Triangles: Monte Carlo simulation of the electron backscattering coefficient η for a carbon film 400 Å thick as a function of the beam primary energy E 0 . Empty circles: Monte Carlo backscattering coefficient of pure C. Filled circles: Monte Carlo backscattering coefficient of pure Al. The stopping power was taken from Ashley [14] (dielectric theory). The elastic scattering cross-section was calculated using the relativistic partial wave expansion method (Mott theory) [16]

There are a lot of technological uses for carbon films, as carbon characteristics are very useful in many fields. Carbon films deposited on polymeric substrates can be used to replace the metallic coatings on plastic materials used for food packaging. Carbon films are also widely employed in medical devices. Biomedical investigators have demonstrated that permanent thin films of pure carbon show excellent biocompatibility: They are used, in particular, as a coating for the stainless steel stents to be implanted in coronary arteries. When investigating the behavior of the backscattering coefficient as a function of the primary energy for various film thicknesses, the appearance, for carbon film thicknesses exceeding ∼100 Å on aluminum, of minima can be observed. These features are presented in Fig. 8.5 for a carbon film 400 Å -thick and in Fig. 8.6 for a carbon film 800 Å -thick. The backscattering coefficient reaches a minimum and then increases up to the aluminum backscattering coefficient. It then follows the almost constant (slightly decreasing) trend typical of the backscattering coefficient of aluminum. An interesting characteristic of the minima is that their position shifts towards higher energies as the film thickness increases. This is quite reasonable because one expects that the backscattering coefficient of the system should approach the behavior of the backscattering coefficient of aluminum for very thin carbon films and should approach the energy dependence of the backscattering coefficient of carbon for thick carbon films. So, as the film thickness increases, the positions of the minima shift towards higher energies while the peaks broaden.

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8 Backscattering Coefficient

Backscattering coefficient

0.3

0.2

0.1

0.0 0

2000

4000

6000

8000

10000

E0 (eV)

Fig. 8.6 Triangles: Monte Carlo simulation of the electron backscattering coefficient η for a carbon film 800 Å thick as a function of the beam primary energy E 0 . Empty circles: Monte Carlo backscattering coefficient of pure C. Filled circles: Monte Carlo backscattering coefficient of pure Al. The stopping power was taken from Ashley [14] (dielectric theory). The elastic scattering cross-section was calculated using the relativistic partial wave expansion method (Mott theory) [16]

The linear best fit of the behavior of the Monte Carlo simulated position of the minimum E min (in eV) of the backscattering curve as a function of the film thickness t (in Å)—for C thin films deposited on an Al substrates in the range 100 Å-1000 Å–is given by E min = m t + q, where m = (2.9 ± 0.1) eV/Å and q = (900 ± 90) eV [28].

8.2.2 Gold Overlayers The behavior described above has been observed both numerically [27–31] and experimentally [30, 31] also for other materials. In all the cases, the backscattering coefficient of the system ranges from the value of the backscattering coefficient of the over-layer (for very low primary energy) to the value of the backscattering coefficient of the substrate (for very high primary energy). In the case of gold deposited on silicon, the backscattering coefficient reaches a maximum and then decreases to the silicon backscattering coefficient. In general, the backscattering coefficient of any system should resemble the behavior of the backscattering coefficient of the substrate for very thin films and approach the behavior of the backscattering coefficient of the material constituting the over-layer for thick films. So, as the film thickness increases,

Normalized backscattering coefficient

8.2 Electrons Backscattered from One Layer Deposited on Semi-infinite Substrates

127

1.0

0.5

0

5000

10000

15000

20000

25000

30000

E0 (eV)

Fig. 8.7 Comparison between normalized experimental and present Monte Carlo backscattering coefficient as a function of the primary electron energy of an Au thin film deposited on a Si substrate [31]. Empty symbols: Experiment. Filled symbols: Monte Carlo. The Au over-layer nominal thickness is 250 Å. Stopping power was calculated by using the Kanaya and Okayama semi-empirical formula [32]. Layer deposition: Courtesy of Michele Crivellari. Experimental data: Courtesy of Nicola Bazzanella and Antonio Miotello

the position of the maximum (or minimum, depending on the constituting materials of over-layer and substrate) shifts towards higher energies while the peaks broaden [27–31]. Figures 8.7 and 8.8 display the data points for the experimental backscattering coefficient and the relevant Monte Carlo results for two samples of gold layers deposited on silicon [31]. The nominal thickness of the gold films were, respectively, 250 and 500 Å. Data were normalized by dividing the curves by their respective maxima. The experimental and the Monte Carlo approaches provide similar results. Similarly to the case of carbon on aluminum, Monte Carlo simulations predict that the energy position of the maximum, E max , linearly depends on the gold overlayer thickness. The linear best fit of E max as a function of the Au film thickness for Au/Si systems is presented in Fig. 8.9 demonstrating that the Monte Carlo method makes it possible to evaluate the overlayer film thickness with nearly 20% uncertainty (estimated from the statistical fluctuations in the energy maximum) [31]. In view of the non-destructiveness, the proposed approach is definitely adding new potentiality to SEM-based experimental methods.

8 Backscattering Coefficient

Normalized backscattering coefficient

128

1.0

0.5

0

5000

10000

15000

20000

25000

30000

E0 (eV)

Fig. 8.8 Comparison between normalized experimental and present Monte Carlo backscattering coefficient as a function of the primary electron energy of an Au thin film deposited on a Si substrate [31]. Empty symbols: Experiment. Filled symbols: Monte Carlo. The Au over-layer nominal thickness is 500 Å. Stopping power was calculated by using the Kanaya and Okayama semi-empirical formula [32]. Layer deposition: Courtesy of Michele Crivellari. Experimental data: Courtesy of Nicola Bazzanella and Antonio Miotello 20000

Emax (eV)

15000

10000

5000

0 0

500

1000

1500

2000

t( )

Fig. 8.9 Linear best fit of Monte Carlo simulated E max (in eV) as a function of the film thickness t (in Å) for Au thin films deposited on a Si substrates in the range 250 Å-2000 Å [31]. E max = m t + q, where m = 5.8 eV/Å (standard error = 0.4 eV/Å) and q = 3456 eV (standard error = 373 eV) [31]

8.3 Electrons Backscattered from Two Layers Deposited on Semi-infinite Substrates

129

8.3 Electrons Backscattered from Two Layers Deposited on Semi-infinite Substrates We are now interested in the calculation of the backscattering coefficient from two layers of different materials and thicknesses deposited on semi-infinite substrates. In particular, we will examine the backscattering from Cu/Au/Si and C/Au/Si systems. In Fig. 8.10, Monte Carlo electron backscattering coefficients of Cu/Au/Si samples are represented. The Monte Carlo simulation code considers the Si substrate as a semi-infinite bulk, while the thickness of the intermediate Au layer is set at 500 Å. The behavior of η as a function of the primary energy, in the 1000–25000 eV range, is represented for different values of the Cu first layer thickness, in the 250–1000 Å range. Stopping power is calculated using the dielectric response theory. Figure 8.11 shows the same quantities, obtained with the same conditions and calculated with the Monte Carlo code based on the Kanaya and Okayama semiempirical formula. The general trends obtained with the two codes are in good qualitative agreement: Both codes predict that the general structure of the curves presents a minimum and a maximum. Furthermore both the minimum and the maximum shift towards higher primary energies as the Cu first layer thickness increases. This behavior is typical of the particular combination of the selected materials and of their thicknesses. In order to further investigate and better understand the effects of layer thickness, in Figs. 8.12 and 8.13 the Monte Carlo backscattering coefficients, obtained with the

0.45

Backscattering coefficient

0.40

0.35

0.30

0.25

250 Cu 500 Cu 750 Cu 1000 Cu

0.20

0.15 0

5000

10000

15000

20000

25000

E0 (eV)

Fig. 8.10 Present Monte Carlo simulation of electron backscattering coefficient η of Cu/Au/Si samples. The Si substrate is semi-infinite, while the thickness of the intermediate Au layer is 500 Å. The behavior of η as a function of the primary energy is represented for different values of the Cu first layer thickness. Stopping power is calculated using the dielectric response theory

130

8 Backscattering Coefficient 0.45

Backscattering coefficient

0.40

0.35

0.30

0.25

250 Cu 500 Cu 750 Cu 1000 Cu

0.20

0.15 0

5000

10000

15000

20000

25000

E0 (eV)

Fig. 8.11 Present Monte Carlo simulation of electron backscattering coefficient η of Cu/Au/Si samples. The Si substrate is semi-infinite, while the thickness of the intermediate Au layer is 500 Å. The behavior of η as a function of the primary energy is represented for different values of the Cu first layer thickness. Stopping power is calculated using the Kanaya and Okayama semi-empirical formula 0.45

Backscattering coefficient

0.40

0.35

0.30

0.25

250 Au 500 Au 750 Au 1000 Au

0.20

0.15 0

5000

10000

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E0 (eV)

Fig. 8.12 Present Monte Carlo simulation of electron backscattering coefficient η of Cu/Au/Si samples. The Si substrate is semi-infinite, while the thickness of the first Cu layer is 500 Å. The behavior of η as a function of the primary energy is represented for different values of the Au intermediate layer thickness. Stopping power is calculated using the dielectric response theory

8.3 Electrons Backscattered from Two Layers Deposited on Semi-infinite Substrates

131

0.45

Backscattering coefficient

0.40

0.35

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0.25

250 Au 500 Au 750 Au 1000 Au

0.20

0.15 0

5000

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E0 (eV)

Fig. 8.13 Present Monte Carlo simulation of electron backscattering coefficient η of Cu/Au/Si samples. The Si substrate is semi-infinite, while the thickness of the first Cu layer is 500 Å. The behavior of η as a function of the primary energy is represented for different values of the Au intermediate layer thickness. Stopping power is calculated using the Kanaya and Okayama semiempirical formula

dielectric response and the semi-empirical approach, respectively, have been represented for the case in which the thickness of the first Cu layer is fixed (500 Å) while the intermediate Au film thickness ranges in the 250–1000 Å interval. Also in this case, the general behaviors obtained with the two approaches are in good qualitative agreement. The characteristic features present a trend different with respect to the previous one: While the position of the maximum shifts toward higher primary energies as the intermediate film thickness increases, the position of the minimum remains practically unchanged. In order to study the agreement between the two codes, Figs. 8.14 , 8.15 and 8.16 compare the calculation of the backscattering coefficients for various combinations of materials and thicknesses, obtained using the two Monte Carlo programs. The codes give practically indistinguishable results for the cases corresponding to the C/Au/Si combination, while some difference can be observed, for the lowest energies, in the case of the Cu/Au/Si combinations.

132

8 Backscattering Coefficient 0.40

Backscattering coefficient

0.35

0.30

0.25

0.20

0.15

0.10

0.05 0

5000

10000

15000

20000

25000

E0 (eV)

Fig. 8.14 Present Monte Carlo simulation of electron backscattering coefficient η of C/Au/Si samples. The Si substrate is semi-infinite, the thickness of the first C layer is 500 Å, and that of the intermediate Au layer is 250 Å. Backscattering coefficients η obtained using the dielectric response theory (filled symbols) and the semi-empirical Kanaya and Okayama approach (empty symbols), respectively, for calculating the stopping power, are compared 0.40

Backscattering coefficient

0.35

0.30

0.25

0.20

0.15 0

5000

10000

15000

20000

25000

E0 (eV)

Fig. 8.15 Present Monte Carlo simulation of electron backscattering coefficient η of Al/Au/Si samples. The Si substrate is semi-infinite, the thickness of the first Al layer is 500 Å, and that of the intermediate Au layer is 500 Å. Backscattering coefficients η obtained using the dielectric response theory (filled symbols) and the semi-empirical Kanaya and Okayama approach (empty symbols), respectively, for calculating the stopping power, are compared

8.4 A Comparative Study of Electron and Positron Backscattering …

133

0.45

Backscattering coefficient

0.40

0.35

0.30

0.25

0.20

0.15 0

5000

10000

15000

20000

25000

E0 (eV)

Fig. 8.16 Present Monte Carlo simulation of electron backscattering coefficient η of Cu/Au/Si samples. The Si substrate is semi-infinite, the thickness of the first Cu layer is 250 Å, and that of the intermediate Au layer is 500 Å. Backscattering coefficients η obtained using the dielectric response theory (filled symbols) and the semi-empirical Kanaya and Okayama approach (empty symbols), respectively, for calculating the stopping power, are compared

8.4 A Comparative Study of Electron and Positron Backscattering Coefficients and Depth Distributions To conclude this chapter, we will compare the Monte Carlo simulations of the backscattering coefficients and the depth distributions of electrons and positrons. Just to provide an example, we will consider the case of penetration of electrons and positrons in silicon dioxide. The presented results were obtained using the Ashley theory for calculating the stopping power and the Mott cross-section for the computation of the differential elastic scattering cross-section [33]. The differences in the inelastic and elastic scattering cross-sections of low energy electrons and positrons explain the results presented in Fig. 8.17. The depth profiles P(z) of electrons and positrons are different even for the highest energy examined (10 keV), for each particle reduces its energy during its travel in the solid, reaching the low values corresponding to significant differences in the cross-sections and stopping powers of electrons and positrons. Indicating with R the maximum penetration range, for any given energy E 0 , R can be easily determined by the curves in Fig. 8.17. From the presented depth profiles it is clear that the maximum penetration range in silicon dioxide is approximately the same for electrons and positrons, in the examined primary energy range.

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8 Backscattering Coefficient

0.0010

P(z) (1/ )

0.0008

0.0006

0.0004

0.0002

0.0000 0

5000

10000

z( )

Fig. 8.17 Monte Carlo simulation of the depth profiles P(z) of electrons (empty symbols) and positrons (filled symbols) in SiO2 as a function of the depth inside the solid measured from the surface, z. E 0 is the primary energies of the particles. 3 keV (squares), 5 keV (circles) and 10 keV (triangles)

For each primary energy E 0 , the integration of the function P(z) from z = 0 to z = R gives the absorption coefficient 1 − η, where we indicated with η the backscattering coefficient. As the primary energy increases, the differences in the depth distributions of electrons and positrons grow smaller. While the maximum ranges of electrons and positrons are similar, the backscattering coefficients present very different trends (see Fig. 8.18). In the primary energy range from 1 keV to 10 keV, the positron backscattering coefficient does not depend on the primary energy and is always smaller than the electron backscattering coefficient. Instead, the electron backscattering coefficient is a decreasing function of the primary energy. Figure 8.19 represents the rescaled depth profiles z m P(z) for 3–10 keV electrons and positrons in SiO2 obtained with the Monte Carlo method. Aers [34] proposed the following equation to fit the rescaled depth profiles:  z m P(z) = N

z/z m C

l

    z/zm m exp − C

(8.31)

where z m is the mean range defined as R z m = 0 R 0

z P(z)dz P(z)dz

R =

0

z P(z)dz . 1−η

(8.32)

8.4 A Comparative Study of Electron and Positron Backscattering …

135

Backscattering coefficient

0.25

0.20

0.15

0.10

0.05

0.00 0

2000

4000

6000

8000

10000

E0 (eV)

Fig. 8.18 Monte Carlo simulation of the backscattering coefficient η of electrons (empty symbols) and positrons (filled symbols) in SiO2 as a function of the primary energy E 0 0.8 0.7 0.6

zm P(z)

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

z/zm

Fig. 8.19 Monte Carlo calculation of the rescaled depth profiles z m P(z) of electrons (empty symbols) and positrons (filled symbols) in SiO2 as a function of z/z m , i.e., the depth z inside the solid measured from the surface divided by the mean range z m . The primary energies E 0 of the particles are: 3 keV (squares), 5 keV (circles) and 10 keV (triangles). The solid line (electrons) and the dashed line (positrons) represent the depth profiles calculated by the use of the Aers equation (8.31) [34]. Electrons: N = 0.85 ± 0.01, C = 1.696 ± 0.005, l = 0.53 ± 0.01, m = 6.2 ± 0.1. Positrons: N = 1.00 ± 0.01, C = 1.642 ± 0.004, l = 0.69 ± 0.01, m = 9.9 ± 0.3

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8 Backscattering Coefficient

The values of the fitting parameters N , C, l and m found out with the present Monte Carlo calculation of 3–10 keV electrons and positrons in SiO2 , represented as lines in the Fig. 8.19 (solid line: Electrons, dashed line: Positrons) are the following: For electrons: N = 0.85 ± 0.01, C = 1.696 ± 0.005, l = 0.53 ± 0.01, m = 6.2 ± 0.1. For positrons: N = 1.00 ± 0.01, C = 1.642 ± 0.004, l = 0.69 ± 0.01, m = 9.9 ± 0.3.

8.5 Summary In this chapter, the Monte Carlo method was used for evaluating the backscattering coefficient of electrons (and positrons) impinging upon bulks and over-layers. In particular, for the case of surface films, it was calculated as a function of the thicknesses of the layers, their nature, and the electron primary energy. The code used in this chapter utilizes the Mott cross-section for elastic scattering calculation and the continuous-slowing-down approximation for energy loss simulation. For the calculation of the stopping power, the Ritchie dielectric response theory [15] and the analytical semi-empirical formula proposed by Kanaya and Okayama [32] were used. Electron backscattering coefficients from several different combinations of layers and substrates were simulated. The main features of the backscattering coefficient as a function of the electron primary energy, which are represented by minima and maxima whose positions in energy depend on the particular combination of materials and thicknesses, are reproduced in similar ways using the two different stopping powers. The last section of this chapter presented a comparative study of electron and positron depth distributions and backscattering coefficients.

References 1. M. Dapor, Phys. Rev. B 46, 618 (1992) 2. M. Dapor, Electron-Beam Interactions with Solids: Application of the Monte Carlo Method to Electron Scattering Problems (Springer, Berlin, 2003) 3. R. Cimino, I.R. Collins, M.A. Furman, M. Pivi, F. Ruggiero, G. Rumolo, F. Zimmermann, Phys. Rev. Lett. 93, 014801 (2004) 4. M.A. Furman, V.H. Chaplin, Phys. Rev. Spec. Top. Accel. Beams 9, 034403 (2006) 5. T. Everhart, J. Appl. Phys. 31, 1483 (1960) 6. J.J. Thomson, Conduction of Electricity Through Gases, 2nd edn. (Cambridge University Press, Cambridge, England, 1906) 7. G.D. Archard, J. Appl. Phys. 32, 1505 (1961) 8. G. Neubert, S. Rogaschewski, Phys. Stat. Sol. (a) 59, 35 (1980) 9. S.G. Tomlin, Proc. Phys. Soc. London 82, 456 (1963) 10. H. Niedrig, J. Appl. Phys. 53, R15 (1982) 11. M. Vicanek, H.M. Urbassek, Phys. Rev. B 44, 7234 (1991) 12. P.G. Coleman, L. Albrecht, K.O. Jensen, A.B. Walker, J. Phys. Condens. Matter 4, 10311 (1992) 13. M. Dapor, J. Appl. Phys. 79, 8406 (1996)

References 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

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J.C. Ashley, J. Electron Spectrosc. Relat. Phenom. 46, 199 (1988) R.H. Ritchie, Phys. Rev. 106, 874 (1957) N.F. Mott, Proc. R. Soc. London Ser. 124, 425 (1929) H.E. Bishop, Proc. 4ème Congrès International d’Optique des Rayons X et de Microanalyse (1967), pp. 153–158 H.J. Hunger, L.G. Küchler, Phys. Status Solidi A 56, K45 (1979) I.M. Bronstein, B.S. Fraiman, Vtorichnaya Elektronnaya Emissiya (Nauka, Moskva, 1969) L. Reimer, C. Tolkamp, Scanning 3, 35 (1980) T. Koshikawa, R. Shimizu, J. Phys. D. Appl. Phys. 6, 1369 (1973) D.C. Joy, (2008), http://web.utk.edu/~srcutk/htm/interact.htm S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interface Anal. 37, 978 (2005) R. Böngeler, U. Golla, M. Kussens, R. Reimer, B. Schendler, R. Senkel, M. Spranck, Scanning 15, 1 (1993) J. Ch. Kuhr, H.J. Fitting, Phys. Status Solidi A 172, 433 (1999) M.M. El Gomati, C.G. Walker, A.M.D. Assa’d, M. Zadraˇzil, Scanning 30, 2 (2008) M. Dapor, J. Appl. Phys. 95, 718 (2004) M. Dapor, Surf. Interface Anal. 38, 1198 (2006) M. Dapor, Surf. Interface Anal. 40, 714 (2008) M. Dapor, N. Bazzanella, L. Toniutti, A. Miotello, S. Gialanella, Nucl. Instrum. Methods Phys. Res. B 269, 1672 (2011) M. Dapor, N. Bazzanella, L. Toniutti, A. Miotello, M. Crivellari, S. Gialanella, Surf. Interface Anal. 45, 677 (2013) K. Kanaya, S. Okayama, J. Phys. D. Appl. Phys. 5, 43 (1972) M. Dapor, J. Electron Spectrosc. Relat. Phenom. 151, 182 (2006) G.C. Aers, J. Appl. Phys. 76, 1622 (1994)

Chapter 9

Secondary Electron Yield

Electron beams impinging upon solid targets induce the emission of secondary electrons. These are the electrons extracted from the atoms in the solid by the electrons of the incident beam or the other secondary electrons traveling in the solid. Some secondary electrons, after a number of elastic and inelastic interactions with the atoms in the solid, reach the solid surface satisfying the conditions to emerge from it. As we already know, the spectrum of the secondary electrons is contaminated by a contribution of the backscattered primary electrons. As this contamination can be safely neglected in the great majority of the practical situations that investigators encounter in laboratory experiments, this effect is usually ignored, at least as a first approximation. The process of secondary-electron emission can be divided into two phenomena. The first one concerns the generation of secondary electrons as a consequence of the interaction between the incident electron beam and electrons bound in the solid. The second one is represented by the cascade, where the secondary electrons diffusing in the solid extract new secondary electrons generating a shower of electrons. As each secondary electron loses energy while traveling in the solid, the whole process proceeds until the energy of the secondary electron is no more sufficient to extract further secondary electrons or until it reaches the surface with enough energy to emerge. The number of the emitted secondary electrons divided by the number of the incident electrons is called secondary electron emission yield. The secondary electron emission yield is measured as the integral of the electron energy distribution over the energy range from 0 to 50 eV.1 Secondary electron emission plays a fundamental role in scanning electron microscope imaging. 1

Please note that the total electron emission yield, i.e., the sum of the secondary electron emission yield and of the backscattering coefficient, is measured as the integral of the electron energy distribution over the whole energy range from 0 to the initial value of the energy of the electron beam, E0 . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1_9

139

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9 Secondary Electron Yield

9.1 Secondary Electron Emission In this chapter, secondary electron emission from Au, polymethyl methacrylate (PMMA), and aluminum oxide (Al2 O3 ) will be modeled quantitatively using the Monte Carlo code. This chapter is aimed at comparing, with the available experimental data, the results of the computational approaches described in the chapter devoted to the main features of the transport Monte Carlo modeling, i.e., the scheme based on the energy-straggling (ES) strategy and the method based on the continuousslowing-down approximation (CSDA). In this way, it will be possible to understand the validity limits of the methods and to face the question of the CPU time costs in evaluating which approach is more convenient in different circumstances. We will learn, on the one hand, that the use of the simple continuous-slowing-down approximation for the calculation of the secondary electron yield allows getting an agreement with the experiment similar to the one we can obtain with the more accurate (but CPU time-consuming) energy-straggling strategy. If, on the other hand, the energy distribution of the secondary electrons is required, the energy-straggling strategy becomes the only choice. Secondary-electron emission involves very complex phenomena and a numerical treatment requires detailed knowledge of the main interactions of the electrons with the solid target. The most important processes that occur in the target are related to the production of individual electron transitions from the valence to the conduction band, to plasmon generation, and to the elastic collisions with the screened potentials of the ions in the solids. If its energy is high enough, the electron can be subject to inelastic collisions with inner-shell electrons. Secondary electrons of very low energy also interact losing (and gaining) energy with phonons. In insulating materials, they can be trapped in the solid (polaronic effects). Each secondary electron can produce further secondary electrons during its travel inside the solid and, in order to obtain quantitative results, following the whole cascade is the best choice [1–3].

9.2 Monte Carlo Approaches to the Study of Secondary Electron Emission The Monte Carlo calculation of the secondary electron emission yield can be performed either by taking into account all the details of the many mechanisms of the electron energy loss [1, 3–5] or by assuming a continuous-slowing-down approximation [6–8]. The use of the first approach has a stronger physical basis but, due to the detailed description of all the collisions in the secondary electron cascade, it is a very time-consuming scheme. The continuous-slowing-down approximation represents instead an approach that saves a lot of CPU time although its physical foundation is more questionable.

9.3 Specific MC Methodologies for SE Studies

141

We report about the MC simulations of the secondary electron emission from PMMA and Al2 O3 obtained with the two approaches. These simulations demonstrate that, if we just calculate the yield as a function of the primary energy, the two Monte Carlo schemes give equivalent results for any practical purposes. Summarizing what we have discussed so far, the secondary electron yields calculated by means of the two approaches are very close. What is more, the two MC schemes give results in satisfactory agreement with the experiment. This means that, for the calculation of the secondary electron yield, the continuous-slowing-down approximation should be preferred, being much faster (more than ten times) than the more detailed scheme. If, on the other hand, secondary electron energy distributions are required, the continuous-slowing-down approximation cannot be used—for it is not able to describe in a realistic way all the energy loss processes—and the detailed scheme becomes mandatory, even if, as far as CPU is concerned, it is much more time consuming [9, 10].

9.3 Specific MC Methodologies for SE Studies 9.3.1 Continuous-Slowing-Down Approximation (CSDA Scheme) In the case of CSDA, as we know, the step length is calculated according to the equation Δs = −λel ln μ—where μ is a random number uniformly distributed in the range [0, 1]—while the energy loss ΔE along the segment of trajectory Δs is approximated by the equation ΔE = (d E/ds) Δs. Compared to the description we gave in the chapter devoted to the Monte Carlo method, the secondary electron yield calculation using CSDA requires a few more pieces of information. The secondary electron yield is calculated, according to Dionne [11], Lin and Joy [6], Yasuda et al. [7], and Walker et al. [8] assuming that (i) the number dn of secondary electrons generated along each step length ds, corresponding to the energy loss d E, is given by dE 1 dE ds = (9.1) dn = s ds s where s is the effective energy necessary to generate a single secondary electron and (ii) the probability Ps (z) that a secondary electron generated at depth z will reach the surface and emerge from it follows the exponential decay law Ps (z) = e−z/λs ,

(9.2)

where λs is the effective escape depth. Thus the secondary electron emission yield is given by

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9 Secondary Electron Yield

 δ =

Ps (z) dn =

1 s



e−z/λs d E .

(9.3)

9.3.2 Energy-Straggling (ES Scheme) Since we described the details of the energy straggling strategy in a previous chapter, here we will discuss the features of the scheme specific for the study of secondary electron emission only. For further information about the adopted simulation methods, see also [1, 5, 9, 10]. If μ is a random number uniformly distributed in the interval [0, 1], every step length Δs of each electron traveling in the solid is calculated by adopting the exponential density of probability, so that Δs = −λ ln μ. In this equation, λ is the electron mean free path including all the scattering mechanisms involved. Its reciprocal, i.e., the so-called inverse mean free path, can be expressed as the sum total of all the inverse mean free paths of the interactions of the electrons with the target: In particular, it is necessary to take into account the inverse mean free path due to the elastic interactions among the incident electrons and the screened atomic nuclei, λ−1 el , and the one due to the inelastic interactions among the incident electrons and the atomic −1 −1 = λ−1 ones, λ−1 inel , so that λ el + λinel . This is the case of metals [12–14]. When the target is an insulator, also the inverse mean free path due to the electronphonon interactions, λ−1 phonon (which becomes important when the electron energy becomes smaller than approximately three times the gap), and the one due to the polaronic effects, λ−1 pol , have to be taken into account. The problem related to the excessively high values of the Mott elastic scattering cross-section observed at very low energy was discussed by Ganachaud and Mokrani [1] who suggested that the elastic scattering cross-section should be multiplied by a cut-off function (see (6.14)) such that, when the electron energy is very low (so that electron-phonon scattering cross-section becomes important), the elastic scattering cross-section will become negligible. For higher energies, on the other hand, when electron-phonon scattering becomes negligible, the cut-off function approaches 1 in order to recover the behavior of the Mott cross-section.2 As already noted, if the collision is inelastic, the energy loss is calculated according to the specific inelastic scattering cross-section while if the collision is elastic, the scattering angle is calculated according to the Mott cross-section. It should be noted that the electron deflections also depend on electron-electron and electron-phonon interactions. The Monte Carlo scheme has to take into account the entire cascade of secondary electrons [2, 4, 5, 9, 10, 12–17]. The initial position of a secondary electron is assumed to be at the site of the electron-atom inelastic collision. In the calculations presented in this chapter, the polar and azimuthal angles of secondary electrons are calculated, following the Shimizu and Ding Ze-Jun approach [18], assuming a 2

Actually, Ganachaud and Mokrani treated the elastic scattering cross-section using the nonrelativistic partial wave expansion method [1].

9.4 Secondary Electron Yield of Metals

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random direction of the secondary electrons. This hypothesis of the random direction of the generated secondary electrons implies that slow secondary electrons should be generated with spherical symmetry [18]. As this assumption violates momentum conservation rules—within the classical binary-collision model—a study will be provided in the next chapter that compares with experimental data the results obtained adopting spherical symmetry and those obtained applying momentum conservation within the classical binary-collision model. This study demonstrates that the determination of the energy distribution of the secondary electrons emitted by solid targets as well as the secondary emission yield are in better agreement with the experiment assuming that slow secondary electrons are generated with spherical symmetry [4].

9.4 Secondary Electron Yield of Metals Secondary electron yield plays an important role in a variety of applications, such as materials characterization techniques (scanning electron microscopy, electron spectroscopies), and in affecting the performance of many of electron devices. The Monte Carlo schemes described above account for the main interactions occurring to the secondary electrons along their travel in materials (insulating targets, semiconductors, and metals). The study of the energy dependence of the secondary electron yield of metals is particularly important today for particle accelerators. In many practical situations, the emission of secondary electrons must be suppressed. Indeed, when the current of reemitted electrons grows close to the vacuum tube walls, detrimental effects are observed on the machine stability (beam loss) [12]. Another important motivation to explore the secondary electrons yield of metals using the electron transport via MC method is related to the realization of drag-free spacecraft (with applications ranging from space exploration to missions aiming to test the laws of fundamental physics) [14]. In Fig. 9.1, we present Monte Carlo simulations (using the energy straggling strategy) of the total yield σ (secondary electron yield δ + backscattering coefficient η) of electrons impinging on Au as a function of the energy E 0 (in the range from 25 eV to 1000 eV). The energy loss function was calculated by utilizing the ab-initio method described in Sect. 4.3.6 (see Fig. 4.37). Elastic scattering was calculated using Mott theory. The experimental data by Gonzales et al. [19] were also presented for comparison. As the primary energy increases, both the secondary and the total electron emission yields increase until a maximum is reached. Then, both yields decrease as the primary energy increases.

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σ=δ+η

2

1

0

0

200

400

600

800

1000

E0 (eV) Fig. 9.1 Comparison between present Monte Carlo calculations and experimental data of Au total electron yield σ (secondary electron yield δ + backscattering coefficient η) as a function of the primary electron energy E 0 . Solid line represents Monte Carlo calculations based on the energystraggling strategy and obtained with work function χ = 4.7 eV. The Monte Carlo data were obtained integrating the curves of energy distribution including all the electrons emerging with energy from 0 to the E 0 . Experimental data from [19] (circles)

9.5 Secondary Electron Yield: PMMA and Al2 O3 In the following subsections results obtained with the two schemes, i.e., the energy straggling and the continuous-slowing-down approximation methods, will be presented and a comparison with the available experimental data will be provided about secondary electron emission from PMMA and Al2 O3 .

9.5.1 Secondary Electron Emission Yield as a Function of the Energy As already observed, experimental evidence and MC simulations show that, as the primary energy increases, the secondary electron emission yield increases until a maximum is reached. Then the yield decreases as the primary energy increases. It is quite easy to provide a simple qualitative explanation of this behavior. At very low primary energy, few secondary electrons are generated and, as the primary energy increases, so does the number of secondary electrons emerging from the surface. The

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145

average depth at which the secondary electrons escaping from the surface are generated also increases as the primary energy increases. When the energy becomes higher than a threshold that depends on the target, the average depth of secondary electron generation becomes so deep that just a small amount of the generated secondary electrons are able to reach the surface satisfying the condition necessary to emerge from the sample and to be detected. We will show that both the MC schemes (ES and CSDA) confirm this behavior by providing consistent quantitative explanations of it.

9.5.2 Comparison Between ES Scheme and Experiment Even if the physical meaning of the parameters used by the ES MC code and appearing in the laws describing the interactions is clear—so that they are, at least in principle, measurable—in practice they can be determined only through an analysis of their influence on the simulated results and a comparison to the available experimental data. Through such an analysis, the values of the parameters for PMMA were determined and, in Figs. 9.2 and 9.3, we report the comparison with the available experimental data of the simulated results obtained using the detailed Monte Carlo scheme based on the energy-straggling strategy (ES scheme). We found out, for PMMA, the best fit with the following parameters values: χ = 4.68 eV, Wph = 0.1 eV, C = 0.01 Å−1 , and γ = 0.15 eV−1 . It should be noted that performing a similar analysis, Ganachaud and Mokrani found out, for amorphous Al2 O3 , the following values of the parameters: χ = 0.5 eV, Wph = 0.1 eV, C = 0.1 Å−1 , and γ = 0.25 eV−1 [1]. It should be noted as well that the yield strongly depends on all these parameters. While both the electron affinity, χ, and the electron energy loss due to phonon creation, Wph , are quantities that have been measured for many materials and whose values can be found in the scientific literature, less information is available concerning the two parameters C and γ (relative to polaronic effects).

9.5.3 Comparison Between CSDA Scheme and Experiment The Monte Carlo code based on the CSDA scheme, also depends on two parameters: The effective escape depth, λs , and the effective energy necessary to generate a single secondary electron, s . The comparison of the results of the CSDA code with the available Al2 O3 experimental data [24] is shown in Fig. 9.4. The values of the physical parameters used by the MC code based on the CSDA (i.e., the effective escape depth, λs , and the effective energy necessary to generate a single secondary electron, s ) for Al2 O3 —in reasonable agreement with other physics reference data [1, 6]—are λs = 15.0 Å and s = 6.0 eV.

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δ

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500

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E0 (eV) Fig. 9.2 Comparison between present Monte Carlo calculations and experimental data of polymethyl methacrylate (PMMA) secondary electron yield δ as a function of the primary electron energy. Squares represent Monte Carlo calculations based on the energy-straggling strategy and obtained with χ = 4.68 eV, Wph = 0.1 eV, C = 0.01 Å−1 , and γ = 0.15 eV−1 . The Monte Carlo data were obtained by integrating the curves of energy distribution including all the electrons emerging with energy from 0 to 50 eV. Experimental data from [7, 20] (circles)

The calculated value of the statistical distribution χ2s , considered to quantitatively evaluate the agreement between the CSDA Monte Carlo simulated data of Al2 O3 (obtained using the parameters given above, i.e., λs = 15.0 Å and s = 6.0 eV) and the examined experimental data is 0.905. The number ν of degrees of freedom is 11. The lower critical value of the χ2s distribution for ν = 11 is 3.053, and the probability p of exceeding this critical value is 0.99. As the calculated χ2s is significantly smaller than the critical one, this means that, in the hypothesis that the Monte Carlo data approximate the experimental ones (the so-called null hypothesis), there is a probability greater than 99% that the observed discrepancies are due to statistical fluctuations. Similar results were obtained also comparing the experimental results to the ES Monte Carlo simulated data presented above.

9.5.4 CPU Time The computation time needed for the ES code is much higher than the time of computation necessary for the CSDA code. For a typical simulation (1 keV electrons impinging upon PMMA), the CSDA scheme is more than ten times faster than the

9.5 Secondary Electron Yield: PMMA and Al2 O3

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3

σ=δ+η

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E0 (eV) Fig. 9.3 Comparison between present Monte Carlo calculations and experimental data of polymethyl methacrylate (PMMA) total electron yield σ (secondary electron yield δ + backscattering coefficient η) as a function of the primary electron energy. Squares represent Monte Carlo calculations based on the energy-straggling strategy and obtained with χ = 4.68 eV, Wph = 0.1 eV, C = 0.01 Å−1 , and γ = 0.15 eV−1 . The Monte Carlo data were obtained integrating the curves of energy distribution including all the electrons emerging with energy from 0 to the E 0 . Experimental data from [21] (circles) and from [22, 23] (triangles)

ES one. The reason for this great difference in CPU time is related to the secondary electron cascade. The ES MC strategy requires that the entire cascade is followed. The CSDA MC code, instead, is able to establish the number of secondary electrons produced at each step of every primary electron trajectory. Note that a further advantage of the CSDA MC strategy is the reduced number of parameters (only two against the four quantities required by the energy-straggling strategy, in the case of insulating materials). Of course, the ES MC code is based on a stronger physical background and allows one to calculate other important properties such as the secondary electron energy distribution and the lateral, angular, and depth distributions which are not accessible using the CSDA approximation (see next chapter). The advantage of using the CSDA code, in practical terms, with respect to just performing an empirical fit to the experimental data is related, of course, to other predictive capabilities of the Monte Carlo simulations. If it is true that, at the moment, the CSDA model requires a fit to existing data or to the results of the detailed simulation to calculate its free parameters, one should take into account that if the parameters were known for a large number of materials, they could be used for investigating many problems, different from the one we have used to find out the values of the

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δ

4 3 2 1 0

0

1000

2000

3000

E0 (eV) Fig. 9.4 Comparison between present Monte Carlo calculations and experimental data of Al2 O3 secondary electron yield as a function of the primary electron energy. Squares represent Monte Carlo calculations based on the continuous-slowing-down approximation and obtained with λs = 15.0 Å and s = 6.0 eV. Circles are the Dawson experimental data [24]

parameters, such as, for example, the dependence of the secondary electron yield on the angle of incidence for any given primary energy, or the secondary electron emission from unsupported thin films (on both sides of the film), or the secondary electron emission yield from thin films deposited on the bulk of different materials, and so on. Of course, all these possibilities are not accessible to a simple empirical fit to the experimental data. In conclusion, the very fast CSDA MC code can be used for the calculation of the secondary electron yield. If secondary electron energy, lateral, and depth distributions, or detailed descriptions of the physics involved in the process are required, the ES MC strategy should be preferred, even if it requires much longer CPU time.

9.6 Summary In this chapter, the transport Monte Carlo method has been applied to the evaluation of the secondary and total electron emission from metals (Au), polymers (PMMA), and insulating materials (Al2 O3 ). The code has been validated by comparing the secondary electron yield obtained by Monte Carlo simulation with the available experimental data. In particular, an analysis of the results of two different approaches

References

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(energy-straggling scheme and continuous-slowing-down approximation scheme) for the determination of the yield of the secondary electrons was presented. We have demonstrated that the two approaches give similar results concerning the secondary electron emission yield as a function of the electron primary energy. Furthermore the simulated results are in good agreement with the available experimental data. The evaluation of the secondary electron energy distribution requires, on the contrary, the energy-straggling scheme in order to accurately take into account all the details of the energy loss mechanisms (see next chapter).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

J.P. Ganachaud, A. Mokrani, Surf. Sci. 334, 329 (1995) M. Dapor, B.J. Inkson, C. Rodenburg, J.M. Rodenburg, Europhys. Lett. 82, 30006 (2008) J.Ch. Kuhr, H.J. Fitting, J. Electron Spectrosc. Relat. Phenom. 105, 257 (1999) M. Dapor, Nucl. Instrum. Methods Phys. Res. B 267, 3055 (2009) M. Dapor, M. Ciappa, W. Fichtner, J. Micro/Nanolith, MEMS MOEMS 9, 023001 (2010) Y. Lin, D.C. Joy, Surf. Interface Anal. 37, 895 (2005) M. Yasuda, K. Morimoto, Y. Kainuma, H. Kawata, Y. Hirai, Jpn. J. Appl. Phys. 47, 4890 (2008) C.G. Walker, M.M. El-Gomati, A.M.D. Assa’d, M. Zadrazil, Scanning 30, 365 (2008) M. Dapor, Nucl. Instrum. Methods Phys. Res. B 269, 1668 (2011) M. Dapor, Progress Nucl. Sci. Technol. 2, 762 (2011) G.F. Dionne, J. Appl. Phys. 44, 5361 (1973) M. Azzolini, M. Angelucci, R. Cimino, L. Larciprete, N.M. Pugno, S. Taioli, M. Dapor, J. Phys.: Condens. Matter 31, 055901 (2019) M. Azzolini, OYu. Ridzel, P.S. Kaplyac, V. Afanas’ev, N.M. Pugno, S. Taioli, M. Dapor, Comput. Mater. Sci. 173, 109420 (2020) S. Taioli, M. Dapor, F. Dimiccoli, M. Fabi, V. Ferroni, C. Grimani, M. Villani, W.J. Weber, Class. Quantum Grav. 40, 075001 (2023) C. Rodenburg, M.A.E. Jepson, E.G.T. Bosch, M. Dapor, Ultramicroscopy 110, 1185 (2010) M. Ciappa, A. Koschik, M. Dapor, W. Fichtner, Microelectron. Reliab. 50, 1407 (2010) A. Koschik, M. Ciappa, S. Holzer, M. Dapor, W. Fichtner, Proc. SPIE 7729, 77290X–1 (2010) R. Shimizu, D. Ze-Jun, Rep. Prog. Phys. 55, 487 (1992) L.A. Gonzalez, M. Angelucci, R. Larciprete, R. Cimino, AIP Adv. 7, 115203 (2017) L. Reimer, U. Golla, R. Bongeler, M. Kassens, B. Schindler, R. Senkel, Fiz. Tverd. Tela Adad. Nauk 1, 277 (1959) M. Boubaya, G. Blaise, Eur. Phys. J. Appl. Phys. 37, 79 (2007) L. Reimer, U. Golla, R. Bongeler, M. Kassens, B. Schindler, R. Senkel, Optik (Jena, Ger.) 92, 14 (1992) É.I. Rau, E.N. Evstaf’eva, M.V. Adrianov, Phys. Solid State 50, 599 (2008) P.H. Dawson, J. Appl. Phys. 37, 3644 (1966)

Chapter 10

Electron Energy Distributions

As we know, the study of the electronic and optical properties of matter is paramount for our comprehension of physical and chemical processes which occur in nanoclusters and solids [1]. Radiation damage, investigation of chemical composition, and electronic structure study represent a few examples of the role played by the electronmatter interaction mechanisms. Electron spectroscopy and electron microscopy are fundamental tools to examine how electrons interact with matter [2]. Electron spectroscopy includes a wide range of techniques. Low energy reflection electron energy loss spectroscopy and Auger electron spectroscopy, in particular, uses electron beams to analyze the surface of materials. Both reflection electron energy loss spectroscopy and Auger electron spectroscopy are applications of the scattering theory [3]. They are based on scattering processes in which the initial state consists of electrons impinging upon solid-state targets, and the final states are characterized by a few non-interacting fragments. The analysis of the energy distribution of the fragments constitutes the main feature of these spectroscopies, as it provides insight into the properties of the examined system. The spectrum, i.e., the plot of the intensity of the emitted electrons as a function of either the kinetic energy or the energy loss, represents a fingerprint of the investigated material [4–6].

10.1 Monte Carlo Simulation of the Spectrum The numerical results we are going to present were obtained with detailed modeling which takes into account all the described mechanisms of energy loss (ES scheme) and applies the Mott cross-section to deal with the elastic scattering events. Furthermore, the entire cascade of secondary electrons is followed. In order to illustrate the general features of a spectrum, the whole Monte Carlo spectrum representing the energy distribution of electrons emerging from a PMMA

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1_10

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a

Intensity (eV -1)

0.15

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Energy (eV) 3x10-3

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b 2x10-3

1x10-3

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Fig. 10.1 Monte Carlo calculation (ES scheme) of the spectrum representing the energy distribution of electrons emerging from a PMMA sample due to a 200 eV electron beam irradiation. Panel a: Entire spectrum including the secondary electron peak (at low energy) and the elastic peak at 200 eV. Panel b: Zoom of the spectrum including the plasmon peak and the elastic peak. The elastic peak, or zero loss peak, whose maximum is located at the energy of the primary beam, represents the electrons that suffered only elastic scattering collisions. The plasmon peak represents the electrons of the primary electron beam that emerge from the surface after having suffered a single inelastic collision with a plasmon. Multiple collisions with plasmons are also present in the spectrum. The secondary electrons energy distribution presents a pronounced peak in the very low energy region of the spectrum, typically below 50 eV [6]

sample due to a 200 eV electron beam irradiation is presented in Fig. 10.1 [6]. Similar spectra have been observed for metals as well [4, 5]. For the purposes of the current calculation, the target is considered semi-infinite. It should be noted that the case of thin films requires special attention, particularly when the samples are thinner than the mean free path. Many electrons of the primary electron beam can be backscattered, after having interacted with the atoms and electrons in the target. A fraction of them conserves the original kinetic energy, having suffered only elastic scattering collisions with the atoms of the target. We already know that these electrons constitute the so-called

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elastic peak, or zero-loss peak, whose maximum is located at the energy of the primary beam. In the simulation presented in Fig. 10.1 it is the narrow peak located at 200 eV. The plasmon peak in Fig. 10.1, located at ≈178 eV (≈22 eV from the elastic peak), collects the electrons of the primary electron beam that emerge from the surface after having suffered a single inelastic collision with a plasmon. Multiple collisions with plasmons are also present in the spectrum, but they are of very low intensity. For example, for the specific case of the PMMA spectrum, a peak corresponding to a double inelastic scattering with plasmons located at ≈156 eV (≈44 eV from the elastic peak) is scarcely visible in the panel b of Fig. 10.1. Electron-phonon energy losses are also present, but they are not visible in Fig. 10.1, for (i) their intensities are much lower than those of the elastic peak and (ii) they are very close to the much more intense elastic peak whose width is, on the other hand, rather wide (≈0.5 eV). Then they are not resolved. The spectrum also includes core-electron excitations and, due to the presence of doubly ionized atoms, Auger electron peaks. They also are not visible on this scale. Finally, as we already know, the secondary electrons produced by a cascade process are those electrons that have been extracted from the atoms by inelastic electronelectron collisions and are able to emerge from the target surface. Their Monte Carlo energy distribution presents a pronounced peak in the very low energy region of the spectrum, typically below 50 eV, and it is clearly visible in the simulated spectrum in Fig. 10.1.

10.2 Plasmon Losses and Electron Energy Loss Spectroscopy In order to briefly discuss the main features of the plasmon losses, a numerical simulation concerning graphite will be first presented. Use has been made of experimental data taken from the literature to calculate the dielectric function and, hence, the energy loss function [7–11]. Note that similar semi-empirical approaches to the calculation of the plasmon losses can be performed for other materials as well. In fact, experimental data concerning the energy loss function can be found in the literature for many materials. See, for example, [7, 8, 12] for the case of electron energy losses in SiO2 , also described in this section.

10.2.1 Plasmon Losses in Graphite Let us consider the graphite electron energy-loss spectrum. It includes the zero-loss peak, the π and π + σ plasmon loss peaks, due to inelastic scattering by outer shell electrons, and additional peaks at multiples of the π + σ plasmon energy, due to

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multiple scattering. π plasmon peak is at around 7 eV while the first, the second, and the third π + σ plasmon loss peaks are located around 27 eV, 54 eV, and 81 eV respectively. Graphite is a uniaxial anisotropic crystal with a layered structure. For such uniaxial anisotropic crystals, the dielectric function is a tensor with only two different diagonal elements, one perpendicular and the other parallel to the c-axis. The former is associated with the response for a momentum-transfer perpendicular to the c-axis (in-plane excitations), the latter with the response for a momentum transfer parallel to the c-axis (out-of-plane excitations). To describe the graphite inelastic scattering, we assume that a single inelastic collision is ruled either by the differential inelastic scattering cross-section corresponding to the parallel component of the dielectric tensor or by the one corresponding to the perpendicular component of the dielectric tensor. We then introduce a numerical parameter, f . According to [13], we used, for the present simulations, f = 0.4. During the simulation of electron transport in graphite, for each inelastic collision, a random number uniformly distributed in the range [0,1] is generated. If the random number is lower than the value of f , the collision is ruled by the differential inelastic scattering cross-section corresponding to the parallel component of the dielectric tensor, otherwise, it is ruled by the differential inelastic scattering cross-section corresponding to the perpendicular component of the dielectric tensor. In Fig. 10.2 a comparison of the Monte Carlo results with experimental data for 500 eV electrons incident on graphite is presented [13]. The Monte Carlo results we present here were obtained using the graphite dielectric functions (parallel and perpendicular to the c-axis) taken from experimental data. For energies lower than 40 eV we used the data presented in [9–11] while, for energies higher than 40 eV, we used the optical data by Henke et al. [7, 8].

10.2.2 Plasmon Losses in Silicon Dioxide Let us now consider the Monte Carlo simulation of the SiO2 electron energy-loss spectrum (see Fig. 10.3). The incident electron energy is given by E 0 = 2000 eV. Dielectric function used in this simulation was taken by Buechner experimental data for energies lower than 33.6 eV [12] and obtained by the Henke et al. optical data for higher energies [7, 8]. The Monte Carlo simulated spectrum presents two plasmon loss peaks at ≈23 eV and at ≈46 eV corresponding to the single inelastic scattering and to the double inelastic scattering, respectively [14]. The main peak and its shoulders in the current calculations can be interpreted as interband transitions from the valence bands and the conduction band. The main plasmon loss at ≈23 eV and the shoulder located at ≈19 eV are attributed to excitations of the bonding bands, while the shoulders located at ≈15 eV and ≈13 eV are due to excitations of a nonbonding band [15].

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(a)

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Energy loss (eV) Fig. 10.2 Comparison between the experimental (black line) and Monte Carlo simulated (red line) line-shape for the π (panel a) and the π + σ (panel b) plasmon peaks [13]. The primary electron energy is 500 eV. The plasmon loss peaks are normalized to a common height after subtracting a linear background. The optical data were taken from Henke et al. [7, 8], while the ELF data presented in [9–11] were utilized for energies lower than 40 eV. The Monte Carlo code was based on the ES scheme. Experimental data [highly oriented pyrolytic graphite (HOPG)]: Courtesy of Lucia Calliari and Massimiliano Filippi

In Fig. 10.3 an energy range is also shown, located between the main energy loss peak and the elastic peak, in which no backscattered electrons can be observed. Indeed, as the target is an insulator, electrons cannot transfer energies smaller than the value of the energy gap E G between the valence and the conduction bands. As a consequence, no electrons of the primary beam with energy between E 0 − E G and E 0 (energy losses between 0 and E G ) can emerge from the target surface [14].

10.3 Surface Phenomena Several theories have been developed that allow estimating the surface excitation probability. The so-called surface excitation parameter (SEP) represents the average number of surface excitations an electron experiences when crossing the surface once. Knowledge of the SEP allows evaluating the probability Ps of a given number of surface excitations in a surface crossing [16]. Based on the theoretical work of Tung et al. [17], Werner et al. [16] proposed the following formula for the calculation of Ps for an electron crossing the surface with an angle θ with respect to the surface normal:

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Intensity (arb. units)

0.04

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Fig. 10.3 Experimental (black line) and Monte Carlo simulated (red line) energy-loss spectrum of 2000 eV electrons incident on SiO2 [14]. The experimental and Monte Carlo spectra are normalized to a common area of the elastic peak. The optical data were taken from Henke et al. [7, 8], while the Buechner experimental energy loss function [12] was utilized for energies lower than 33.6 eV. The Monte Carlo code was based on the ES scheme. Experimental data: Courtesy of Lucia Calliari and Massimiliano Filippi

Ps (θ, E) =

√

1

a E cos θ + 1

,

(10.1)

where a is a parameter that varies depending on the material (a ≈ 0.12 for Al). Dapor et al. [18] calculated the energy loss spectra of Al and Si under the assumption that experimental spectra arise from electrons undergoing a single large-angle elastic scattering event (so-called V-type trajectories [19]) and describing inelastic scattering by the Chen and Kwei differential inverse inelastic mean free path, dependent on the distance from the surface (in the solid and in the vacuum) and on the surface-crossing angle [20]. Reasonable agreement was found between these calculations and available experimental data for bulk and surface plasmon peaks in the energy range from 500 eV to 2000 eV [18] (see, for example, Fig. 4.41 for 1000 eV in Al and Fig. 4.42 for 1000 eV in Si). The surface is an interface between the bulk of the material and the vacuum. Since the dielectric function depends on the material, we expect the mean energy of the plasmons characterizing the surface (surface plasmons) to be different from the mean energy of the bulk plasmons. Please note the existence of the begrenzungs effect, i.e., a reduction of the intensity of the bulk plasmon due to the boundary, which was discussed by Ritchie [21]. In the limit of large target thickness, Ritchie demonstrated that the total transition

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probability is given by two terms. The first one is just the target thickness times the transition probability per unit path length in an infinite target. The second one is proportional to     4 1 (1 − ε)2 = Im −1− . Im − ε(1 + ε) ε+1 ε

(10.2)

The begrenzungs effect is represented by Im[− 1 − 1/ε] [22]. Chiarello et al. [23], observed that the REEL spectrum can be described by the combination of two terms, arising from surface and bulk inelastic scattering. According to these authors’ approach, the bulk and surface excitations can be described by the combined effect of the bulk energy loss function and the surface energy loss function which, according to Chiarello et al. [23], Ohno [24], and Yubero et al. [25] can be described as   1 (10.3) f s (k, ω) = Im ε(k, ω) + 1 Attempts to include the combination of surface and bulk energy loss functions in Monte Carlo simulations, as proposed by Chiarello et al. [23], can be found in [26, 27]. In Figs. 10.4 and 10.5, we present both experimental and Monte Carlo REEL spectra of 1000 eV electrons impinging on Al and Si, respectively. The Monte Carlo results were obtained using the Chiarello et al. approach [23]. Details of the Monte Carlo simulation can be found in [27]. The number of simulated Monte Carlo trajectories was 109 . The Monte Carlo simulations used the experimental conditions. The angle between the sample surface normal and the incident electron beam was 30◦ . The entrance aperture of the analyzer ranged from 36◦ to 48◦ . The sputter-induced amorphization in the surface region can modify the density of the samples. As a consequence, we used the experimentally determined values of the densities in the calculations. Plasmon peaks up to the 5th order of scattering for Al and up to the 4th order of scattering for Si can be observed in both the experimental and simulated spectra. Please note the presence of surface plasmon peaks and bulk plasmon peaks in the spectra.

10.4 Energy Losses of Auger Electrons The Monte Carlo code can be used to model the energy losses occurring in the Auger electrons during their travel in the solid—before emerging from the surface. In this context, the Monte Carlo code can be used to calculate the changes caused to the original electron energy distribution by energy losses suffered by the Auger electron on its way out of the solid. The original electron distribution was computed by using ab initio calculations of non-radiative decay spectra obtained by the program suite

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Fig. 10.4 Experimental (black line) and Monte Carlo simulated (red line) REEL spectra for 1000 eV primary electrons impinging on Al are shown. The spectra were normalized to a common area of the zero-loss peak. Experimental data: Courtesy of Lucia Calliari and Massimiliano Filippi

0.05

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Energy loss (eV) Fig. 10.5 Experimental (black line) and Monte Carlo simulated (red line) REEL spectra for 1000 eV primary electrons impinging on Si are shown. The spectra were normalized to a common area of the zero-loss peak. Experimental data: Courtesy of Lucia Calliari and Massimiliano Filippi

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Energy (eV) Fig. 10.6 O K-LL Auger spectrum in SiO2 . Comparison between the quantum mechanical theoretical data (blue line), the Monte Carlo results (red line), and the original experimental data (black line) [28]. Quantum mechanical theoretical data: Courtesy of Stefano Simonucci and Simone Taioli. Experimental data: Courtesy of Lucia Calliari and Massimiliano Filippi

SURface PhotoelectRon and Inner Shell Electron Spectroscopy (SURPRISES). The physics of SURPRISES can be found in [28–30]. It is a program that performs ab initio calculations of photoionization and non-radiative decay spectra in nano-clusters and solid-state systems. The comparison between simulated and experimental Auger spectra requires properly taking into account the changes caused, to the Auger electron energy distribution, by the energy losses which the Auger electrons suffer along their travel in the solid toward the surface. Due to that, we use previously calculated ab initio Auger probability distribution as a source of electrons suffering inelastic processes. The theoretical Auger spectrum previously calculated using ab initio calculations was adopted to describe the initial energy distribution of the escaping electrons. Auger electron generation is calculated assuming a constant depth distribution whose thickness, according to [31], was set to 40 Å. A plot of the calculation compared to the original experimental data—i.e., the experimental data presented without any deconvolution of energy losses—is given in Fig. 10.6. The original theoretical spectrum (ab initio calculation) is also provided. One can see that the Monte Carlo energy loss calculation increases and broadens the Auger probability. The large broadening of the K-L1 L23 peak after the Monte Carlo treatment is due to the main plasmon of SiO2 whose distance from the zero loss peak (≈23 eV: See Fig. 10.3) is the same as the distance between the K-L23 L23 and K-L1 L23 features in the Auger spectrum.

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10 Electron Energy Distributions

A satisfactory agreement between the experiment and the combination of the ab initio calculation with the Monte Carlo simulation can be found: In particular, good accordance can be recognized in the energy position, in the relative intensities of the peaks, and in the background contribution over the entire investigated energy range.

10.5 Elastic Peak Electron Spectroscopy (EPES) The analysis of the line shape of the elastic peak is known as elastic peak electron spectroscopy (EPES) [32, 33]. The energy of electrons of the elastic peak is reduced by the energy transferred to the atoms of the target (the so-called recoil energy). Since the lightest elements present the largest energy shifts, EPES is, among electron spectroscopies, the only one able to detect hydrogen in polymers and hydrogenated carbon-based materials [34–41]. Hydrogen detection is obtained by measuring the energy difference between the position of the carbon (or the carbon + oxygen) elastic peak and that of the hydrogen elastic peak: This difference between the energy positions of the elastic peaks—for incident electron energy in the range 1000–2000 eV—is in the range from ≈2 eV to ≈4 eV, and increases as the kinetic energy of the incident electrons increases. The reason is that the mean recoil energy E R for an atom of mass m A is given by: ER =

ϑ 4m E 0 sin2 . mA 2

(10.4)

Monte Carlo simulated PMMA elastic peak electron spectra are presented in Figs. 10.7 (E 0 = 1500 eV) and 10.8 (E 0 = 2000 eV). The H elastic peak position is shifted in energy with respect to the C+O elastic peak, the difference in energy being an increasing function of the primary energy E 0 .

10.6 Spectrum of Secondary Electrons Another important feature of the electron energy spectrum is represented by the secondary-electron emission distribution, i.e., the energy distribution of those electrons that, once extracted from the atoms by inelastic collisions and having traveled in the solid, reach the surface with the energy sufficient to emerge. The energy distribution of the secondary electrons is confined in the low energy region of the spectrum, typically well below 50 eV. A very pronounced peak characterizes it, which is due to a cascade process in which every secondary electron generates, along its trajectory, further secondary electrons, so that a shower of secondary electrons is created.

10.6 Spectrum of Secondary Electrons

161

0.04

Intensity (arb. units)

0.03

0.02

0.01

0.00 -2

0

2

4

6

Energy loss (eV)

Fig. 10.7 Monte Carlo simulation of EPES for PMMA, showing the C+O (left) and the H (right) elastic peaks. E 0 = 1500 eV 0.04

Intensity (arb. units)

0.03

0.02

0.01

0.00 -2

0

2

4

6

Energy loss (eV)

Fig. 10.8 Monte Carlo simulation of EPES for PMMA, showing the C+O (left) and the H (right) elastic peaks. E 0 = 2000 eV

162

10 Electron Energy Distributions

10.6.1 Wolff Theory Electron beams impinging on solid targets stimulate the emission of secondary electrons, i.e., electrons extracted from the atoms in the solid, as a consequence of interaction with primary beam electrons or with the other energetic secondary electrons traveling in the solid. Some secondary electrons, following a number of elastic and inelastic collisions with the atoms of the solid, reach the surface of the solid with enough energy to emerge from it. Even some of the primary electrons, after several elastic and inelastic collisions with the target atoms, emerge from the surface. So the spectrum of the secondary electrons is contaminated by the contribution of the backscattered primary electrons. In the following paragraphs, we will focus our attention on the secondary electrons, dismissing the backscattered electrons. The process of secondary-electron emission can be conceptually divided into two phenomena: The first concerns the production of secondary electrons due to interaction between primary electrons and electrons bound in the solid; The second is represented by the cascade, whereby the secondary electrons, diffusing in the solid, extract new secondary electrons in a cascade process [42, 43]. Since every secondary electron loses energy during its travel within the solid, the process goes on until either the secondary-electron energy has decreased to the point that it is not sufficient to extract new secondary electrons or the electron has reached the surface of the solid with sufficient energy to emerge from it. Let us introduce some relevant physical quantities. We will indicate by N (r, Ω, E, t) the number of electrons at time t between r and r + dr, between Ω and Ω + dΩ, and between E and E + d E, where Ω represents a unit vector in the direction of the velocity of electron v. λ(E) is the electron mean free path and F(Ω, E; Ω  , E  ) the probability that an electron at Ω  , E  is found, after scattering, at Ω, E. Let us indicate by S(r, Ω, E, t) the source, i.e., the density of the secondary electrons due to the bombardment of primary particles. The electron cascade is governed by the Boltzmann diffusion equation, ∂N vN + v · ∇N = − +S ∂t λ  v  N (r, Ω  , E  , t) F(Ω, E; Ω  , E  ) . (10.5) + d E  dΩ  λ(E  ) Considering a geometry in which the primary particles are incident normally on the target surface, the problem has azimuthal symmetry and involves the distance to surface z, energy E, and the angle between the velocity of the secondary electron and the direction normal to the surface, θ. If we focus our attention on the steady-state condition ∂N =0, ∂t

(10.6)

10.6 Spectrum of Secondary Electrons

163

then vN +S v · ∇N = − λ  v  N (r, Ω  , E  , t) + d E  dΩ  F(Ω, E; Ω  , E  ) . λ(E  )

(10.7)

Let us indicate by Θ the angle between Ω and Ω  and expand in spherical harmonics the three functions N , F and S: ∞ 1  N (z, cos θ, E) = (2l + 1)Nl (z, E)Pl (cos θ) , 4π l=0

S(z, cos θ, E) =

∞ 1  (2l + 1)Sl (z, E)Pl (cos θ) , 4π l=0

F(Ω, E; Ω  , E  ) = F(cos Θ; E, E  ) ∞ 1  = (2l + 1)Fl (E, E  )Pl (cos Θ) . 4π l=0

(10.8)

(10.9)

(10.10)

Let us introduce the function ψl = vNl /λ(E) ,

(10.11)

and note that v · ∇N = = = = = =

∂Nl 1  cos θ Pl (cos θ) (2l + 1)v 4π l ∂z   λ(E)  ∂ vNl cos θ Pl (cos θ) (2l + 1) 4π l ∂z λ λ  ∂ψl cos θ Pl (cos θ) (2l + 1) 4π l ∂z λ  ∂ψl [(l + 1)Pl+1 (cos θ) + l Pl−1 (cos θ)] 4π l ∂z λ  ∂ψl−1 λ  ∂ψl+1 l Pl (cos θ) + (l + 1)Pl (cos θ) 4π l ∂z 4π l ∂z   1  l + 1 ∂ψl+1 l ∂ψl−1 + Pl (cos θ) . (2l + 1)λ(E) 4π l 2l + 1 ∂z 2l + 1 ∂z (10.12)

164

10 Electron Energy Distributions

Then,  ψl (z, E) = Sl (z, E) + d E  ψl (z, E  )Fl (E, E  )   l + 1 ∂ψl+1 l ∂ψl−1 + . + λ(E) 2l + 1 ∂z 2l + 1 ∂z

(10.13)

Once ψl (z, E) is known, the function Nl can be calculated as λ(E)ψl (z, E) Nl = = v



m λ(E)ψl (z, E) , 2E

(10.14)

where m is the electron mass. Let us assume now that the secondary electrons be uniformly produced throughout the depth, so the functions ψl depend only on E and are independent of z. If the energy is sufficiently low, the distribution of the secondary electrons is spherically symmetric, so one can neglect all the harmonics higher than l = 0. Within this approximation, the integro-differential equation (10.13) with S0 = 0 becomes  ∞ d E  ψ0 (E  ) F0 (E, E  ) . (10.15) ψ0 (E) = E

Note that, once ψ0 is known, the secondary-electron energy distribution can be calculated as1 1

Concerning F0 (E, E  ), let us note that  2π sin Θ dΘ F(cos Θ; E, E  )Pk (cos Θ) 

∞ 1  (2l + 1)Fl (E, E  )Pl (cos Θ)Pk (cos Θ) 4π 0 l=0  1 ∞ 1 = (2l + 1)Fl (E, E  ) Pl (u)Pk (u) du 2 −1

= 2π

π

sin Θ dΘ

l=0

∞ 2 1 (2l + 1)Fl (E, E  ) = δlk 2 2l + 1 l=0

= Fk (E, E  ) , so that F0 (E, E  ) = 2π



sin Θ dΘ F(cos Θ; E, E  ) .

Therefore, within the approximation of spherical symmetry of the secondary-electron distribution and ignoring all the harmonics higher than l = 0, (10.15) can also be written as follows:  ∞  π ψ0 (E) = 2π d E  ψ0 (E  ) sin Θ dΘ F(cos Θ; E, E  ) . E

0

10.6 Spectrum of Secondary Electrons

165

j (E) = N0 v = ψ0 (E)λ(E) .

(10.16)

In (10.15) F0 (E, E  ) represents the total scattering probability between the energies E and E , independently of the angle. Let us assume that, on average, electrons of energy E  < 100 eV lose half of their energy at each collision. Let us introduce the average energy after scattering, ¯ defined by E, E¯ = γ E  .

(10.17)

Here E  is the energy before the scattering and, for energies four times greater than the Fermi energy, γ is ∼ 1/2. Within this approximation, ¯ = 2 δ(E − γ E  ) . F0 (E, E  ) = 2 δ(E − E)

(10.18)

Note factor 2, which takes into account the fact that there are two electrons after each electron collides. Equation (10.15) therefore becomes  ψ0 (E) = 2 δ(E − γ E  )ψ0 (E  ) d E  , (10.19) which, as the reader can easily verify, produces the following solution:   E 2 . ψ0 (E) = ψ0 γ(E) γ(E)

(10.20)

Let us now define the function x(E)2 : 2[γ(E)]x(E)−1 = 1 .

(10.21)

The solution of (10.20) is proportional to E −x : ψ0 (E) ∝ E −x .

(10.22)

The secondary electron current is thus given by j (E) ∝ P(E)

λ(E) . Ex

(10.23)

where P(E) is a factor providing the probability that an electron reaching the surface with energy E has normal velocity sufficient to overcome the barrier χ (work

2

Note that for E higher than 2E F (where E F is the Fermi energy), x(E) becomes almost constant (x(E) ≈ 2.1).

166

10 Electron Energy Distributions

function/electron affinity) between the vacuum level and the bottom of the conduction band. According to Wolff [42]  χ + EF . (10.24) P(E) = 1 − E

10.6.2 Other Formulas Describing the Spectrum of the Secondary Electrons Amelio [43] proposed the following formula, based on a Streitwolf equation [44]: E − W j (E  ) = (10.25) × 4    

E 5 3 E W2 × −  ψ0 (E  + W ) − ψ2 (E  + W ) 1− , E + W 2 2 (E  + W )2 E +W

where W = E F + χ and E  = E − W , while Chung and Everhart [45] devised the following expression: j (E) ∝

E − EF − χ . (E − E F )4

(10.26)

10.6.3 Initial Polar and Azimuthal Angle of the Secondary Electrons The question we wish now to face is whether secondary electrons emerge from the collision with spherical symmetry or if they are emitted conserving momentum and energy according to the classical binary-collision theory (see Sect. 6.2.5). Note that the classical binary-collision theory neglects the environment where the collision occurs and where energy can be transferred to other particles. Since the energy distributions resulting from these two approaches (spherical symmetry and classical binary-collision theory) are different, two versions of the Monte Carlo code were considered. One of the two versions adopts spherical symmetry for the angular distribution of the secondary electron emission at the site of generation, while the other one is based on the classical binary-collision theory and the corresponding conservation laws. The results of the two Monte Carlo codes are compared to experimental data, in order to decide which one better describes the phenomena. We will show that the hypothesis of spherical symmetry provides results in better agreement with the experimental evidences than that of the binary-collision theory. This is in agreement with the suggestion given by Shimizu and Ding Ze Jun

10.6 Spectrum of Secondary Electrons

167

[46] to use spherical symmetry in the Monte Carlo simulation of secondary electron generation when Fermi sea excitations are involved in the process. The initial polar angle θs and the initial azimuth angle φs of each secondary electron can be calculated in two different ways. In the first one, based on the hypothesis that the secondary electrons emerge with spherical symmetry, their initial polar and azimuthal angles are randomly determined as (10.27) θs = arccos(2μ1 − 1) , φs = 2 π μ2

(10.28)

where μ1 and μ2 are random numbers uniformly distributed in the range [0, 1]. MCSS is the name attributed, in the present context, to the Monte Carlo code based on this method. A second code is proposed in which conservation laws according to the classical binary-collision theory are taken into account so that, if θ and φ are, respectively, the polar and azimuthal angle of the incident electron, then sin θs = cos θ ,

(10.29)

φs = π + φ .

(10.30)

We will indicate, in the present context, with MCMC the Monte Carlo code corresponding to this second approach. The results of the MCSS and MCMC codes will be compared with theoretical and experimental data [47].

10.6.4 Comparison with Theoretical and Experimental Data Secondary Electron Energy Distribution of Silicon and Copper We present, in Figs. 10.9 and 10.10, the energy distributions of the secondary electrons emitted by silicon and copper targets, respectively. The energy of the primary electron beam is E 0 = 1000 eV for silicon, while it is E 0 = 300 eV for copper. The Monte Carlo calculations obtained with the two different methods described above (MCSS and MCMC) are compared with the Amelio theoretical and experimental results [43]. Using the theory of Amelio for the comparison, the MCSS scheme gives results that show a much better agreement than the MCMC code. Indeed, in the range of primary energies examined and for both the materials considered, it is clear that, using the MCSS code, the position of the maxima and the general trend of the energy distributions are in excellent agreement with the Amelio theoretical calculations. On the other hand, the use of the MCMC code generates electron energy distributions that are not in such a good agreement with the Amelio theoretical data. The position

168

10 Electron Energy Distributions 1

(b)

1

(dN/dE)/(dN/dE)max

(a)

0

0 0

5

10

0

15

5

10

15

E (eV)

E (eV)

Fig. 10.9 Energy distribution of the secondary electrons emitted by a silicon target. The Monte Carlo calculations (red lines) [47] are compared with the Amelio theoretical results (black lines) [43]. Data are normalized to a common maximum. The initial electron energy is 1000 eV. The zero of the energy scale is located at the vacuum level. The primary electron beam is normal to the surface. Electrons are accepted over an angular range from 0◦ to 90◦ integrated around the full 360◦ azimuth. Panel a: MCSS code. Panel b: MCMC code (see details in the text) 1

(b)

1

(dN/dE)/(dN/dE)max

(a)

0

0 0

5

10

E (eV)

15

0

5

10

15

E (eV)

Fig. 10.10 Energy distribution of the secondary electrons emitted by a copper target. The Monte Carlo calculations (gray lines) [47] are compared with the Amelio theoretical results (black lines) [43]. Data are normalized to a common maximum. The initial electron energy is 300 eV. The zero of the energy scale is located at the vacuum level. Electrons are accepted over an angular range from 0◦ to 90◦ integrated around the full 360◦ azimuth. The primary electron beam is normal to the surface. Panel a: MCSS code. Panel b: MCMC code (see details in the text)

10.6 Spectrum of Secondary Electrons

169

Table 10.1 Monte Carlo most probable energy (MPE) and full width at half maximum (FWHM) of secondary electron energy distributions obtained with two different schemes (MCSS and MCMC). The experimental data were reported by Amelio [43]. Calculations and measurements were carried out on bulks of Si irradiated by streams of electrons in the +z direction. The primary energy of the incident electron beam was 1000 eV Si (1000 eV) MCSS MCMC Experiment MPE (eV) FWHM (eV)

1.8 5.3

2.8 8.5

1.7 5.0

Table 10.2 Monte Carlo most probable energy (MPE) and full width at half maximum (FWHM) of secondary electron energy distributions obtained with two different schemes (MCSS and MCMC). The experimental data were reported by Amelio [43]. Calculations and measurements were carried out on bulks of Cu irradiated by streams of electrons in the +z direction. The primary energy of the incident electron beam was 300 eV Cu (300 eV) MCSS MCMC Experiment MPE (eV) FWHM (eV)

2.8 9.2

3.5 12

2.8 10

of the maxima are shifted to higher energies and the shapes of the distributions are quite different from the Amelio energy distributions. It should be noted that experimental data concerning secondary electron energy distributions were reported by Amelio as well. In Tables 10.1 and 10.2 the main features regarding the energy distributions [i.e., the most probable energy (MPE) and the full width at half maximum (FWHM)] obtained with MCSS and MCMC are compared with the experimental data reported by Amelio. The agreement of the MCSS results with both the theoretical and experimental Amelio data [43] can be attributed to the isotropy of the low-energy secondary electron emission due to: (i) Post-collisional effects and consequent random energy and momentum transfer among secondary electrons; (ii) Interactions with the conduction electrons, just after secondary electrons are emitted. In conclusion, the results of the present investigation suggest that slow secondary electrons should be generated, in MC codes, with spherical symmetry. Secondary Electron Energy Distribution of Polymethyl Methacrylate The energy distribution, obtained using the MCSS code, of the electrons emerging from polymethyl methacrylate irradiated by an electron beam with primary energy E 0 = 1000 eV is presented in Fig. 10.11 [48]. The spectrum is simulated assuming that the primary electron beam is normal to the surface. The zero of the energy is located at the vacuum level. In the same figure, a comparison of the Monte Carlo simulated spectrum to the Joy et al. experimental electron energy distribution [49] is shown. The same conditions of the experiment are used for the simulation, i.e., acceptance angles in the range from 36◦ to 48◦ integrated around the full 360◦ azimuth (Cylindrical Mirror Analyzer geometry). The Monte Carlo calculation describes very

170

10 Electron Energy Distributions

(dN/dE)/(dN/dE)max

1.0

0.5

0.0 0

5

10

15

E (eV) Fig. 10.11 Energy distribution of the secondary electrons emitted by a PMMA target. The Monte Carlo calculations (red line) [48] are compared with the Joy et al. experimental data (black line) [49]. Data are normalized to a common maximum. The initial electron energy is 1000 eV. The zero of the energy scale is located at the vacuum level. The primary electron beam is normal to the surface. Electrons are accepted, according to the Joy et al. experimental conditions, over an angular range from 36◦ to 48◦ integrated around the full 360◦ azimuthal angle

well the initial increase of the spectrum, the energy position of the maximum, and the full width at half maximum. The present Monte Carlo simulation is not able, on the other hand, to describe the observed fine structure of the peak, in particular the shoulder on the left of the maximum.

10.7 Summary In this chapter, reflection electron energy loss spectra, Auger electron spectra, elastic peak electron spectra, and secondary electron energy spectra were simulated by the Monte Carlo method. The results of the simulation of the spectra were compared with the available experimental data.

References

171

References 1. R.M. Martin, Electronic Structure. Basic Theory and Practical Methods (Cambridge University Press, Cambridge, 2004) 2. M.D. Crescenzi, M.N. Piancastelli, Electron Scattering and Related Spectroscopies (World Scientific, Singapore, 1996) 3. R.G. Newton, Scattering Theory of Wave and Particle (Springer, New York, 1982) 4. R. Cimino, I.R. Collins, M.A. Furman, M. Pivi, F. Ruggiero, G. Rumolo, F. Zimmermann, Phys. Rev. Lett. 93, 014801 (2004) 5. M.A. Furman, V.H. Chaplin, Phys. Rev. Special Topics - Accel Beams 9, 034403 (2006) 6. M. Dapor, Appl. Surf. Sci. 391, 3 (2017) 7. B.L. Henke, P. Lee, T.J. Tanaka, R.L. Shimabukuro, B.K. Fujikawa, At. Data Nucl. Data Tables 27, 1 (1982) 8. B.L. Henke, P. Lee, T.J. Tanaka, R.L. Shimabukuro, B.K. Fujikawa, At. Data Nucl. Data Tables 54, 181 (1993) 9. J. Daniels, C.V. Festenberg, H. Raether, K. Zeppenfeld, Springer Tracts Modern Phys. 54, 78 (1970) 10. H. Venghauss, Phys. Status Solidi B 71, 609 (1975) 11. A.G. Marinopoulos, L. Reining, A. Rubio, V. Olevano, Phys. Rev. B 69, 245419 (2004) 12. U. Buechner, J. Phys. C: Solid State Phys. 8, 2781 (1975) 13. M. Dapor, L. Calliari, M. Filippi, Nucl. Instrum. Methods Phys. Res. B 255, 276 (2007) 14. M. Filippi, L. Calliari, M. Dapor, Phys. Rev. B 75, 125406 (2007) 15. M.H. Reilly, J. Phys. Chem. Solids 31, 1041 (1970) 16. W.S.M. Werner, W. Smekal, C. Tomastik, H. Stoöri, Surf. Sci. 486, L461 (2001) 17. C.J. Tung, Y.F. Chen, C.M. Kwei, T.L. Chou, Phys. Rev. B 49, 16684 (1994) 18. M. Dapor, L. Calliari, S. Fanchenko, Surf. Interface Anal. 44, 1110 (2012) 19. A. Jablonski, C.J. Powell, Surf. Sci. 551, 106 (2004) 20. Y.F. Chen, C.M. Kwei, Surf. Sci. 364, 131 (1996) 21. R.H. Ritchie, Phys. Rev. 106, 874 (1957) 22. R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 3rd edn. (Springer, Berlin, 2011) 23. G. Chiarello, E. Colavita, M. De Crescenzi, S. Nannarone, Phys. Rev. B 29, 4878 (1984) 24. Y. Ohno, Phys. Rev. B 39, 8209 (1989) 25. F. Yubero, J.M. Sanz, E. Elizalde, L. Galán, Surf. Sci. 237, 173 (1990) 26. L. Calliari, M. Dapor, M. Filippi, Surf. Sci. 601, 2270 (2007) 27. M. Dapor, Front. Mater. 9, 1068196 (2022) 28. S. Taioli, S. Simonucci, L. Calliari, M. Dapor, Phys. Rep. 493, 237 (2010) 29. S. Taioli, S. Simonucci, L. Calliari, M. Filippi, M. Dapor, Phys. Rev. B 79, 085432 (2009) 30. S. Taioli, S. Simonucci, M. Dapor, Comput. Sci. Discovery 2, 015002 (2009) 31. G.A. van Riessen, S.M. Thurgate, D.E. Ramaker, J. Electron Spectrosc. Relat. Phenom. 161, 150 (2007) 32. G. Gergely, Progr. Surf. Sci. 71, 31 (2002) 33. A. Jablonski, Progr. Surf. Sci. 74, 357 (2003) . Surf. Sci. 74, 357 (2003) 34. D. Varga, K. Tökési, Z. Berènyi, J. Tóth, L. Kövér, G. Gergely, A. Sulyok, Surf. Interface Anal. 31, 1019 (2001) 35. A. Sulyok, G. Gergely, M. Menyhard, J. Tóth, D. Varga, L. Kövér, Z. Berènyi, B. Lesiak, A. Jablonski, Vacuum 63, 371 (2001) 36. G.T. Orosz, G. Gergely, M. Menyhard, J. Tóth, D. Varga, B. Lesiak, A. Jablonski, Surf. Sci. 566–568, 544 (2004) 37. F. Yubero, V.J. Rico, J.P. Espinós, J. Cotrino, A.R. González-Elipe, Appl. Phys. Lett. 87, 084101 (2005) 38. V.J. Rico, F. Yubero, J.P. Espinós, J. Cotrino, A.R. González-Elipe, D. Garg, S. Henry, Diam. Relat. Mater. 16, 107 (2007)

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10 Electron Energy Distributions D. Varga, K. Tökési, Z. Berènyi, J. Tóth, L. Kövér, Surf. Interface Anal. 38, 544 (2006) M. Filippi, L. Calliari, Surf. Interface Anal. 40, 1469 (2008) M. Filippi, L. Calliari, C. Verona, G. Verona-Rinati, Surf. Sci. 603, 2082 (2009) P.A. Wolff, Phys. Rev. 95, 56 (1954) G.F. Amelio, J. Vac. Sci. Technol. 7, 593 (1970) H.W. Streitwolf, Ann. Physik 3, 183 (1959) M.S. Chung, T.E. Everhart, J. Appl. Phys. 45, 707 (1974) R. Shimizu, D. Ze-Jun, Rep. Prog. Phys. 55, 487 (1992) M. Dapor, Nucl. Instrum. Methods Phys. Res. B 267, 3055 (2009) M. Dapor, G.I.T. Imaging & Microscopy 2, 38 (2016) D.C. Joy, M.S. Prasad, H.M. Meyer III., J. Microsc. 215, 77 (2004)

Chapter 11

Applications

In this chapter, we will discuss some important applications of the Monte Carlo method. We will describe, in particular, (1) the line-scan calculations of resist materials with given geometrical cross-sections deposited on silicon substrates, (2) the energy selective SEM for image contrast in silicon p-n junctions, (3) the density of energy radially deposited by secondary electrons around the tracks of ions through polymers and bio-molecular systems.

11.1 Linewidth Measurement in Critical Dimension SEM A very important application of the MC calculations of the secondary electron yield is related to nanometrology and linewidth measurement in critical dimension SEM [1– 5]. In [1, 6] this problem has been investigated using an approach based on the energystraggling strategy, described earlier in this work, and on the detailed description of all the main mechanisms of scattering (elastic electron-atom interactions, quasi-elastic electron-phonon interactions, inelastic electron-plasmon interactions, and polaronic effects) [7–9]. The corresponding energy straggling Monte Carlo module has been then included in the PENELOPE code [10–12].

11.1.1 Critical Dimension SEM In order to provide metrics for the complementary metal-oxide-semiconductor (CMOS) technologies, critical dimension measurements with sub-nanometer uncertainty have to be performed, in particular for the line-width measurement of photoresist lines (PMMA lines, for example) used in electron beam lithography. The physics of image formation in scanning electron microscopy have to be understood and

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1_11

173

174

11 Applications

Fig. 11.1 Dielectric material (e.g., PMMA) with trapezoidal cross-section on silicon substrate. Linescans are acquired perpendicularly to the structures

modeled. Monte Carlo simulation of the generation and transport of secondary electrons—produced by low primary energy electrons—is today the most accurate method for obtaining information about image formation in scanning electron microscopy. Typical structures of interest for CMOS technologies are dielectric lines (e.g., PMMA lines) on silicon substrates with trapezoidal cross-section (see Fig. 11.1). In SEM measurements with sub-nanometer uncertainty, the critical dimensions to be investigated are the bottom line width, the top line width, the slope of the rising edge, and the slope of the falling edge.

11.1.2 Lateral and Depth Distributions The lateral and depth resolutions of secondary electron imaging are related to the diffusion of the secondary electrons in the solid. It seems therefore important, before proceeding, to investigate the extent of lateral and depth distributions of the emerging electrons. Figure 11.2 shows the lateral distribution of the secondary electrons emerging from PMMA, for a primary energy of 1000 eV, adopting a delta-shaped beam spot. Figure 11.3 presents, for the same energy, the depth distribution of the sites where the secondary electrons that were able to emerge from the sample surface have originated. Both Figs. 11.2 and 11.3 provide an evaluation of the lateral and

11.1 Linewidth Measurement in Critical Dimension SEM 0.03

dN/dx (arb. units)

Fig. 11.2 Monte Carlo simulation of the lateral distribution d N /d x of the secondary electrons emerging from PMMA [6]. The electron primary energy E0 is 1000 eV

175

0.02

0.01

0.00 -40

-30

-20

-10

0

10

20

30

40

x( )

0.04

dN/dz (arb. units)

Fig. 11.3 Monte Carlo simulation of the depth distribution d N /dz of the sites where the secondary electrons that were able to emerge from the PMMA sample surface have been generated [6]. The electron primary energy E0 is 1000 eV

0.03

0.02

0.01

0.00 0

10

20

30

40

50

z( )

depth resolution of secondary electron emission. In agreement with this theoretical model, the lateral and depth distributions of emerging secondary electrons have an extent smaller than ∼50 Å.

11.1.3 Linescan of a Silicon Step Where the surface is flat, the secondary electron emission yield corresponds to that of normal incidence. Approaching the step on the negative x positions, a shadowing effect is observed that is due to the interception of the trajectories of the emerging secondary electrons by the step and, at the bottom edge of the step, the signal reaches its minimum value. The signal maximum is observed at the top edge of the step. An

176

11 Applications

SE emission signal (arb. units)

1.2

1.0

0.8

0.6

0.4

-800

-400

0

400

800

Position ( )

Fig. 11.4 Monte Carlo simulation of a linescan from a silicon step with side wall angle equal to 10◦ and incident electron energy equal to 700 eV. The shape of the signal is determined by the angle of incidence of the primary electrons, their energy, positions of incidence, and geometry. The simulation has been performed by means of a pencil beam. Courtesy of Mauro Ciappa and Emre Ilgüsatiroglu, ETH, Zurich

intermediate level within the transition is observed in Fig. 11.4, according to the behavior of the secondary electron yield as a function of the angle of incidence.

11.1.4 Linescan of PMMA Lines on a Silicon Substrate Figure 11.5 shows linescans obtained by simulating three adjacent PMMA lines on a silicon substrate. Additional geometry shadowing effects due to the neighboring lines can be observed in the internal edge signals.

11.2 Application to Energy Selective Scanning Electron Microscopy Monte Carlo simulations of secondary electron energy distribution and yield have important applications in the design and characterization of semiconductor devices, particularly in the investigation of the distribution of dopant atom concentrations at the nanometer scale. This application is known as two-dimensional dopant mapping. Two-dimensional dopant mapping based on secondary electron emission is a technique that allows a fast investigation of dopant distributions in semiconductors. The dopant contrast can be explained by taking into account the electron affinity in

11.2 Application to Energy Selective Scanning Electron Microscopy

3.5

SE emission signal (arb. units)

Fig. 11.5 Monte Carlo simulation of a linescan from PMMA lines (height: 200 Å , bottom width: 160 Å , top width: 125 Å , side wall angle: 5◦ ) on a Si substrate scanned with a pencil-like electron beam at 500 eV. Courtesy of Mauro Ciappa and Emre Ilgüsatiroglu, ETH, Zurich

177

3.0

2.5

2.0

1.5

1.0

0.5 -600

-400

-200

0

200

400

600

Position ( )

Monte Carlo simulations to calculate the secondary electron emission from doped silicon [13].

11.2.1 Doping Contrast A reliable method for mapping quantitatively the distribution of dopant atom concentrations at the nanometre scale is the use of secondary-electron contrast in the SEM [14–20]. Secondary-electron yield changes across a p-n junction [21]. The p-type region emits more secondary electrons than n-type region. As a consequence, the p-type is brighter in the SEM image. The contrast Cpn can be calculated by Cpn = 200

Ip − In , Ip + In

(11.1)

where Ip and In represent the secondary electron emission yields of the p-type and n-type regions, respectively. Increasing the electron affinity reduces the intensity of the lower end of the secondary electron emission spectrum, as electrons approaching the surface from the bulk meet an increased potential barrier. The integral of the secondary electron emission spectrum provides the secondary-electron emission yield so that the latter decreases as the electron affinity increases. Since the Fermi levels equilibrate at the p-n junction, the p-type region has lower electron affinity than the n-type region. Thus the p-type region emits more secondary electrons than the n-type region, as confirmed by the Monte Carlo simulation. Actually, in a p-n junction, it is the difference in the electron affinities—the socalled built-in potential eVbi —rather than the absolute values of the electron affinities,

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11 Applications

what determines the contrast. The bulk built-in potential can be easily calculated for a simple p-n junction. In the case of full ionization, it is given by [22]: eVbi = kB T ln

Na Nd , Ni2

(11.2)

where kB is the Boltzmann constant, T is the absolute temperature, and Na , Nd , and Ni are, respectively, the acceptor dopant carrier concentration, the donor dopant carrier concentration, and the intrinsic carrier concentration. Equation (11.2) allows to calculate the built-in potential if the doping levels of n- and p- regions in contact are known. The value of the electron affinity of pure (un-doped) Si was reported to be 4.05 eV [23]. The Monte Carlo calculations were performed with χ = 3.75 eV for the ptype sample and χ = 4.35 eV for the n-type sample, and for an electron energy E 0 = 1000 eV. Using a specimen with doping levels of p and n consistent with these values of the electron affinities, and the same electron energy, a contrast of Cpn = (16±3)% was reported by Elliott et al. [20]. The Monte Carlo simulated contrast is Cpn = (17±3)% [13], in reasonable agreement with the Elliott et al. experimental observation.

11.2.2 Energy Selective Scanning Electron Microscopy Rodenburg et al. [24] experimentally demonstrated that the image contrast in p-n junctions, obtained considering only the low energy electrons, is significantly higher than the one obtained under standard conditions (where secondary electrons of all energies contribute to the formation of the image). As a consequence, selecting secondary electrons in a given (low) energy window, instead of permitting all the secondary electrons to form the image, allows more accurate quantification of the contrast. It is well known that the built-in potential is smaller close to the surface—with respect to the value that one can calculate using (11.2)—due to surface band bending [25]. Thus secondary electrons emitted from the surface region, where surface band bending reduces the built-in potential, will not see the full built-in voltage of the bulk. For a pure silicon specimen, the Monte Carlo simulated spectra are presented in Fig. 11.6. The maximum of the distribution of the secondary electrons that are generated at a depth of 1 Å is located at ≈10 eV. We can observe that the distribution of the secondary electrons generated within a depth of up to 20 Å presents a maximum located at an energy ≈2 eV. Thus, secondary electrons leaving the material with lower kinetic energy were generated deeper in the specimen than those with higher energy. The Monte Carlo results, relative to Si, are similar to those published by other authors relative to Cu [26] and to SiO2 [27]. According to the experimental observations, the larger the Vbi , the higher the contrast. Since the high energy electrons

11.3 Energy Density Radially Deposited Along Ion Tracks 40

1Å 2Å 5Å 20 Å

35

Intensity (arb. units)

Fig. 11.6 Monte Carlo calculated contributions of secondary electrons originating from different depths to secondary electron spectrum in Si [24]

179

30 25 20 15 10 5 0 0

5

10

15

20

25

30

E (eV)

are generated close to the surface, where the built-in potential is smaller, selecting secondary electrons of low energy increases the dopant contrast. This is in perfect agreement with the experimental observations provided by Rodenburg et al. [24].

11.3 Energy Density Radially Deposited Along Ion Tracks The ionization yield of ion tracks in polymers and bio-molecular systems reaches a maximum, known as the Bragg peak, close to the end of the ion trajectory. Many electrons are generated along the path of MeV ions in the materials. They produce a cascade of further ionizations and, consequently, a shower of secondary electrons. These low-energy secondary electrons can produce damage to the biomolecules. By following in detail the motion of all the secondary electrons, we can find the energy density they radially deposit in the material along the ion track.

11.3.1 Ion Track Simulation and Bragg Peak The interaction of energetic ions with matter is present in many aspects of our daily lives. Cosmic radiation contains an enormous amount of projectiles [28]. 99% of primary cosmic rays are atomic nuclei and about 1% are electrons. 90% of the atomic nuclei are hydrogen nuclei (protons), 9% alpha particles, and 1% nuclei of heavier elements. They produce showers of secondary particles that penetrate the atmosphere and can reach the surface of the Earth. Many of these particles can also reach delicate microelectronic devices present in spatial missions, as well as the crew of manned spatial missions.

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11 Applications

Besides these potential hazards, energetic projectiles properly used can be employed to characterize and modify the properties of materials. Also, swift ion beams represent a useful tool in radiotherapy [29–37]. In particular, ion-beam cancer therapy is based on the characteristic pattern of the energy deposited by ion beams in condensed targets. The depth-dose profile presents a sharp and narrow maximum, called Bragg’s peak, at the end of the particle trajectory, close to the ions’ maximum range. This particular feature of the energy deposition pattern of ion beams is exploited to maximize the damage in the tumor regions minimizing the effects of the irradiation on the healthy tissues near the diseased cells [38].

11.3.2 Damage in the Biomolecules by Ionizations, Excitations, and Dissociative Electron Attachment From the modeling point of view, the interaction of charged particles with condensed matter represents a very tricky and complicated multi-scale problem. In particular, hadron therapy concerns biological materials. The relevant bio-physical processes are very complex, as they involve reactions of nuclear fragmentation, secondary electron emission, damages to the cells, and repair mechanisms of macromolecules (e.g., DNA, proteins, etc.), among other things. It is known that a relevant part of the biological damage is due to secondary electrons. Ionizations, excitations, and dissociative electron attachment produce damage to the biomolecules [39–46].

11.3.3 Simulation of Electron Transport and Further Generation The first step for the buildup of the radial energy deposition profile is the generation of secondary electrons by the primary energetic ion. The motion of the swift ions through the target is accounted for in detail by means of the SEICS code (Simulation of Energetic Ions and Clusters through Solids), which has been described elsewhere [47, 48]. By combining molecular dynamics and Monte Carlo techniques, the SEICS code incorporates the electronic energy loss of the projectile (including stochastic fluctuations), multiple Coulomb scattering and elastic energy loss, electron charge exchange processes, as well as the nuclear fragmentation reactions induced by the incoming projectile. The energy distribution of the electrons generated by each ion is obtained from the corresponding cross-sections evaluated according to a semiempirical model [49]. Due to the interaction with the target atoms (electrons and nuclei), the projectile energy degrades as it moves along the target. Therefore the ion beam has an energy distribution that broadens as it reaches deeper regions in the target. As the ionization cross-section is a function of the projectile energy, the energy spectra of the electrons

11.3 Energy Density Radially Deposited Along Ion Tracks

181

generated along the ion track are obtained by convoluting the energy distribution of the primary ion beam at each depth (obtained with the SEICS code) with the differential ionization cross-section with respect to the electron ejection energy. As a consequence of the energy delivered by a swift ion along its path, electrons are emitted due to the ionization of the target atoms. These electrons move away from the region where they originated experiencing elastic and inelastic scattering with the target constituents, generating secondary electrons. These, in turn, produce new ionizations, which result in an avalanche of electrons. The simulation of the secondary electron emission follows the entire cascade of electrons. As described above, electrons lose energy via the three following processes: (i) Secondary electron emission. In this process, no energy is deposited into the material. The energy lost by the originating electron is transferred to the secondary electron and transported away. (ii) Phonon creation. This process is responsible for energy deposition into the material. (iii) Trapping of the electron. This process is responsible for energy deposition into the material as well. The energy transferred to each secondary electron is transported away from the site where the collision occurred, with energy being deposited into the material only where each secondary electron interacted with phonons or was trapped in the solid [50, 51].

11.3.4 Radial Distribution of the Energy Deposited in PMMA by Secondary Electrons Generated by Energetic Proton Beams Swift protons are projectiles used for cancer radiotherapy. The energy distributions of the electrons generated by each proton were evaluated according to a semi-empirical model described in [49]. Their cumulative probabilities were used to calculate the initial energy of each electron. Then the secondary electron avalanche was followed in order to calculate the energy density radially deposited in the PMMA target along the proton track. The Monte Carlo simulated energy density radially deposited by the secondary electrons for a 3 MeV proton beam impinging on PMMA is presented in Fig. 11.7, where the Udalagama et al. results [52] are also shown for comparison.

11.3.5 Radial Distribution of the Energy Deposited in Liquid H2 O by Secondary Electrons Generated by Energetic Carbon Ion Beams Swift carbon ions are also utilized as projectiles for cancer radiotherapy. Since biological media are mainly composed of liquid water, we are interested in describing how the energy of carbon ions is deposited in water. As already noted, the Monte

182

11 Applications Normalized Deposited Energy Density

1E-4

1E-5

1E-6

1E-7

1E-8

1E-9

1E-10 200

400

600

800

1000

Radial Distance (Å)

Fig. 11.7 Normalized energy density deposited by the secondary electrons at a given radial distance away from a 3 MeV proton track in PMMA. Black line: Udalagama et al. data [52]. Red line: Current Monte Carlo data. The initial energy distribution of the secondary electrons generated by the proton impact with PMMA was calculated according to a semi-empirical model described in [49] and kindly provided by the authors 1E8 1E7

Dose (Gy)

1000000 100000 10000 1000 100 10 1 1

10

100

1000

10000

Radial distance (Å)

Fig. 11.8 Energy density deposited by the secondary electrons at a given radial distance away from a 2 MeV/u carbon ion track in liquid H2 O. Stars: Waligórski et al. [53]. Squares: Liamsuwan and Nikjoo [54]. Circles: Incerti et al. [55] (Geant4-DNA). Black line: de Vera et al. analytical model [56]. Red line: Current Monte Carlo data

Carlo method allows us to simulate the energy carried at the nanometre scale by the generated secondary electrons around the ion’s path [41–46]. Damages to DNA are strongly related to the particle track structure [46]. The Monte Carlo method enables the simulation of the distributions of clustered events in volumes similar to the dimensions of DNA convolutions. It has been observed that, in a wide range of energies, approximately 70% of the damaging events due to

References

183

energetic carbon ion beams interacting with DNA molecules correspond to ionization processes [43]. In Fig. 11.8, the Monte Carlo-simulated radial dose for 2 MeV/u carbon ions penetrating liquid water is compared with other Monte Carlo simulated results [53– 55] and calculated data [56].

11.4 Summary In this chapter, we described some applications of the Monte Carlo method in nanometrology, doping contrast, and hadron therapy. In particular, some fundamental aspects related to the possible use of the Monte Carlo code for line-scan measurements by secondary electron imaging at very low primary beam energy were discussed. In order to extract information about critical dimensions for accurate nanometrology in CMOS processes, we described Monte Carlo investigation of the physics of image formation in scanning electron microscopy [6]. Furthermore, Monte Carlo simulation demonstrated that secondary electrons leaving the material with lower kinetic energy are generated deeper in the specimen compared to those with higher energy. As a consequence, as the built-in potential is smaller close to the surface, selecting low energy secondary electrons—instead of permitting all the secondary electrons to form the image—enhances contrast, thus improving the quality of the measurement of two-dimensional dopant atom distributions [24]. In the end, starting from a realistic description of the energy and angular distribution of the electrons ejected by protons moving through PMMA [49, 51] and carbon ions moving through H2 O [43–45], the simulation of the emission of secondary electrons was used to calculate electron-energy radial deposition (due to the entire cascade of the generated secondary electrons) around the ion track.

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Appendix A

The First Born Approximation and the Rutherford Cross-Section

The Rutherford formula represents the differential elastic scattering cross-section obtained solving the Schrödinger equation in the first Born approximation [1].

A.1 The Elastic Scattering Cross-Section The solid angle dΩ depends on the scattering angle [θ, θ + dθ] and the azimuthal angle [φ, φ + dφ]. The differential elastic scattering cross-section dσel /dΩ is defined as the ratio between the flux of particles per unit of time emerging in the solid angle dΩ (divided by dΩ) and the incident flux. The flux of particles per unit of time emerging after the collision in the solid angle dΩ depends on the component jr of the current density in the outgoing direction from the center of the atomic nucleus. The number of electrons emerging in the solid angle dΩ per unit time is given by jr r 2 dΩ (note that r 2 dΩ is the cross-sectional area normal to the radius). Let us consider a beam of incident electrons in the direction z normalized to one particle per unit volume. Let K = mv/ be the electron momentum in the z direction, where v is the electron velocity, m the electron mass and  the Planck constant divided by 2π. This beam can be represented by the plane wave exp(i K z). Since the incident beam has been normalized to one particle per unit volume, then the electron velocity v is the incident flux. As a consequence, jr r 2 dΩ jr r 2 dσel = = . dΩ v dΩ v

(A.1)

At a large distance from the atomic nucleus, the potential V (r ) is negligible and the scattered particles can be described by a spherical wave, i.e., a function proportional to exp(i K r )/r . If f (θ, φ) is the constant of proportionality (scattering amplitude), © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1

187

188

Appendix A: The First Born Approximation and the Rutherford Cross-Section

then the wave function Ψ (r) of the whole scattering process (i.e., of the incident and the scattered electrons) satisfies the boundary conditions Ψ (r ) ∼ exp(i K z) + f (θ, φ) r →∞

exp(i K r ) . r

(A.2)

The electron position probability density P is given by |Ψ |2 = Ψ ∗ Ψ , and the current density j (r, t) is j (r, t) =

i [(∇Ψ ∗ )Ψ − Ψ ∗ (∇Ψ )]. 2m

(A.3)

Let us now calculate the radial component of the current density j , jr :   exp(i K r ) ∂ exp(−i K r ) f ∗ (θ, φ) r ∂r r   exp(i K r ) exp(−i K r ) ∂ f (θ, φ) − f ∗ (θ, φ) r ∂r r 2 v| f (θ, φ)| = . r2

jr =

i 2m



f (θ, φ)

(A.4)

Comparing this equation with (A.1), we can see that the differential elastic scattering cross-section is the square of the modulus of f (θ, φ): dσel = | f (θ, φ)|2 . dΩ

(A.5)

A.2 The First Born Approximation The first Born approximation is a high-energy approximation. If E is the incident electron energy, e the electron charge, a0 the Bohr radius, and Z the target atomic number, according to Bohm [2] the main criterion of validity of the first Born approximation, for the case of electron-atom elastic scattering, is E

e2 2 Z . 2a0

(A.6)

Appendix A: The First Born Approximation and the Rutherford Cross-Section

189

A.3 Integral-Equation Approach Let us now introduce the Green function and the integral-equation approach. Starting from the Schrödinger equation, (∇ 2 + K 2 )Ψ (r) =

2m V (r )Ψ (r), 2

(A.7)

with the boundary condition expressed by (A.2), it is possible to show that this is a problem equivalent to the following integral equation:  2m Ψ (r) = exp(i K z) + 2 (A.8) d 3r  g(r, r  )V (r  )Ψ (r  ),  in which g(r, r  ) = −

exp(i K |r − r  |) 4π|r − r  |

(A.9)

is the Green function of the operator ∇ 2 + K 2 . As is known, this operator satisfies the equation (∇ 2 + K 2 )g(r, r  ) = δ(r − r  ),

(A.10)

where δ(r − r  ) is the Dirac delta function. In order to demonstrate that (A.8) follows from (A.7), let us apply the operator ∇ 2 + K 2 to the function Ψ (r) defined by the integral equation (A.8): (∇ 2 + K 2 )Ψ (r) = (∇ 2 + K 2 ) exp(i K z)  2m + 2 d 3r  (∇ 2 + K 2 )g(r, r  )V (r  )Ψ (r  ). 

(A.11)

The application of the operator ∇ 2 to the plane wave exp(i K z) gives ∇ 2 exp(i K z) =

∂2 exp(i K z) = −K 2 exp(i K z) ∂z 2

(A.12)

and, as a consequence, we can write (∇ 2 + K 2 ) exp(i K z) = 0. Therefore,

(A.13)

190

Appendix A: The First Born Approximation and the Rutherford Cross-Section

 2m d 3r  (∇ 2 + K 2 )g(r, r  )V (r  )Ψ (r  ) 2  2m = 2 d 3r  δ(r − r  )V (r  )Ψ (r  )  2m = 2 V (r)Ψ (r). 

(∇ 2 + K 2 )Ψ (r) =

(A.14)

For the boundary conditions (r → ∞), we have |r − r  | =



r 2 − 2r · r  + r 2  r 2 2ˆr · r  + 2 = r 1− r r

1 rˆ · r  . +O ∼ r 1− r r2

(A.15)

Note that in the last equation, rˆ =

r . r

(A.16)

Let us introduce K, the wave number in the direction of the outgoing unit vector rˆ , K ≡ K rˆ .

(A.17)

So the Green function for the operator ∇ 2 + K 2 , expressed by (A.9), has the following asymptotic behavior: g(r, r  ) ∼ − r →∞

exp(i K r − iK · r  ) . 4πr

(A.18)

Let us now introduce the asymptotic behavior of the Green function (A.18) into the integral equation (A.8):

Ψ (r) ∼ exp(i K z) − r →∞

2m 2



d 3r 

exp(i K r − iK · r  ) V (r  )Ψ (r  ). 4πr

(A.19)

From the equation 

exp(i K r − iK · r  ) V (r  )Ψ (r  ) 4πr  exp(−iK · r  ) exp(i K r ) V (r  )Ψ (r  ), = d 3r  r 4π d 3r 

(A.20)

Appendix A: The First Born Approximation and the Rutherford Cross-Section

we can conclude that, if the scattering amplitude is given by  m f (θ, φ) = − d 3r exp(−iK · r) V (r)Ψ (r), 2π2

191

(A.21)

then the boundary conditions are satisfied. In (A.21), K is the wave number of the scattered particles, and Ψ (r) is the scattering wave function. Let us suppose that the ratio between the electron kinetic energy and the atomic potential energy is high enough to make the scattering weak and Ψ (r) not very different from the incident plane wave exp(i K z). This is the assumption that is the basis of the first Born approximation, i.e., Ψ (r) = exp(i K z) = exp(i K · r).

(A.22)

Utilizing the first Born approximation, expressed by (A.22), the previous equation (A.21) becomes  m (A.23) f (θ, φ) = − d 3r exp(−iK · r) V (r) exp(i K · r). 2π2 If we use q to indicate the momentum lost by the incident electron, q = (K − K),

(A.24)

for fast particles, we can write that f (q) = f (θ, φ) = −

m 2π2

 d 3 r exp(i q · r) V (r).

(A.25)

As we are interested in a central potential, then V (r) = V (r ),

(A.26)

and, as a result, f (θ, φ) = f (θ)  ∞  2π  π m dϕ sin ϑ dϑ r 2 dr exp(iqr cos ϑ) V (r ). (A.27) = − 2π2 0 0 0 We carry out the integrations over ϕ and over ϑ and obtain  2m ∞ sin(qr ) V (r ) r dr. f (θ) = − 2 q 0

(A.28)

192

Appendix A: The First Born Approximation and the Rutherford Cross-Section

A.4 The Rutherford Formula Let us calculate the differential elastic scattering cross-sections in the first Born approximation for a screened Coulomb potential, such as a Wentzel-like potential [3], V (r ) = −

 r Z e2 exp − . r a

(A.29)

The exponential factor here represents a rough approximation of the screening of the nucleus by the orbital electrons, while the a parameter is a=

a0 , Z 1/3

where a0 = 2 /me2 is the Bohr radius. Let us calculate the scattering amplitude:   r 2m Z e2 ∞ dr. sin(qr ) exp − f (θ) = 2  q 0 a

(A.30)

(A.31)

As the equation 

 r q dr = 2 sin(qr ) exp − a q + (1/a)2

∞ 0

(A.32)

holds, we can conclude that dσel Z 2 e4 4m 2 = | f (θ)|2 = 4 . dΩ  [q 2 + (1/a)2 ]2

(A.33)

On the other hand, |K | = |K| and q = K − K, and, as a consequence q 2 = (K − K) · (K − K) = K 2 + K2 − 2K K cos θ = 2K 2 (1 − cos θ),

(A.34)

where θ is the scattering angle. The electron kinetic energy is given by E=

2 K 2 , 2m

(A.35)

Appendix A: The First Born Approximation and the Rutherford Cross-Section

193

so that the differential elastic scattering cross-section for the collision of an electron beam with a Wentzel-like atomic potential is given in the first Born approximation by 1 dσel Z 2 e4 . (A.36) = 2 dΩ 4E (1 − cos θ + α)2 In the last equation, α is the screening parameter, given by α=

1 me4 π 2 Z 2/3 . = 2K 2 a 2 h2 E

(A.37)

The well-known classical Rutherford formula, dσel Z 2 e4 1 = , 2 dΩ 4E (1 − cos θ)2 can be obtained by imposing α = 0 in (A.36). The total elastic scattering cross-section can be obtained from  dσ dΩ. σel = dΩ

(A.38)

(A.39)

For a Wentzel-like potential, the total elastic scattering cross-section can be easily calculated as follows:  π  Z 2 e4 2π 1 σel = dφ sin ϑ dϑ 2 4E 0 (1 − cos ϑ + α)2 0 2 4 1 πZ e . (A.40) = 2 E α(2 + α) If α → 0 and, as a consequence, the differential elastic scattering cross-section is given by the classical Rutherford formula, the total elastic scattering cross-section diverges, reflecting the long-range nature of the pure Coulomb potential. The elastic mean free path is the reciprocal of the total elastic scattering crosssection divided by the number N of atoms per unit volume in the target: λel =

1 α(2 + α)E 2 = . N σel N πe4 Z 2

(A.41)

A.5 Summary In this chapter, we have deduced the Rutherford formula by solving the Schrödinger equation in a central field in the first Born approximation.

194

Appendix A: The First Born Approximation and the Rutherford Cross-Section

References 1. J.E.G. Farina, The International Encyclopedia of Physical Chemistry and Chemical Physics, vol. 4 (Pergamon, Oxford, 1973) 2. D. Bohm, Quantum Theory (Dover, New York, 1989) 3. G. Wentzel, Z. Phys. 40, 590 (1927)

Appendix B

Partial Wave Expansion Method

The partial wave expansion method allows us to calculate the differential elastic scattering cross-section solving the Schrödinger equation in a central field [1] and avoiding the approximations required for calculating the screened Rutherford formula [2]. Please note that the partial wave expansion method is based on the non-relativistic quantum theory [3].

B.1 The Radial Equation The wave function describing the scattering process satisfies the Schrödinger equation   2 2 − ∇ + V Ψ = EΨ , 2m where V = V (r) is the atomic potential energy. It is convenient to introduce the quantity U (r) defined as U (r) =

2m V (r) . 2

(B.1)

Thus, Schrödinger equation can be rewritten in the following simpler form (∇ 2 + K 2 ) Ψ = U Ψ .

(B.2)

where K = mv/ is the electron momentum, v is the electron velocity, m is the electron mass, and  is the Planck constant divided by 2π. When the problem is spherically symmetric, the potential energy V and, consequently, the quantity U depend only on the distance from origin r , so that V = V (r ) © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1

195

196

Appendix B: Partial Wave Expansion Method

and U = U (r ). Since the wave function Ψ must be axially symmetric around the electron beam, the former does not depend on the azimuthal angle φ and, hence, ψ = ψ(r, cos θ). Legendre polynomials constitute a complete set, so we can expand Ψ as follows Ψ (r, cos θ) =

∞

Cl

l=0

Fl (r ) Pl (cos θ) . r

(B.3)

Here Cl are constants to be determined, Fl (r ) are functions of r to be determined, and Pl (cos θ) are the Legendre polynomials. Please note that the functions Fl (r ) are called partial waves and, in particular, F0 (r ) is known as the s-wave, F1 (r ) as the p-wave, and F2 (r ) as the d-wave. Let us write (∇ 2 + K 2 − U )

∞

Cl

l=0

Fl (r ) Pl (cos θ) = 0 . r

(B.4)

Let us now differentiate the sum term by term, so that ∞

(∇ 2 + K 2 − U ) Cl

l=0

Fl (r ) Pl (cos θ) = 0 . r

(B.5)

The Laplacian operator ∇ 2 is expressed, in spherical coordinates, by ∇2 =

1 ∂2 2 ∂ + 2 Λ, + 2 ∂r r ∂r r

(B.6)

where the Λ operator depends only on the angular variables θ (polar angle) and φ (azimuthal angle): Λ =

1 ∂2 cos θ ∂ ∂2 + + . ∂θ2 sin θ ∂θ sin2 θ ∂φ2

(B.7)

Therefore, ∞

l=0

Cl

1 ∂2 2 ∂ + 2 Λ + K2 − U + ∂r 2 r ∂r r



Fl (r ) Pl (cos ϑ) = 0 . r

Legendre polynomials satisfy the following differential equation   2 d 2 d + l(l + 1) Pl (u) = 0 (1 − u ) 2 − 2u du du

(B.8)

(B.9)

Appendix B: Partial Wave Expansion Method

197

and are eigenfunctions of operator Λ defined by (B.7) with eigenvalues −l(l + 1). In fact Λ Pl (cos θ) =

2 cos θ ∂ ∂ 1 ∂2 Pl (cos θ) = + = + ∂θ2 sin θ ∂θ sin2 θ ∂φ2 2

d cos θ d + = Pl (cos θ) = dθ2 sin θ dθ

d 1 d sin θ Pl (cos θ) = = sin θ dθ dθ

1 d 1 d 2 Pl (cos θ) . = sin θ sin θ dθ sin θ dθ Let us introduce the new variable u = cos θ. Since sin2 θ = 1 − u 2 and du = − sin θdθ, we have   d d (1 − u 2 ) Pl (u) , (B.10) ΛPl (u) = du du or, equivalently,  (1 − u 2 )

 d2 d − Λ Pl (u) = 0 . − 2u du 2 du

(B.11)

Comparing (B.9) with (B.11), we obtain Λ Pl (u) = −l(l + 1) Pl (u) .

(B.12)

So the eigenvalues of the operator Λ are −l(l + 1) with eigenvectors Pl . We can thus rewrite (B.8) as follows: ∞

l=0

 Cl

∂2 l(l + 1) 2 ∂ − + + K2 − U 2 ∂r r ∂r r2



Fl (r ) Pl (cos θ) = 0 . (B.13) r

Since the Legendre polynomials Pl (cos θ) are linearly independent, all the coefficients of the Legendre polynomials in this equation can be equated to zero  2  d l(l + 1) 2 d Fl (r ) 2 − = 0. (B.14) + + K − U dr 2 r dr r2 r

198

Appendix B: Partial Wave Expansion Method

From d 2 Fl (r ) = dr 2 r

d Fl (r ) 1 d Fl (r ) = − 2 + = dr r r dr 1 d 2 Fl (r ) 2 2 d Fl (r ) + = 3 Fl (r ) − 2 r r dr r dr 2 and 2 d Fl (r ) = r dr r 2 2 d Fl (r ) , = − 3 Fl (r ) + 2 r r dr we have

d2 2 d + 2 dr r dr



Fl (r ) 1 d 2 Fl (r ) . = r r dr 2

Thus (B.14) can be rewritten as follows:  2  d l(l + 1) 2 Fl (r ) = 0 . − − U (r ) + K dr 2 r2

(B.15)

Equation (B.15) involves only r , the radial distance from the center of scattering and it is known as the radial equation. Please note that, when r → ∞, U (r ) → 0 and the radial equation assumes the form

d2 + K2 dr 2

Fl (r ) = 0 .

(B.16)

In this case, the solutions are simply given by sinusoidal functions: Fl (r ) ∼ Al sin(K r + δl ) .

(B.17)

r →∞

Let us now define the phase shifts ηl ηl = δl +

πl , 2

(B.18)

Appendix B: Partial Wave Expansion Method

so that

199

πl + ηl . Fl (r ) ∼ Al sin K r − 2 r →∞

(B.19)

Since the Cl constants are not yet specified, we can choose Al = 1 for each l = 0 , . . . , ∞ and write

πl (B.20) + ηl . Fl (r ) ∼ sin K r − 2 r →∞ The case of a free particle corresponds to U (r ) = 0. In this case, the partial waves are indicated with the symbols Fl0 (r ), the phase shifts with the symbols ηl0 , and the radial equations (l = 0, . . . , ∞) are given by:  2  d l(l + 1) 2 Fl0 (r ) = 0 . − + K (B.21) dr 2 r2 The solutions of these equations are proportional to the regular spherical Bessel functions, jl (K r ), multiplied by K r . The constant of proportionality and the phase shifts ηl0 can be found considering the limit r → ∞

πl , (B.22) K r jl (K r ) ∼ sin K r − 2 r →∞ and comparing these functions with the partial waves Fl0 (r ) in the same limit

πl 0 0 (B.23) + ηl . Fl (r ) ∼ sin K r − 2 r →∞ The comparison demonstrates that, for each l = 0, . . . , ∞, the constants of proportionality are equal to 1 and ηl0 = 0. Thus, if the potential is null, the partial waves are simply given by the product of K r by the regular spherical Bessel functions jl (K r ): Fl0 (r ) = K r jl (K r ) .

(B.24)

B.2 Expansion of the Plane Wave Let us now expand the plane wave exp(i K z) on the complete set of Legendre polynomials: exp(i K r cos θ) = exp(i K z) =

∞

l=0

Cl0

Fl0 (r ) Pl (cos θ) . r

(B.25)

200

Appendix B: Partial Wave Expansion Method

Using (B.24), we obtain exp(i K z) =

∞

Cl0 K jl (K r ) Pl (cos θ) =

l=0

∞

cl jl (kr ) Pl (cos θ) ,

(B.26)

l=0

where cl = Cl0 K are constants to be determined. Introducing the two new variables s ≡ K r and u ≡ cos θ, we have exp(isu) =

∞

cl jl (s)Pl (u).

(B.27)

l=0

From the equation 

1 −1

Pm (u) Pn (u) du =

2 δmn , 2n + 1

(B.28)

we obtain 

+1

−1

∞

cn jn (s) Pn (u) Pl (u) du =

n=0

2 cl jl (s) , 2l + 1

(B.29)

and, as a consequence, 2 cl jl (s) = 2l + 1



+1

−1

exp(isu) Pl (u) du .

(B.30)

Let us integrate by parts, obtaining +1  +1 exp(isu)  exp(isu) Pl (u) Pl (u) du = − is is −1 −1 −1 exp(is) Pl (1) − exp(−is) Pl (−1) + h(s) , = is



+1



exp(isu) Pl (u) du =

where we have defined the function h(s) as  h(s) ≡ −

+1

−1

exp(isu)  Pl (u) du . is

Since Pl (1) = 1 and Pl (−1) = (−1)l , we have

2i l πl 2 jl (s) cl = sin s − + h(s) . 2l + 1 s 2

(B.31)

Appendix B: Partial Wave Expansion Method

201

As s → ∞, by repeated integrations by parts we see that h(s) → 0 faster than 1/s. As s → ∞ we also have

πl 1 sin s − . (B.32) jl (s) ∼ s 2 Comparing the last two equations in the limit s → ∞ we therefore obtain cl = (2l + 1) i l ,

(B.33)

so that

exp(i K r cos θ) = exp(i K z) =

∞

(2l + 1) i l jl (K r ) Pl (cos θ) .

(B.34)

l=0

B.3 Scattering Amplitude When r → ∞ the plane wave can be expressed as exp(i K z) ∼ r →∞



∞

πl (2l + 1) i l sin K r − Pl (cos θ) . Kr 2 l=0

(B.35)

In the same limit, (B.3) assumes the form: ∞

Ψ (r, cos θ) ∼

r →∞ l=0

Cl

πl 1 sin K r − + ηl Pl (cos θ) , r 2

(B.36)

where we have taken into account (B.20). Therefore, introduced two functions F(θ) and G(θ) defined as

F(θ) =

∞

l=0

G(θ) =

∞

l=0

 Pl (cos θ)

 

 (2l + 1)i l πl πl Cl − exp i − + ηl exp −i , 2i 2 2i K 2 (B.37)

 

  (2l + 1)i l πl πl Cl + exp i − ηl exp i , Pl (cos θ) − 2i 2 2i K 2 (B.38)

202

Appendix B: Partial Wave Expansion Method

we can write exp(i K r ) exp(−i K r ) F(θ) + G(θ) , r r r →∞

Ψ (r, cos θ) − exp(i K z) ∼

(B.39)

On the other hand, at a large distance from the atomic nucleus where the potential V (r ) is negligible, the following boundary condition has to be satisfied (see Appendix A): Ψ (r, cos θ) − exp(i K z) ∼ r →∞

exp(i K r ) f (θ) . r

(B.40)

Comparing (B.39) with (B.40) we conclude, in particular, that G(ϑ) = 0 . Therefore, from (B.38), we obtain



 (2l + 1)i l Cl πl πl = . exp i − ηl exp i 2i 2 2i K 2

(B.41)

(B.42)

We can thus express the coefficients Cl as a function of the phase shifts ηl , for l = 0, . . . , ∞: Cl =

(2l + 1)i l exp(iηl ) . K

(B.43)

Another consequence of the comparison between (B.39) and (B.40) is that F(θ) = f (θ) .

(B.44)

Consequently, taking into account (B.43), the scattering amplitude can be expressed as the following expansion on the complete set of Legendre’s polynomials: ∞ 1 (2l + 1) [exp(2iηl − 1)] Pl (cos θ) , f (θ) = 2i K l=0

(B.45)

∞ 1 (2l + 1) exp(iηl ) sin ηl Pl (cos θ) . f (θ) = K l=0

(B.46)

or

Appendix B: Partial Wave Expansion Method

203

B.4 Differential and Total Elastic Scattering Cross-Section. Optical Theorem The differential elastic scattering cross-section dσel /dΩ is the squared modulus of the scattering amplitude: dσel = | f (θ)|2 = dΩ ∞ ∞ 1

= 2 (2m + 1) (2n + 1) × K m=0 n=0 × exp[i(ηm − ηn )] sin ηm sin ηn Pm (cos θ) Pn (cos θ) . The total elastic scattering cross-section,  dσel dΩ , σel = dΩ

(B.47)

(B.48)

is thus given by σel =

∞ 4π (2l + 1) sin2 ηl . K 2 l=0

(B.49)

Please note that the imaginary part of the forward scattering amplitude f (0) is given by Im f (0) =

∞ 1 (2l + 1) sin2 ηl , K l=0

(B.50)

so that σel =

4π Im f (0) . K

(B.51)

This result is known as the “optical theorem”.

B.5 Summary In this chapter, we described the partial wave expansion method that allows us to calculate the electron elastic scattering cross-section by solving the Schrödinger equation in a central field.

204

Appendix B: Partial Wave Expansion Method

References 1. J.E.G. Farina, The International Encyclopedia of Physical Chemistry and Chemical Physics, vol. 4 (Pergamon, Oxford, 1973) 2. D. Bohm, Quantum Theory (Dover, New York, 1989) 3. M. Dapor, Electron-Atom Collisions. Quantum Relativistic Theory and Exercises (de Gruyter, Berlin, 2022)

Appendix C

The Mott Theory

The original version of the Mott theory (also known as the relativistic partial wave expansion method, RPWEM) can be found in [1]. See, for details and several applications [2–6]. The elastic scattering process can be described by calculating the phase shifts. If we indicate with r the radial coordinate, since the large-r asymptotic behavior of the radial wave function is known, the phase shifts can be computed by solving the Dirac equation for a central electrostatic field up to a large radius where the atomic potential can be neglected.

C.1 The Dirac Equation in a Central Potential In order to appropriately treat the quantum-relativistic scattering theory, we need to know the form assumed by the Dirac equation for an electron in the presence of a central electrostatic field described by a central potential energy eϕ(r ) = V (r ). The natural units  = c = 1 are used here as they are particularly convenient for the quantum-relativistic equations. Let us first remind our readers that the Dirac hamiltonian describing a spin-1/2 particle in the presence of an electromagnetic field described by the four-potential (ϕ, A) is H = eϕ + α · ( p − e A) + βm, where the 4 × 4 matrices α and β are given by

0 σj , αj = σj 0 β=

I 0 0 −I

(C.1)

(C.2)

.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1

(C.3) 205

206

Appendix C: The Mott Theory

Note that (σx , σ y , σz ) = (σ 1 , σ 2 , σ 3 ) are the 2 × 2 Pauli matrices and I the 2 × 2 identity matrix. Let us now introduce the operator K defined by K ≡ β(1 + σ · L),

(C.4)

where L is the electron orbital angular momentum. For an electron in a central electrostatic field, it is possible to show that1 2

dσ dL =− . dt dt

(C.5)

As a consequence, the total angular momentum, defined as J = L + (1/2)σ, is a constant of the motion. On the other hand, 3 J 2 − L2 = σ · L + , 4

(C.6)

K = β( J 2 − L 2 + 1/4).

(C.7)

so we can conclude that

K commutes with H and is, as a consequence, a constant of the motion. 1

For a central electrostatic field, we have, V (r) = V (r ) = eϕ(r ),

and H = α · p + β m + V (r ). From ∇V (r ) =

∂V (r ) r ∂r r

it follows that r × ∇V (r ) = 0 and, hence, [V (r ), r × p] = 0. As a consequence, dL = i[H, L] = α × p. dt On the other hand, it is also easy to see that dσ = i[H, σ] = −2(α × p), dt so that −

1 dσ dL = . 2 dt dt

Appendix C: The Mott Theory

207

Let us now define the radial-momentum operator pr , where pr ≡ −i

r · p−i 1 ∂ r= , r ∂r r

(C.8)

and introduce the radial component αr of the α operator, where αr =

α·r . r

(C.9)

This obeys the relation αr2 = 1.

(C.10)

For any pair of vectors a and b, the following equations hold: (σ · a)(σ · b) = a · b + iσ · a × b

(C.11)

(α · a)(α · b) = (σ · a)(σ · b).

(C.12)

(α · r)(α · p) = r pr + iβK.

(C.13)

and

As a consequence,

This last equation is equivalent to (α · p) = αr

pr +

iβK . r

(C.14)

As a result, the Dirac equation [α · p + βm + V (r )] Ψ = E Ψ, can be rewritten as the following: 

 iβK αr pr + + βm + V (r ) Ψ = E Ψ. r

(C.15)

(C.16)

The operators β, K, L 2 and Jz are mutually commutative. In the following, X indicates an eigenvector common to those operators, so that βX = X ,

(C.17)

KX = −kX ,

(C.18)

208

Appendix C: The Mott Theory

L 2 X = l(l + 1)X ,

(C.19)

Jz X = m j X .

(C.20)

Y = −αr X ,

(C.21)

Introducing the function Y, where

we can observe that it has the following properties: X = −αr Y,

(C.22)

KY = −kY,

(C.23)

βY = αr βX = −Y.

(C.24)

Let us now consider the following linear combination of X and Y: Ψ = F(r )Y + i G(r )X ,

(C.25)

which is an eigenvector common to H , K and Jz and thus the spinor we are looking for. Our objective is to determine the functions F(r ) and G(r ). The eigenvalues of L 2 are ( j ± 1/2)( j ± 1/2 + 1). As K = β( J 2 − L 2 + 1/4), we have, for the case j = l + 1/2 (spin up),

1 = −(l + 1). (C.26) k=− j+ 2 In the other case, where j = l − 1/2 (spin down),

1 = l. k= j+ 2

(C.27)

We are now able to find the equations corresponding to the radial behavior of the functions F and G. In order to do this, let us consider the Dirac equation (C.16) and observe the following:   d F(r ) F(r ) X, (C.28) αr pr F(r )Y = i + dr r  iαr pr G(r )X = −

 dG(r ) G(r ) + Y, dr r

(C.29)

Appendix C: The Mott Theory

209

i iαr βK F(r )Y = − F(r )kX , r r

(C.30)

iαr βK 1 i G(r )X = − G(r )kY, r r

(C.31)

βm F(r )Y = −m F(r )Y,

(C.32)

βmi G(r )X = imG(r )X .

(C.33)

Hence, since X and Y belong to different eigenvalues of β and are, consequently, linearly independent,2 the fundamental equations of the theory of the elastic scattering of electrons (and positrons) by atoms are [E + m − V (r )]F(r ) +

dG(r ) 1 + k + G(r ) = 0, dr r

− [E − m − V (r )]G(r ) +

d F(r ) 1 − k + F(r ) = 0. dr r

(C.34) (C.35)

C.2 Relativistic Partial Wave Expansion Method The fundamental equation of relativistic quantum mechanics is the Dirac equation. The wave function, as is well known, is a four-component spinor, and the asymptotic forms of the four components of the scattered wave are Ψi ∼ ai exp(i K z) + bi (θ, φ) r →∞

exp(i K r ) , r

(C.36)

where K = mv/ is the electron momentum in the z direction. The differential elastic scattering cross-section is given by |b1 |2 + |b2 |2 + |b3 |2 + |b4 |2 dσel = dΩ |a1 |2 + |a2 |2 + |a3 |2 + |a4 |2 |b1 |2 + |b2 |2 + c|b1 |2 + c|b2 |2 = |a1 |2 + |a2 |2 + c|a1 |2 + c|a2 |2 |b1 |2 + |b2 |2 = , |a1 |2 + |a2 |2 2

(C.37)

Please note that this is a general theorem. Let us consider a linear operator O with eigenvalues a and b. Indicating with x the eigenvector of a and with y the eigenvector of b, this means that O x = ax and O y = by. Suppose that a = b. Let us now assume that x and y are linearly dependent so that y = cx for some scalar c. Thus O(cx) = bcx, or cO x = cbx. This means that O x = bx and so a = b. This contradicts the assumption that a and b are different.

210

Appendix C: The Mott Theory

where c is a constant of proportionality which takes into account the fact that the ai and the bi coefficients are not all independent. Indeed, asymptotically the scattered wave is made up of plane waves proceeding, from the center, in various directions. The coefficients of the solutions for a plane wave are not all independent. If the spin is parallel to the direction of incidence (spin up), a1 = 1, a2 = 0, b1 = f + (θ, φ), b2 = g + (θ, φ), where f + and g + are two scattering amplitudes. The asymptotic behavior is described by the following equations: Ψ1 ∼ exp(i K z) + f + (θ, φ) r →∞

Ψ2 ∼ g + (θ, φ) r →∞

exp(i K r ) , r

exp(i K r ) . r

(C.38)

(C.39)

The case of spin antiparallel to the direction of incidence (spin down) corresponds to a1 = 0, a2 = 1, b1 = g − (θ, φ), b2 = f − (θ, φ). The asymptotic behavior is now given by Ψ1 ∼ g − (θ, φ) r →∞

exp(i K r ) , r

Ψ2 ∼ exp(i K z) + f − (θ, φ) r →∞

exp(i K r ) . r

(C.40)

(C.41)

The Dirac equations for an electron in a central field are given by the following (see previous section): [E + m − V (r )]Fl± (r ) +

− [E − m − V (r )]G l± (r ) +

dG l± (r ) 1 + k ± + G l (r ) = 0, dr r

d Fl± (r ) 1 − k ± + Fl (r ) = 0. dr r

(C.42)

(C.43)

The superscript “+” refers to the electrons with spin up (k = −(l + 1)) while “−” refers to electrons with spin down (k = l). Once the new variables μ(r ) ≡ E + m − V (r ),

(C.44)

ν(r ) ≡ E − m − V (r ),

(C.45)

Appendix C: The Mott Theory

211

have been introduced, and indicated with μ the derivative of μ(r ) μ =

dμ , dr

(C.46)

the Dirac equations become Fl± (r )

1 =− μ



dG l± 1+k ± + Gl dr r

(C.47)

and

d Fl± μ dG l± 1+k ± = 2 + Gl dr μ dr r

1 d 2 G l± 1+k ± 1 + k dG l± − . − + G l μ dr 2 r dr r2

(C.48)

Therefore, after simple algebraic manipulations, we obtain the following: d 2 G l± + dr 2



2 μ − r μ



dG l± k(k + 1) 1 + k μ G l± = 0. − + μν − dr r2 r μ

(C.49)

Let us now introduce the function Gl± , where r G±. μ1/2 l

(C.50)

K 2 = E 2 − m2,

(C.51)

μν = K 2 − 2E V + V 2 .

(C.52)

Gl± ≡ Upon observing that

it is possible to see that

We conclude that, once the function Ul± (r ) (effective Dirac potential) has been defined, i.e., − Ul± (r ) = −2E V + V 2 −

1 μ 3 μ2 k μ + − , r μ 2 μ 4 μ2

(C.53)

the following equation holds: 

 d2 k(k + 1) ± 2 − + K − Ul (r ) Gl± = 0. dr 2 r2

(C.54)

212

Appendix C: The Mott Theory

For large values of r , Gl± is essentially sinusoidal. Indeed, when r is large enough, V (r ) and Ul± are negligible and the solution of the equation is therefore a linear combination of the regular and irregular spherical Bessel functions multiplied by K r . Taking account the fact that Gl± = (r/μ1/2 ) G l± , we can therefore conclude that G l± ∼ jl (K r ) cos ηl± − n l (K r ) sin ηl± .

(C.55)

r →∞

Here ηl± (phase shifts) are constants to be determined. Taking into account the asymptotic behavior of the Bessel functions,

1 lπ , (C.56) jl (K r ) ∼ sin K r − Kr 2 r →∞

lπ 1 n l (K r ) ∼ − cos K r − , Kr 2 r →∞

(C.57)

we can conclude that

G l±





1 1 lπ lπ ± ∼ sin K r − cos ηl + cos K r − sin ηl± . K r 2 K r 2 r →∞

(C.58)

Therefore,

lπ 1 sin K r − + ηl+ , Kr 2

(C.59)

lπ 1 − sin K r − + ηl . ∼ Kr 2 r →∞

(C.60)

G l+ ∼ r →∞

and G l−

The phase shifts ηl± represent the effect of the potential V (r ) on the phases of the scattered waves. In Appendix B we demonstrated the following equation: exp(i K r cos θ) =

∞

(2l + 1)i l jl (K r )Pl (cos θ) ,

(C.61)

l=0

where Pl (cos θ) are the Legendre polynomials and jl (K r ) the spherical Bessel functions. Equation (C.61) can be also demonstrated according to the considerations

Appendix C: The Mott Theory

213

below. We know that a plane wave describing a free particle with the z axis in the direction of K may be expressed as an expansion in a series of Legendre polynomials Pl (cos θ): exp(i K z) = exp(i K r cos θ) =

∞

cl jl (K r )Pl (cos θ).

(C.62)

l=0

Let us define the two new variables s ≡ K r and t ≡ cos θ, to have

cl jl (s)Pl (t). exp(ist) =

(C.63)

l

Differentiating with respect to s, we obtain it exp(ist) =

itcl jl (s)Pl (t) =

l

l

cl

d jl (s) Pl (t). ds

(C.64)

Recalling the properties of the Legendre polynomials, we have: Pl (t) =

(l + 1)Pl+1 (t) + l Pl−1 (t) . t (2l + 1)

(C.65)

Therefore,

it exp(ist) =

itcl jl (s)

l

=

l

 i Pl (t)

(l + 1)Pl+1 (t) + l Pl−1 (t) t (2l + 1)

 l l +1 cl−1 jl−1 (s) + cl+1 jl+1 (s) . (C.66) 2l − 1 2l + 3

On the other hand, it is well known that l d jl (s) l +1 = jl−1 (s) − jl+1 (s), ds 2l + 1 2l + 1

(C.67)

and as a consequence, it exp(ist) =

l



 l l +1 jl−1 (s) − jl+1 (s) . cl Pl (t) 2l + 1 2l + 1

Consequently, from (C.66) and (C.68), the following is obtained:

(C.68)

214

Appendix C: The Mott Theory

icl−1 cl − 2l + 1 2l − 1 l

 icl+1 cl + = 0. − jl+1 (s)(l + 1) 2l + 1 2l + 3

 Pl (t) jl−1 (s)l

(C.69)

The Legendre polynomials Pl (t) are linearly independent and therefore

icl−1 cl − jl−1 (s) l 2l + 1 2l − 1

cl icl+1 = jl+1 (s) (l + 1) + . 2l + 1 2l + 3

(C.70)

Every value of s satisfies the last equation if and only if i 1 cl = cl−1 . 2l + 1 2l − 1

(C.71)

In order to obtain an explicit expression for cl we need to know the value of the first coefficient of the set, i.e., c0 . Imposing r = 0 in (C.62), we obtain

cl jl (0)Pl (cos θ). (C.72) exp(0) = 1 = l

Since jl (0) = 0 for any l = 0, while j0 (0) = 1 and P0 (cos θ) = 1, we may conclude that c0 = 1. The recursive repetition of the relation (C.71) allows us to obtain the values of the coefficients, cl = (2l + 1)i l ,

(C.73)

and the expansion of the plane wave in Legendre polynomials, exp(i K r cos θ) = exp(i K z) =

∞

(2l + 1)i l jl (K r )Pl (cos θ).

(C.74)

l=0

Let us remind our readers that we are looking for functions Ψ1 and Ψ2 which satisfy the asymptotic conditions. So, we must begin by expanding them in spherical harmonics: Ψ1 =

∞

[Al G l+ + Bl G l− ]Pl (cos θ),

(C.75)

l=0

Ψ2 =

∞

[Cl G l+ + Dl G l− ]Pl1 (cos θ) exp(iφ), l=1

(C.76)

Appendix C: The Mott Theory

215

where Pl1 (cos θ) are the associated Legendre polynomials Pl1 (x) = (1 − x 2 )1/2

d Pl (x) . dx

(C.77)

The coefficients Al , Bl , Cl and Dl can be determined by considering the asymptotic behaviors of the functions involved. Let us begin with the function Ψ1 and observe that Ψ1 − exp(i K z) =

∞

[Al G l+ + Bl G l− − (2l + 1)i l jl (K r )]Pl (cos θ). l=0

(C.78) As exp(i K r ) + f (θ, φ), r r →∞

Ψ1 − exp(i K z) ∼

(C.79)

the following occurs:



∞  lπ lπ 1 + − Al sin K r − + ηl + Bl sin K r − + ηl K r l=0 2 2

 lπ l Pl (cos θ) −(2l + 1)i sin K r − 2 exp(i K r ) + f (θ, φ). = r

(C.80)

As a consequence,

∞ exp(i K r ) lπ exp −i 2i K r l=0 2 × [Al exp(iηl+ ) + Bl exp(iηl− ) − (2l + 1)i l ]Pl (cos θ)

∞ exp(−i K r ) lπ − exp i 2i K r 2 l=0 × [Al exp(−iηl+ ) + Bl exp(−iηl− ) − (2l + 1)i l ]Pl (cos θ) exp(i K r ) + f (θ, φ). = r

(C.81)

216

Appendix C: The Mott Theory

The asymptotic conditions are satisfied if Al exp(−iηl+ ) + Bl exp(−iηl− ) = (2l + 1)i l .

(C.82)

Al = (l + 1)i l exp(iηl+ ),

(C.83)

Bl = li l exp(iηl− ),

(C.84)

With the choices

Equation (C.82) is satisfied. Proceeding in a similar way for the Ψ2 function, we obtain (C.85) Cl exp(−iηl+ ) + Dl exp(−iηl− ) = 0 , so that we can choose Cl = −i l exp(iηl+ ),

(C.86)

Dl = i l exp(iηl− ),

(C.87)

In conclusion, for electrons with spins parallel to the direction of incidence, we have Ψ1 =

∞

[(l + 1) exp(iηl+ )G l+ + l exp(iηl− )G l− ]i l Pl (cos θ),

(C.88)

∞

[exp(iηl− )G l− − exp(iηl+ )G l+ ]i l Pl1 (cos θ) exp(iφ),

(C.89)

l=0

Ψ2 =

l=1

and, by using (C.81), f + (θ, φ) = f + (θ) ∞ 1 = {(l + 1)[exp(2iηl+ ) − 1] 2i K l=0 +l[exp(2iηl− ) − 1]}Pl (cos θ), ∞ 1 g (θ, φ) = [exp(2iηl− ) − exp(2iηl+ )]Pl1 (cos θ) exp(iφ). 2i K l=1 +

(C.90) (C.91)

Appendix C: The Mott Theory

217

For electrons with spins antiparallel to the direction of incidence (spin down), where we indicate the scattering amplitudes by f − and g − , we can see that f − (θ, φ) = f + (θ, φ)

(C.92)

g − (θ, φ) = −g + (θ, φ) exp(−2iφ).

(C.93)

and

It is therefore convenient to define the functions f (θ) =

∞

Al Pl (cos θ),

(C.94)

Bl Pl1 (cos θ),

(C.95)

l=0

g(θ) =

∞

l=0

where Al =

1 {(l + 1)[exp(2iηl+ ) − 1] + l[exp(2iηl− ) − 1]}, 2i K

(C.96)

1 {exp(2iηl− ) − exp(2iηl+ )}. 2i K

(C.97)

Bl = With this notation, we have

f + = f − = f,

(C.98)

g + = g exp(iφ)

(C.99)

g − = −g exp(−iφ).

(C.100)

and

For an arbitrary spin direction, the electron incident plane wave will be given by Ψ1 = A exp(i K z) and Ψ2 = B exp(i K z), and as a consequence a1 = A, a2 = B. Furthermore, b1 = A f + + Bg − = A f − Bg exp(−iφ),

(C.101)

b2 = Ag + + B f − = B f + Ag exp(iφ).

(C.102)

218

Appendix C: The Mott Theory

Consequently, dσel dΩ

   AB ∗ exp(iφ) − A∗ B exp(−iφ) , (C.103) = (| f | + |g| ) 1 + i S(θ) |A|2 + |B|2 2

2

where S(θ) is the Sherman function, defined by S(θ) = i

f g∗ − f ∗ g . | f |2 + |g|2

(C.104)

Note that i

AB ∗ exp(iφ) − A∗ B exp(−iφ) = ξ † (σ y cos φ − σx sin φ)ξ, |A|2 + |B|2

(C.105)

where σx , σ y and σz are the Pauli matrices and ξ is the two-component spinor 

A/|A|2 + |B|2 ξ= , (C.106) B/ |A|2 + |B|2  ξ = †



A∗ |A|2 + |B|2



B∗



|A|2 + |B|2

.

(C.107)

As the z axis has been chosen along the incidence direction, the unit vector perpendicular to the plane of scattering is given by ⎛ ⎞ − sin φ nˆ = ⎝ cos φ ⎠ , (C.108) 0 so we can write ˆ ξ † (σ y cos φ − σx sin φ)ξ = P · n,

(C.109)

where P is the initial polarization vector of the electron beam. The differential elastic scattering cross-section can then be recast as follows: dσel ˆ = (| f |2 + |g|2 )[1 + S(θ) P · n]. dΩ

(C.110)

Note that, if the beam is completely unpolarised, then P = 0 and dσel = | f |2 + |g|2 . dΩ

(C.111)

Appendix C: The Mott Theory

219

The total elastic scattering cross-section (σel ) and the transport cross-section (σtr ) are defined by  π dσel sin θ dθ, (C.112) σel = 2 π 0 dΩ 

π

σtr = 2 π

(1 − cos θ)

0

dσel sin θ dθ, dΩ

(C.113)

and can be easily calculated by numerical integration. Note that by assigning ηl− = ηl+ = ηl

(C.114)

in the previous equations, we can obtain the non-relativistic results. Indeed, 1 {(l + 1)[exp(2iηl ) − 1] + l[exp(2iηl ) − 1]} 2i K 1 (2l + 1)[exp(2iηl ) − 1], = 2i K

Al =

Bl = 0,

(C.115) (C.116)

so that f (θ) = =

∞ 1 (2l + 1)[exp(2iηl ) − 1]Pl (cos θ) 2i K l=0 ∞ 1 (2l + 1) exp(iηl ) sin ηl Pl (cos θ), K l=0

(C.117)

g(θ) = 0 ,

(C.118)

dσ = | f |2 . dΩ

(C.119)

and

220

Appendix C: The Mott Theory

C.3 Phase Shift Calculations In order to proceed, it is useful to perform the following transformation (Lin, Sherman, and Percus transformation) [7]: Fl± (r ) = al± (r )

sin φl± (r ) , r

(C.120)

G l± (r ) = al± (r )

cos φl± (r ) . r

(C.121)

After simple algebraic manipulations, (C.42) and (C.43) become [E + m − V (r )] tan φl± (r ) + − tan φl± (r )

1 al± (r )

dφl± (r ) k + = 0, dr r

−[E − m − V (r )] cot φl± (r ) + + cot φl± (r )

dal± (r ) dr (C.122)

1 dal± (r ) al± (r ) dr

dφl± (r ) k − = 0, dr r

(C.123)

and therefore dφl± (r ) k = sin 2φl± (r ) − m cos 2φl± (r ) + E − V (r ), dr r k 1 dal± (r ) = − cos 2φl± (r ) − m sin 2φl± (r ). ± r al (r ) dr

(C.124)

(C.125)

For 0 < r < /mc, the spherical symmetric electrostatic potential experienced by a point charge at distance r from the nucleus, V (r ), may be approximated by the following: V (r ) ∼ − r →0

Z 0 + Z 1r + Z 2r 2 + Z 3r 3 . r

(C.126)

Expressing the electrostatic potential as the product of the potential of a bare nucleus multiplied by a screening function ψ(r ) having the analytical form ψ(r ) =

p

i=1

Ai exp(−αi r ),

(C.127)

Appendix C: The Mott Theory

221 p

Ai = 1,

(C.128)

i=1

we can easily evaluate Z 0 , Z 1 , Z 2 and Z 3 : Z 0 = Z e2

p

Ai = Z e2 ,

(C.129)

i=1

Z 1 = −Z 0

p

αi Ai ,

(C.130)

i=1

Z2 =

p Z0 2 α Ai , 2 i=1 i

Z3 = −

p Z0 3 α Ai . 6 i=1 i

(C.131)

(C.132)

Let us expand φl± as a power series ± ± ± 2 ± 3 + φl1 r + φl2 r + φl3 r + ··· . φl± (r ) = φl0

(C.133)

As we can see see, after simple algebraic manipulations, the relationships between the coefficients of this expansion and Z 0 , Z 1 , Z 2 and Z 3 are the following [9]: ± =− sin 2φl0

± φl3 =

(C.134)

± E + Z 1 − m cos 2φl0 , ± 1 − 2k cos 2φl0

(C.135)

± ± ± sin 2φl0 (m − kφl1 ) + Z2 2φl1 , ± 2 − 2k cos 2φl0

(C.136)

± φl1 =

± φl2 =

Z0 , k

± ± ± ±2 ± ± sin 2φl0 (m − 2kφl1 ) + 2φl1 cos 2φl0 [m − (2/3)kφl1 ] + Z3 2φl2 , ± 3 − 2k cos 2φl0 (C.137)

with the extra conditions ± ≤ 0 ≤ 2φl0

1 π 2

(C.138)

222

Appendix C: The Mott Theory

if k < 0, and ± π ≤ 2φl0 ≤

3 π 2

(C.139)

if k > 0. Let us now calculate the phase shifts, examining (C.121):

G l± =

al± cos φl± (r ) al± a ± cos φl± (r ) − sin φl± (r )φl± (r ) − l , r r r2

(C.140)

G l± al± 1 = − φl± (r ) tan φl± (r ) − , r G l± al±

(C.141)

G l± 1+k . = −(E + m − V ) tan φl± (r ) − r G l±

(C.142)

so that

or

Let us observe that the asymptotic form of the solution in the regions corresponding to large values of r for which V (r ) ≈ 0 is (see (C.55)) G l± = jl (K r ) cos ηl± − n l (K r ) sin ηl± ,

(C.143)

where K 2 = E 2 − m 2 , ηl± are the lth phase shifts, and jl and n l are respectively the regular and irregular spherical Bessel functions. Therefore, G l± K jl (K r ) cos ηl± − K n l (K r ) sin ηl± = . G l± jl (K r ) cos ηl± − n l (K r ) sin ηl±

(C.144)

Taking into account the properties of the Bessel functions jl (x) =

l jl (x) − jl+1 (x), x

(C.145)

n l (x) =

l n l (x) − n l+1 (x), x

(C.146)

we can conclude that tan ηl± =

(l/r ) jl (K r ) − K jl+1 (K r ) − jl (K r )(G l± /G l± ) . (l/r )n l (K r ) − K n l+1 (K r ) − n l (K r )(G l± /G l± )

(C.147)

Appendix C: The Mott Theory

223

Let us define φ˜ l± = lim φl± (r ).

(C.148)

r →∞

For large values of r , (C.142) becomes G l± ˜± 1 + k , ± = −(E + m) tan φl − r Gl

(C.149)

and therefore

tan ηl± =

K jl+1 (K r ) − jl (K r )[(E + m) tan φ˜ l± + (1 + l + k)/r ] . (C.150) K n l+1 (K r ) − n l (K r )[(E + m) tan φ˜ ± + (1 + l + k)/r ] l

Using this last equation, we can calculate the phase shifts of the scattered wave and, therefore, the functions f (θ), g(θ), and the differential elastic scattering crosssection.

C.4 Analytical Approximation of the Mott Cross-Section Taking advantage of the simple closed form of the equations deduced starting by the Rutherford cross-section, it is sometimes convenient to look for an approximation where similar equations are used for the Mott cross-section [9, 10]. For low atomic number elements, the Mott differential elastic scattering cross-section can be roughly approximated by the following equation: B dσel = , dΩ (1 − cos θ + A)2

(C.151)

where the unknown parameters A and B are calculated with the aim to get the best fit of the total and the first transport elastic scattering cross-section previously calculated using the relativistic partial wave expansion method.3

3

Please note that, with A =

me4 π 2 Z 2/3 h2 E

and Z 2 e2 4E 2 Equation (C.151) becomes the screened Rutherford formula. B =

224

Appendix C: The Mott Theory

Once we know the total elastic scattering cross-section and the transport elastic scattering cross-section previously calculated using the Mott theory, we can calculate A and B in order to approximate the Mott theory [10]. From (C.112), it follows that: σel =

4π B . A(A + 2)

(C.152)

As a consequence the differential elastic scattering cross-section can be rewritten as: σel A(A + 2) dσel = dΩ 4 π (1 − cos θ + A)2

(C.153)

Using (C.113) and (C.153) it is possible to obtain the ratio Ξ between the transport and the total elastic scattering cross-sections: 

 A+2 A+2 σtr ln = A Ξ ≡ −1 . (C.154) σel 2 A Once the values of the total and transport elastic scattering cross-sections have been numerically calculated using the relativistic partial wave expansion method, the ratio Ξ is determined as a function of electron kinetic energy E. In such a way it is possible to get the screening parameter A as a function of E (using a bisection algorithm).

C.5 Positron Differential Elastic Scattering Cross-Section The electron differential elastic scattering cross-section shows diffraction-like structures. It is a typical quantum-mechanical phenomenon due to the interaction of the incident electron with the atomic electron cloud. On the contrary, the positron elastic scattering cross-section is a decreasing and monotonic Rutherford-like function [11, 12]. This is due to the different sign of the electric charges of electrons and positrons. Positrons are rejected by the atomic nucleus. As a consequence, they do not penetrate the atomic electron cloud as deeply as electrons do. Since electrons penetrate deeper into the core of atomic electrons, they draw closer to the nucleus than positrons do. Thus, electrons can loop around the nucleus once or more times, and electron outgoing wave—a superposition of the incoming and of the scattered wave—exhibits interference effects. This is just a semi-classical description that qualitatively explains the reason why the atomic electron cloud influences electrons more than positrons.

Appendix C: The Mott Theory

225

C.6 Summary The Mott theory [1] (also known as the relativistic partial wave expansion method) was described in this chapter. It allows to calculate the electron elastic scattering cross-section. Differences among elastic scattering cross-sections of electrons and positrons were discussed.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

N.F. Mott, Proc. R. Soc. Lond. Ser. 124, 425 (1929) A. Jablonski, F. Salvat, C.J. Powell, J. Phys. Chem. Data 33, 409 (2004) F. Salvat, A. Jablonski, C.J. Powell, Comp. Phys. Comm. 165, 157 (2005) M. Dapor, J. Appl. Phys. 79, 8406 (1996) M. Dapor, Electron-Beam Interactions with Solids: Application of the Monte Carlo Method to Electron Scattering Problems (Springer, Berlin, 2003) M. Dapor, Electron-Atom Collisions. Quantum Relativistic Theory and Exercises (de Gruyter, Berlin, 2022) S.-R. Lin, N. Sherman, J.K. Percus, Nucl. Phys. 45, 492 (1963) P.J. Bunyan, J.L. Schonfelder, Proc. Phys. Soc. 85, 455 (1965) J. Baró, J. Sempau, J.M. Fernández-Varea, F. Salvat, Nucl. Instrum. Methods Phys. Res. B 84, 465 (1994) M. Dapor, Phys. Lett. A 333, 457 (2004) M. Dapor, A. Miotello, At. Data Nucl. Data Tables 69, 1 (1998) M. Dapor, J. Electron Spectrosc. Relat. Phenom. 151, 182 (2006)

Appendix D

The Atomic Potential

Since phase shifts depend on the atomic potential, we need to calculate the atomic potential in order to compute the phase shifts and, from here, the differential elastic scattering cross-section. The atomic potential can be approximated as a superimposition of Yukawa potentials which depend on a number of parameters. Such parameters have been determined by analytical fitting of self-consistent fields. Cox and Bonham [1] proposed an analytical approximation of the wavefunctions calculated by Clementi [2] for Z = 1 − 54, while Salvat et al. [3], gave an analytical approximation of their numerical calculation of the atomic potential for Z = 1 − 92.

D.1 The Atomic Screening Function The atomic screening function ψ(r ) is defined as the ratio between the electrostatic potential experienced by a point charge at a distance r from the nucleus and the electrostatic potential of the bare nucleus (assuming spherical symmetry). The screening functions of Cox and Bonham and of Salvat et al. are given by: ψ(r ) =

p

Ai exp(−αi r ) ,

(D.1)

i=1 p

Ai = 1 .

(D.2)

n=1

where p, Ai and αi depend on the element and on the author. Note that such equations have the analytical form originally proposed by Molière in order to approximate the Thomas-Fermi screening function [4].

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1

227

228

Appendix D: The Atomic Potential

D.2 Electron Exchange As electrons are identical particles, exchange effects have to be taken into account, since it can occur that the incident electron is captured by an atom and a new electron is emitted. Exchange effects are well described by adding, to the atomic potential described above, the Furness and McCarthy exchange potential [5]: Vex =

1 1 (E − V ) − [(E − V )2 + 4πρe2 2 /m]1/2 . 2 2

(D.3)

In this equation, E is the electron energy, V the electrostatic potential, ρ the atomic electron density (obtained by Poisson’s equation), and e the electron charge.

D.3 Charge Cloud Polarization Effects The polarization potential can be described, when the incident electron is relatively far from the target atom, by the so-called Buckingham potential [6] Vpol = −

1 αd e2 . 2 (r 2 + d 2 )2

(D.4)

Here αd is the dipole polarizability of the atom. The cut-off parameter d is given by, according to Mittleman and Watson [7], 2 , d 4 = 0.5αd a0 Z −1/3 bpol

(D.5)

where a0 is the Bohr radius and [6] 2 = max[(E − 50)/16, 1] , bpol

(D.6)

with E expressed in eV. According to Salvat et al. [6], in order to obtain a more accurate description of the charge cloud polarization effects, the long-range Buckingham potential has to be combined with the correlation potential. Such a potential can be calculated using the so-called local density approximation (LDA) [8]. According to the LDA, the correlation energy of the electron is calculated assuming that the projectile is in a free-electron gas whose density is given by the local atomic electron density.

Appendix D: The Atomic Potential

229

D.4 Solid-State Effects For atoms bound in solids, the outer orbitals are modified, so that solid-state effects should be introduced. In the so-called muffin-tin model, the potential of every atom in the solid is changed by the nearest-neighbor atoms. If we assume that the nearestneighbor atoms are located at distances equal to 2rws , where rws is the radius of the Wigner–Seitz sphere [9], then the potential can be calculated, for r ≤ rws , as follows Vsolid (r ≤ rws ) = V (r ) + V (2rws − r ) − 2V (rws ) .

(D.7)

It is equal to zero outside the Wigner–Seitz sphere, i.e., Vsolid (r ≥ rws ) = 0 .

(D.8)

The term 2V (rws ) was introduced in (D.7) to shift the energy scale so that Vsolid (r = rws ) = 0. According to Salvat and Mayol, it also has to be subtracted from the kinetic energy of the incident electrons [10].

D.5 Summary Atomic potential was described. Electron exchange, charge cloud polarization effects, and solid-state effects were also introduced.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

H.L. Cox Jr., R.A. Bonham, J. Chem. Phys. 47, 2599 (1967) E. Clementi, Tables of Atomic Functions (IBM Research Publication, 1965) F. Salvat, J.D. Martínez, R. Mayol, J. Parellada, Phys. Rev. A 36, 467 (1987) G. Molière, Z. Naturforsch. Teil A 2, 133 (1947) J.B. Furness, I.E. McCarthy, J. Phys. B 6, 2280 (1973) F. Salvat, A. Jablonski, C.J. Powell, Comp. Phys. Comm. 165, 157 (2005) M.H. Mittleman, K.M. Watson, Ann. Phys. 10, 268 (1960) J.P. Perdew, A. Zunger, Phys. Rev. B 23, 5048 (1981) N. Ashcroft, N.D. Mermin, Solid State Physics (W.B Saunders, 1976) F. Salvat, R. Mayol, Comput. Phys. Commun. 74, 358 (1993)

Appendix E

The Fröhlich Theory

The original version of the Fröhlich theory can be found in [1]. Also see [2] for further details. In his theory of the electron-phonon interaction, Fröhlich [1] considered, in particular, the interaction of free electrons with the longitudinal optical mode lattice vibrations. The interaction was described considering both phonon creation and phonon annihilation, corresponding to electron energy loss and to electrons energy gain, respectively [2]. As the phonon generation probability is much higher than the phonon absorption probability, the latter is often neglected in Monte Carlo simulations. Furthermore, since, according to Ganachaud and Mokrani [3], the dispersion relation of the longitudinal phonons can be neglected in the optical branch, one can use a single phonon frequency. As a matter of fact, the theory adopts a single value of the frequency for all momenta, corresponding to a flat longitudinal optical branch. This approximation is quite reasonable and experimentally confirmed (see, for example, Fujii et al. [4] for the ionic crystal AgBr). In semiconductors, the longitudinal optical branch is flat as well, as demonstrated both experimentally (see, for example, Nilsson and Neil for Ge [5] and Si [6]) and theoretically (see, for example, Gianozzi et al. for Si, Ge, GaAs, AlAs, GaSb, and AlSb [7]).

E.1 Electrons in Lattice Fields. Interaction Hamiltonian According to Fröhlich [1], an electron traveling in a dielectric material polarizes the medium, and the polarization reacts on the charged particle. Let us indicate with P(r) the electric polarization. The only source of the displacement vector, D(r) = E(r) + 4π P(r) (where E(r) is the electric field), are the free charges. If r el represents the position of a single free electron, then ∇ · D = − 4π e δ(r − r el ) ,

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1

(E.1)

231

232

Appendix E: The Fröhlich Theory

where e is the electron charge. The electric polarization can be written as P(r) = P uv (r) + P ir (r) ,

(E.2)

where P uv (r) and P ir (r) are the polarizations corresponding, respectively, to the ultraviolet (atomic polarizability) and the infrared (displacement polarizability) optical absorptions [8]. They satisfy the harmonic oscillator equations d 2 P uv (r) D(r, r el ) 2 , + ωuv P uv (r) = 2 dt δ

(E.3)

d 2 P ir (r) D(r, r el ) . + ω 2 P ir (r) = 2 dt γ

(E.4)

In these equations, ωuv (atomic deformation) and ω (atomic displacement) are the angular frequencies for ultraviolet and infrared optical absorption, respectively, and δ and γ are constants that are related to the dielectric function. To determine these constants, let us firstly consider the static case and indicate with ε0 the static dielectric constant. Since D(r) = ε0 E(r) ,

(E.5)

  1 D(r) . 4πP(r) = 1 − ε0

(E.6)

then

Furthermore, let us assume that the high-frequency dielectric constant ε∞ is determined in the hypothesis that the angular frequency ω∞ of the external field is low compared with the atomic excitation frequencies ωuv and high compared with the 2 P uv (r), and lattice vibrational frequencies ω [8]. Then P ir ≈ 0, d 2 P uv /dt 2 ωuv 1/2 D(r) = ε∞ E(r), where ε∞ is the high frequency dielectric constant (ε∞ is the refraction index). As a consequence, P uv (r) can be approximated assuming that it has the value corresponding to a static field of the same strength [1]   1 D(r) (E.7) 4πP uv (r) = 1 − ε∞ so that  4πP ir (r) =

 1 1 D(r) − ε∞ ε0

Since, in the static case, d 2 P uv /dt 2 = 0 and d 2 P ir /dt 2 = 0,

(E.8)

Appendix E: The Fröhlich Theory

233

P uv (r) =

D(r) , 2 δ ωuv

(E.9)

P ir (r) =

D(r) , γ ω2

(E.10)

1 , 1− ε∞

(E.11)

and hence we obtain 1 ω2 = uv δ 4π 1 ω2 = γ 4π



1 1 − ε∞ ε0

.

(E.12)

In order to describe a slow electron interacting with an ionic lattice field, Fröhlich considered the infrared contribution to the polarization and introduced the complex field B(r) such that 

γω i dP ir (r) P ir (r) + . (E.13) B(r) = 2 ω dt The phonon annihilation operators aq are defined by the equation B(r) =

q exp(i q · r) aq , √ q V q

(E.14)

where B(r) is subject to a boundary condition over a cube of volume V . As a consequence 

2 q i dP ir (r) = aq exp(i q · r) . (E.15) P ir (r) + ω dt γωV q q Taking the Hermitian adjoint we obtain 

2 q † i dP ir (r) = a exp(−i q · r) , P ir (r) − ω dt γωV q q q

(E.16)

so that  P ir (r) =

 qˆ [aq exp(i q · r) + aq† exp(−i q · r)] , 2γω V q

(E.17)

234

Appendix E: The Fröhlich Theory

where q q

(E.18)

2π ni , L

(E.19)

qˆ = and qi =

with n i = 0, ±1, ±2, . . . and L = V 1/3 . The quantities aq† represent the phonon creation operators. In order to write the interaction Hamiltonian, Hinter , let’s observe that the displacement vector D is the external field that determines the crystal polarization. Since D = 0 in the absence of free charges, the potential φir is given by − 4π P ir = E = −∇ φir ,

(E.20)

so that Hinter = e φir =   e2 1 † [a exp(−i q · r) − aq exp(i q · r)] , 4πi 2γω V q q q

(E.21)

with q = 0.

E.2 Electron-Phonon Scattering Cross-Section If we indicate with ω the angular frequency of the longitudinal optical vibrations of the lattice, then the average number of phonons at temperature T is given by the occupation function n(T ) =

1 , exp ( ω/kB T ) − 1

(E.22)

where kB is the Boltzmann constant. Fröhlich theory [1] uses the perturbation approach, assuming that the electron-lattice coupling is weak. If the electron energy, measured with respect to the bottom of the conduction band, is given by Ek =

2 k 2 , 2 m∗

(E.23)

Appendix E: The Fröhlich Theory

235

where m ∗ is the electron effective mass and k the electron wavenumber, then the unperturbed electron wavefunction can be written as | k =

exp (i k · r) , V 1/2

(E.24)

where, as above, we have indicated with V a cubic volume containing the electron. We know that, according to Fröhlich [1], the interaction Hamiltonian is given by (E.21)  e2  1 † [a exp (−i q · r) − aq exp (i q · r)] , Hinter = 4 π i 2γω V q q q where q = 0 is the phonon wavenumber, aq† and aq are, respectively, the operators of creation and annihilation of phonons, and γ is related to the static dielectric constant ε0 and to the high frequency dielectric constant ε∞ by the Eq. (E.12)

1 ω2 1 1 . = − γ 4 π ε∞ ε0 In order to calculate the transition rate Wkk  from the state | k to the state | k , Llacer and Garwin [2] used the standard results of the perturbation theory. In the case of phonon annihilation, corresponding to an electron energy gain, one should consider the frequency β =

Ek − Ek −  ω , 2

(E.25)

while, for phonon creation (electron energy loss), the frequency to be considered is the following β =

Ek − Ek +  ω . 2

(E.26)

With these definitions, the rate is given by Wkk 

 2 | ∂ |Mkk = 2  ∂t



sin2 βt β2

(E.27)

 where Mkk is the matrix element for the transition from the state k to the state k . They can be calculated by using the interaction Hamiltonian, (E.21), taking into account the properties of the creation and annihilation operators,

aq |n = aq† |n =

√

√ n |n − 1 ,

(E.28)

n + 1 |n + 1 ,

(E.29)

236

Appendix E: The Fröhlich Theory

and utilizing the conditions of orthonormality satisfied by the set of vectors |n(T ), n|n = 1 ,

(E.30)

n|n + m = 0 .

(E.31)

where m is any integer = 0. For the case of annihilation of a phonon of wavenumber q (electron energy gain), k = k + q and  √ e2  n(T ) Mkk  = 4 π i , (E.32) 2γω V q while, in the case of the creation of a phonon of wavenumber q (electron energy loss), k = k − q and  √ e2  n(T ) + 1 Mkk  = − 4 π i . (E.33) 2γω V q The total scattering rate from a state k to all the available states k can be obtained by integrating over q. Let us first perform the integration for the case of phonon annihilation: Wk− =  qmax  dq qmin



2π

π

dφ 0

0

16 π 2 e2 n(T ) ∂ sin2 β t V 2 q sin α dα . 2  γ ω V q2 ∂ t β 2 8 π3

(E.34)

Note that α represents in this context the angle between the direction of k and that of q, while we will use the symbol ϑ to indicate the angle between k and k . As k 2 = k 2 + q 2 − 2 k q cos α ,

(E.35)

some simple algebraic manipulations allow as to see that β =

  ω q2 − k q cos α − , 4 m∗ 2 m∗ 2

so that sin α dα =

2 m∗ 1 dβ .  kq

(E.36)

(E.37)

Appendix E: The Fröhlich Theory

237

As a consequence Wk− =





qmax

βmax

dq βmin

qmin

4 m ∗ e2 n(T ) 1 ∂ sin2 β t dβ , 2 γ ω k q ∂ t β2

(E.38)

where βmin =

  ω q2 − kq − , 4 m∗ 2 m∗ 2

(E.39)

βmax =

  ω q2 + kq − , ∗ ∗ 4m 2m 2

(E.40)

and

Now 

βmax

βmin



∂ sin2 β t dβ = ∂ t β2



βmax

βmin

sin 2 β t dβ = β

 2 βmax t sin( 2 β t) sin x 2 t dβ = dx = (2 β t) x βmin 2 βmin t  2 βmax t  2 βmin t sin x sin x = dx − dx . x x 0 0 =

βmax

In order to carry out the calculation, we need to know the limits of integration, qmin and qmax . They can be obtained by using the law of conservation of energy, E k  = E k +  ω, which corresponds to β = 0. Since cos α can take all the values between −1 and +1, the limits of integration satisfy the following equations q2 + 2 k q −

k2  ω = 0, Ek

(E.41)

q2 − 2 k q −

k2  ω = 0, Ek

(E.42)

so that, as q is positive,  qmin = k

 ω 1+ −1 , Ek

(E.43)

 qmax = k

 ω 1+ +1 . Ek

(E.44)

238

Appendix E: The Fröhlich Theory

Inserting these limits of integration in the definition of βmin and βmax , we see that βmin ≤ 0 and βmax ≥ 0. Therefore, 

βmax

lim

t→∞

βmin



∂ sin2 β t dβ = ∂ t β2

 2 βmin t sin x sin x = lim dx − dx = t→∞ x x 0 0  +∞  −∞ sin x sin x = dx − dx = x x 0 0  π π = si(+∞) − si(−∞) = − − = π, 2 2 2 βmax t

so that Wk−

 =

qmax

qmin

4 π m ∗ e2 n(T ) 1 dq . 2 γ ω kq

(E.45)

As a consequence, we conclude that the total scattering rate for the phonon annihilation (electron energy gain) is given by

√ 1 +  ω/E k + 1 4 π e2 m ∗ n(T ) . (E.46) Wk− = ln √ 2 γ ω k 1 +  ω/E k − 1 The case of phonon creation (electron energy loss) √ √ is similar. Remember that, in this case, we have to use n(T ) + 1 instead of n(T ) in the matrix element of the transition of the electron from the state k to the state k . Furthermore, in this case,

β =

1   ω [E k  − (E k +  ω)] = q2 − k q cos α + , ∗ ∗ 2 4m 2m 2

(E.47)

so that 

 qmin = k

1− 

qmax = k

1+

ω 1− Ek



ω 1− Ek

 ,

(E.48)

.

(E.49)



Therefore, Wk+

√

1 + 1 −  ω/E k 4 π e2 m ∗ [n(T ) + 1] . ln = √ 2 γ ω k 1 − 1 −  ω/E k

(E.50)

Appendix E: The Fröhlich Theory

239

Concerning the angular distribution of the scattering, let us consider the angle ϑ between k and k , so that q 2 = k 2 + k 2 − 2 k k  cos ϑ

(E.51)

q dq = k k  sin ϑ dϑ .

(E.52)

and, hence,

The probability of scattering between ϑ and ϑ + dϑ can be calculated by considering the integrand of (E.45): q dq k k  sin ϑ dϑ dq = A = = A kq k q2 k (k 2 + k 2 − 2 k k  cos ϑ) k  sin ϑ dϑ , = A 2 k + k 2 − 2 k k  cos ϑ A

where for phonon annihilation, A =

4 π e2 m ∗ n(T ) . 2 γ ω

(E.53)

Similar considerations hold for phonon creation so that we conclude that the angular distribution is proportional, in both cases, to 1/2

dη =

E k  sin ϑ dϑ E k + E k  − 2 (E k E k  )1/2 cos ϑ

(E.54)

After an electron-phonon collision, the new angle ϑ is obtained inverting this distribution. Indicating with μ the cumulative probability, we have  ϑ dη μ = 0π = 0 dη  ϑ 1/2 E k  sin ϑ dϑ = / E k + E k  − 2 (E k E k  )1/2 cos ϑ 0  π 1/2 E k  sin ϑ dϑ / , E k + E k  − 2 (E k E k  )1/2 cos ϑ 0 and, as a consequence,

(E.55)

240

Appendix E: The Fröhlich Theory

cos ϑ =

E k + E k (1 − B μ ) + B μ , √ 2 E k E k

√ Ek + Ek + 2 Ek Ek B = . √ Ek + Ek − 2 Ek Ek

(E.56)

(E.57)

The relationship between the mean free path λphonon and the total scattering rate from a state k to all the other available states k is λphonon =

1 dP v dt

−1

,

(E.58)

where v is the electron velocity before the electron-phonon collision v =

k m∗

(E.59)

and dP = Wk− + Wk+ . dt The electron-phonon mean free path can then be written as √ 2 E k /m ∗  k/m ∗ λphonon = = , − + − Wk + Wk Wk + Wk+

(E.60)

(E.61)

and, as a consequence,  ε0 − ε∞  ω 1 × ε0 ε∞ Ek 2 √    1 + 1 −  ω/E k × [n(T ) + 1] ln + √ 1 − 1 −  ω/E k √   1 +  ω/E k + 1 + n(T ) ln √ , 1 +  ω/E k − 1 λ−1 phonon

1 = a0



(E.62)

where we have assumed that the electron effective mass m ∗ is equal to that of a free electron, m ∗ = m. The probability of phonon creation is much higher than that of phonon annihilation [2, 3, 9], so that one can safely ignore the electron energy gain due to the phonon annihilation. As a consequence we can write λphonon =

 k/m ∗ , Wk+

(E.63)

Appendix E: The Fröhlich Theory

241

so that, indicating with E = E k the energy of the incident electron and with Wph = ω the energy of the created phonon (and assuming that m ∗ = m) we conclude that the inverse inelastic mean free path for electron energy loss due to phonon creation can be written as [2] λ−1 phonon =

   1 + 1 − Wph /E 1 ε0 − ε∞ Wph n(T ) + 1  = ln . a 0 ε0 ε∞ E 2 1 − 1 − Wph /E

(E.64)

This equation was used in the Monte Carlo simulations of secondary electron emission from insulating materials presented in this book [3, 9, 10].

E.3 Summary The Fröhlich theory [1, 2] concerning the interaction of free electrons with the longitudinal optical mode lattice vibrations was described in this chapter. Both phonon creation and phonon annihilation, corresponding to electron energy loss and to electrons energy gain, respectively, are described by the theory.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

H. Fröhlich, Adv. Phys. 3, 325 (1954) J. Llacer, E.L. Garwin, J. Appl. Phys. 40, 2766 (1969) J.P. Ganachaud, A. Mokrani, Surf. Sci. 334, 329 (1995) Y. Fujii, S. Hoshino, S. Sakuragi, H. Kanzaki, J.W. Lynn, G. Shirane, Phys. Rev. B 15, 358 (1977) G. Nilsson, G. Nelin, Phys. Rev. B 3, 364 (1971) G. Nilsson, G. Nelin, Phys. Rev. B 6, 3777 (1972) P. Giannozzi, S. De Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B 43, 7231 (1991) N. Ashcroft, N.D. Mermin, Solid State Physics (W.B Saunders, 1976) M. Dapor, M. Ciappa, W. Fichtner, J. Micro/Nanolith, MEMS MOEMS 9, 023001 (2010) M. Dapor, Nucl. Instrum. Methods Phys. Res. B 269, 1668 (2011)

Appendix F

The Ritchie Theory

The Ritchie theory describes the relationship between the dielectric function and the electron energy loss in a solid. It allows us to calculate the differential inverse inelastic mean free path, the inelastic mean free path, and the stopping power. The original version of the Ritchie theory can be found in [1]. Also, see [2–6] for further details.

F.1 Energy Loss and Dielectric Function The response of the ensemble of conduction electrons to the electromagnetic disturbance due to electrons passing through a solid and losing energy in it is described by the complex dielectric function ε(k, ω), where k is the wave vector and ω is the frequency of the electromagnetic field. If, at time t, the electron position is r and its speed is v, then, indicating with e the electron charge, the electron that passes through the solid can be represented by a charge distribution given by ρ(r, t) = −e δ(r − vt) .

(F.1)

The electric potential ϕ generated in the medium can be calculated as4 ε(k, ω) ∇ 2 ϕ(r, t) = −4π ρ(r, t) .

(F.2)

4

Please note that the vector potential is zero due to the chosen gauge. Also, please note that the velocities are nonrelativistic. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1

243

244

Appendix F: The Ritchie Theory

In the Fourier space, we have ϕ(k, ω) = −

8π 2 e δ(k · v + ω) . ε(k, ω) k2

(F.3)

In fact, on the one hand, ϕ(r, t) =





1 (2π)4

d 3k

+∞

−∞

dω exp[i(k · r + ω t)] ϕ(k, ω) ,

(F.4)

dω exp[i(k · r + ω t)] k 2 ϕ(k, ω) ,

(F.5)

so that 1 ∇ ϕ(r, t) = − (2π)4





2

3

+∞

d k −∞

and, on the other hand,  ρ(k, ω) =   3 = d r

 3

+∞

d r +∞

−∞

−∞

dt exp[−i(k · r + ω t)] ρ(r, t) =

dt exp[−i(k · r + ω t)] [−e δ(r − vt)] =

 +∞ 1 dt exp[−i(k · v + ω)t] = 2π −∞ = −2πe δ(k · v + ω) , = −2πe

(F.6)

so that

ρ(r, t) =

1 (2π)4



 d 3k

+∞ −∞

dω exp[i(k · r + ω t)] [− 2πe δ(k · v + ω)] . (F.7)

Then, using (F.2), (F.5), and (F.7), we obtain ε(k, ω) k 2 ϕ(k, ω) = −8π 2 e δ(k · v + ω)) ,

(F.8)

which is equivalent to (F.3). We are interested in calculating the energy loss −d E of an electron due to its interaction with the electric field E generated by the electrons passing through the solid. Let us indicate with Fz the z component of the electric force, so that − d E = F · d r = Fz dz .

(F.9)

Appendix F: The Ritchie Theory

245

It should be noted that here and in the following equations, the electric force (and the electric field E = F /e) are considered at r = v t. Since Ez dz =

v·E dz dr dt Ez = · E dt = dz dt dt v

(F.10)

the energy loss −d E per unit path length dz, −d E/dz, is given by −

e dE = v · E. dz v

(F.11)

Since E = − ∇ ϕ(r, t)

(F.12)

and ϕ(k, ω) is the Fourier transform of ϕ(r, t) [see (F.4)], then E =



= −∇

1 (2 π)4



 3

+∞

d k −∞

 dω exp[i(k · r + ω t)] ϕ(k, ω) . (F.13)

As a consequence dE = dz  8π 2 e2 × = Re − (2π)4 v     +∞ δ(k · v + ω)  3 × d k = dω(−∇) exp[i(k · r + ω t)] · v 2 k ε(k, ω)  r = v t −∞  8π 2 e2 = Re − × (2π)4 v     +∞ δ(k · v + ω)  3 × d k = dω(−i k) · v exp[i(k · r + ω t)] k 2 ε(k, ω)  r = v t −∞  i8π 2 e2 = Re × 16π 4 v     +∞ δ(k · v + ω)  × d 3k . (F.14) dω(k · v) exp[i(k · r + ω t)] 2 k ε(k, ω)  r = v t −∞ −

Taking into account (i) that the electric field has to be calculated at r = v t and (ii) of the presence in the integrand of the δ(k · v + ω) distribution, we have

246

Appendix F: The Ritchie Theory

dE = dz  i e2 = Re × 2 π2 v    +∞ δ(k · v + ω) 3 = × d k dω k · v exp[i(k · v t + ω t)] k 2 ε(k, ω) −∞  i e2 = Re × 2 π2 v    +∞ δ(k · v + ω) = × d 3k dω k · v exp[i(−ω t + ω t)] k 2 ε(k, ω) −∞  i e2 × = Re 2 π2 v    +∞ δ(k · v + ω) 3 × d k = dω (−ω) exp[i(−ω t + ω t)] k 2 ε(k, ω) −∞     +∞ −i e2 δ(k · v + ω) 3 . (F.15) = Re k dω ω d 2 π2 v k 2 ε(k, ω) −∞ −

Since     +∞ δ(k · v + ω) 3 Re i d k = dω ω ε(k, ω) −∞     +∞ δ(k · v + ω) , = 2 Re i d 3k dω ω ε(k, ω) 0 we conclude that5 e2 dE = 2 − dz π v





∞

3

d k 0



1 dω ω Im ε(k, ω)



δ(k · v + ω) , k2

(F.16)

or −

dE = dz



∞

dω ω τ (v, ω) ,

(F.17)

0

where τ (v, ω) =

5

e2 π2 v



 d 3 k Im

1 ε(k, ω)



Note that, for any complex number w, Re(i w) = − Im(w).

δ(k · v + ω) k2

(F.18)

Appendix F: The Ritchie Theory

247

is the probability of an energy loss ω per unit distance traveled by a non-relativistic electron of velocity v [1].

F.2 Homogeneous and Isotropic Solids Let us assume now that the solid is homogeneous and isotropic, and ε is a scalar depending on the magnitude of k and not on its direction ε(k, ω) = ε(k, ω)

(F.19)

so that τ (v, ω) =  π  k+  2π e2 dφ dϑ dk k 2 sin ϑ × = 2 π v 0 0 k−   δ(k v cos ϑ + ω) 1 × Im = ε(k, ω) k2     k+ 1 2 e2 π δ(k v cos ϑ + ω) dϑ dk sin ϑ Im = πv 0 ε(k, ω) k−

(F.20)

where  k± =

√

2m E ±



2 m (E −  ω) .

(F.21)

and E = m v 2 / 2. These limits of integration come from conservation laws (see Sect. 6.2.5). Let us introduce the new variable ω  defined as ω  = −k v cos ϑ ,

(F.22)

dω  = k v sin ϑ dϑ

(F.23)

so that

and, hence,    k+  1 dk 2 e2 k v  Im δ(−ω  + ω) = (F.24) dω τ (v, ω) = π v −k v ε(k, ω) k− k v    k+ 2 m e2 dk 1 . = Im π m v 2 k− k ε(k, ω)

248

Appendix F: The Ritchie Theory

We thus can write that τ (E, ω) =

m e2 πE



  dk 1 . Im k ε(k, ω)

k+

k−

(F.25)

Indicating with W = ω the energy loss and with Wmax the maximum energy loss, the inverse electron inelastic mean free path, λ−1 inel , can be calculated as λ−1 inel = =

m e2 π 2 E

1 = π a0 E





Wmax

k+

dω 

0



Wmax

k− k+

dω 0

k−

  1 dk Im = k ε(k, ω)   dk 1 . Im k ε(k, ω)

(F.26)

F.3 Summary The Ritchie theory [1] was described. It allows us to establish the relationship between electron energy loss and dielectric function and to calculate the differential inverse inelastic mean free path, the inelastic mean free path, and the stopping power.

References 1. R.H. Ritchie, Phys. Rev. 106, 874 (1957) 2. H. Raether, Excitation of Plasmons and Interband Transitions by Electrons (Springer, Berlin, 1982) 3. P. Sigmund, Particle Penetration and Radiation Effects (Springer, Berlin, 2006) 4. R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 3rd edn. (Springer, New York, 2011) 5. R.F. Egerton, Rep. Prog. Phys. 72, 016502 (2009) 6. S. Taioli, S. Simonucci, L. Calliari, M. Dapor, Phys. Rep. 493, 237 (2010)

Appendix G

The Chen and Kwei and the Li et al. Theory

The original version of the Chen and Kwei theory can be found in [1] for outgoing projectiles. It was generalized by Li et al. [2] for incoming projectiles. Below, the Chen and Kwei and the Li et al. formulas are rewritten in terms of angular variables, according to [3].

G.1 Outgoing and Incoming Electrons Let us consider the component qx and q y of the momentum transfer parallel to the surface. For outgoing electrons, qx =

mv (θ cos φ cos α + θ E sin α) , 

(G.1)

while, for incoming electrons, qx =

mv (θ cos φ cos α − θ E sin α) . 

(G.2)

For both outgoing and incoming electrons we have qy =

mv θ sin φ . 

(G.3)

In these equations, α is the angle of the electron trajectory with respect to the normal to the target surface, θ and φ indicate the polar and azimuthal angles, and θE =

ω , 2E

(G.4)

where E is the electron energy and  ω the energy loss. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1

249

250

Appendix G: The Chen and Kwei and the Li et al. Theory

G.2 Probability of Inelastic Scattering If z is the coordinate along the normal to the surface target, the probability for inelastic scattering (the differential inverse inelastic mean free path) is given by: 1 Poutside (z, α) = 2 2π a0 E



θcutoff

0

θ dθ 2 θ + θE2



2π

dφ f (z, θ, φ, α)

(G.5)

dφ g(z, θ, φ, α)

(G.6)

0

in the vacuum, and Pinside (z, α) =

1 2π 2 a0 E

 0

θcutoff

θ dθ θ2 + θE2



2π

0

inside the material. The cutoff angle is taken to be the Bethe ridge angle [4]  ω θcutoff = . (G.7) E It should be noted that in the Chen and Kwei approach [1], there is no proper limit for the high-momentum cutoff, while there exists a maximum angle, known as the Bethe ridge angle, only up to which electron excitation is allowed [5]. For outgoing electrons, functions f (z, θ, φ, α) and g(z, θ, φ, α) can be written as:

2 h(z, θ, φ, α) [ p(z, θ, φ, α) − h(z, θ, φ, α)] , (G.8) f (z, θ, φ, α) = Im ε+1



1 [1 − h 2 (z, θ, φ, α)] . ε (G.9) For incoming electrons, the same functions f (z, θ, φ, α) and g(z, θ, φ, α) are given by:

2 h 2 (z, θ, φ, α) , (G.10) f (z, θ, φ, α) = Im ε+1 g(z, θ, φ, α) = Im

2 ε+1



h 2 (z, θ, φ, α) + Im

2 h(z, θ, φ, α) [ p(z, θ, φ, α) − h(z, θ, φ, α)] + g(z, θ, φ, α) = Im ε+1

1 [1 − h(z, θ, φ, α) p(z, θ, φ, α) + h 2 (z, θ, φ, α)] . + Im (G.11) ε

Appendix G: The Chen and Kwei and the Li et al. Theory

251

Functions h(z, θ, φ, α) and p(z, θ, φ, α) are, in turn, given by:

h(z, θ, φ, α) = exp



−|z|

p(z, θ, φ, α) = 2 cos

  mv (θ cos φ cos α + θE sin α)2 + θ2 sin2 φ , 

(G.12)

 mv  |z| (θE cos α − θ cos φ sin α) , 

(G.13)

for outgoing electrons and by:   mv 2 2 2 (θ cos φ cos α − θE sin α) + θ sin φ , h(z, θ, φ, α) = exp −|z|  (G.14)   mv (θE cos α + θ cos φ sin α) , (G.15) p(z, θ, φ, α) = 2 cos |z|  

for incoming electrons. Finally, ε(ω) is the dielectric function. The examples presented in this book about the Chen and Kwey theory use, in the case of aluminum (see Figs. 4.39 and 4.41), the complex dielectric function ε(ω) = 1 −

ω 2p ω 2 − iΓ ω

,

(G.16)

while, in the case of silicon (see Figs. 4.40 and 4.42), ε(ω) = 1 −

ωp2 ω 2 − ωg2 − iΓ ω

,

(G.17)

where ω is the electron energy loss, ωg is the average excitation energy for valence electrons in silicon, Γ is the damping constant, and ωp is the plasmon energy.

G.3 Summary The Chen and Kwei theory for outgoing electrons [1], and its generalization proposed by Li et al. [2] for incoming projectiles, were described and rewritten in terms of angular variables, according to [3].

252

Appendix G: The Chen and Kwei and the Li et al. Theory

References 1. 2. 3. 4.

Y.F. Chen, C.M. Kwei, Surf. Sci. 364, 131 (1996) Y.C. Li, Y.H. Tu, C.M. Kwei, C.J. Tung, Surf. Sci. 589, 67 (2005) M. Dapor, L. Calliari, and S. Fanchenko, Surf. Interface Anal. 44, 1110 (2012) R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 3rd edn. (Springer, New York, 2011) 5. L. Calliari, S. Fanchenko, Surf. Interface Anal. 44, 1104 (2012)

Appendix H

The Mermin Theory and the Generalized Oscillator Strength Method

Dielectric formalism is the most used method for investigating the interaction of swift electrons with solid targets. In this chapter, the Mermin energy loss functiongeneralized oscillator strength method (MELF-GOS method) is briefly described within the framework of dielectric formalism [1–4].

H.1 The Mermin Theory The Mermin dielectric function [1] is given by: ε M (q, ω) = 1 +

(1 + i/ωτ )[ε0 (q, ω + i/τ ) − 1] , 1 + (i/ωτ )[ε0 (q, ω + i/τ ) − 1]/[ε0 (q, 0) − 1]

(H.1)

where q is the momentum, ω the frequency, τ the relaxation time, and ε0 (q, ω) the Lindhard dielectric constant [5] ε0 (q, ω) = 1 +  B(q, ω) =

4π 2 q 2 B(q, ω) , e2

f p+q/2 − f p−q/2 dp . 4π 3 ω − (εp+q/2 − εp−q/2 )/

(H.2) (H.3)

In these equations e is the electron charge, f p is the Fermi-Dirac distribution, and εp the free electron energy. Note that the Lindhard dielectric function [5] can be numerically calculated by using (H.2) and (H.3). The integration can also be carried out in a closed form. The result of the integration is the following [2, 3, 6]: ε0 (q, ω) = 1 +

χ2 [ f 1 (u, z) + i f 2 (u, z)] , z2

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1

(H.4) 253

254

Appendix H: The Mermin Theory and the Generalized Oscillator Strength Method

where u = ω/(qv F ), z = q/(2q F ), and χ2 = e2 /(π  v F ) is a measure of the electron density [6]. In this equation, v F is the Fermi velocity of the valence electrons of the target and q F = mv F /. The functions f 1 (u, z) and f 2 (u, z) are given by f 1 (u, z) =

f 2 (u, z) =

⎧ ⎨ ⎩

1 1 + [g(z − u) + g(z + u)] , 2 8z

π u, 2 π [1 8z

0,

− (z − u)2 ] ,

z+u 1 ,

(H.5)

(H.6)

where   1 + x  .  g(x) = (1 − x ) ln  1−x 2

(H.7)

H.2 The Mermin Energy Loss Function-Generalized Oscillator Strength Method (MELF-GOS) Let us now consider a superposition of free and bound oscillators. For any oscillator:   −ε M2 1 = 2 , (H.8) Im ε M (ωi , γi ; q, ω) ε M1 + ε2M2 where ε M = ε M1 + iε M2

(H.9)

and ωi and γi are, respectively, the frequency and the damping constant associated with each specific oscillator. A linear combination of Mermin-type energy loss functions, one per oscillator, allows to calculate the energy loss function (ELF) for q = 0, for any specific material [2, 3, 4]:    

1 1 = . (H.10) Ai Im Im ε(q = 0, ω) ε M (ωi , γi ; q = 0, ω) i In this equation, Ai , ωi , and γi are determined by looking for the best fit of the available experimental optical ELF. Actually, as both Mermin and Drude–Lorentz oscillators converge on the same values in the optical limit (i.e., for q = 0) [7]

Appendix H: The Mermin Theory and the Generalized Oscillator Strength Method

 Im

1 ε(q = 0, ω)

 =

 1 = ε M (ωi , γi ; q = 0, ω) i  

1 , Ai Im ε D (ωi , γi ; q = 0, ω) i

255



Ai Im

  where the Drude–Lorentz functions Im ε D (ωi ,γi1;q=0,ω) are given by [8]   1 γi ω Im , = ε D (ωi , γi ; q = 0, ω) (ωi2 − ω 2 )2 + (γi ω)2

(H.11)

(H.12)

the best fit can also be obtained using a linear combination of Drude–Lorentz functions, instead of Mermin functions. Once the values of the best-fit parameters have been established (see, for example [4, 9, 10, 11]), the extension beyond the optical domain (q = 0) can be obtained by [2, 3, 4]    

1 1 = . (H.13) Ai Im Im ε(q, ω) ε M (ωi , γi ; q, ω) i Planes et al. [2], Abril et al. [3], and de Vera et al. [4] construct the ELF in the optical limit including the contribution of the electrons from the outermost atomic inner shells as follows: ⎧   1 ⎪ ω < ωi,edge A Im ⎪   i i ⎨ ε M (ω  i ,γi ;q=0,ω) 1 1 Im = ω ≥ ωi,edge i,sh Ai,sh Im ε M (ωi,sh ,γi,sh ;q=0,ω) ⎪ ε(q = 0, ω) ⎪ ⎩ (H.14) where the first term represents the contribution of the outer electrons while the second one includes the electrons of the outermost atomic inner shells.

H.3 Summary In this chapter, after a brief discussion about the Mermin theory [1], the Mermin energy loss function-generalized oscillator strength method (MELF-GOS method), in the framework of the dielectric formalism, was shortly described [2–4].

256

Appendix H: The Mermin Theory and the Generalized Oscillator Strength Method

References 1. N.D. Mermin, Phys. Rev. B 1, 2362 (1970) 2. D.J. Planes, R. Garcia-Molina, I. Abril, N.R. Arista, J. Electron Spectrosc. Relat. Phenom. 82, 23 (1996) 3. I. Abril, R. Garcia-Molina, C.D. Denton, F.J. Pérez-Pérez, N.R. Arista, Phys. Rev. A 58, 357 (1998) 4. P. de Vera, I. Abril, R. Garcia-Molina, J. Appl. Phys. 109, 094901 (2011) 5. J. Lindhard, Kgl. Danske Videnskab. Selskab Mat.-fys. Medd. 28, 8 (1954) 6. P. Sigmund, Particle Penetration and Radiation Effects (Springer, Berlin, 2006) 7. W. de la Cruz, F. Yubero, Surf. Interface Anal. 39, 460 (2007) 8. R.H. Ritchie, A. Howie, Philos. Mag. 36, 463 (1977) 9. R. Garcia-Molina, I. Abril, C.D. Denton, S. Heredia-Avalos, Nucl. Instr. Meth. Phys. Res. B 249, 6 (2006) 10. M. Dapor, Nucl. Instr. Meth. Phys. Res. B 352, 190 (2015) 11. M. Dapor, Front. Mater. 2, 27 (2015)

Appendix I

The Kramers–Kronig Relations and the Sum Rules

In this chapter, we discuss the response of a material to applied electromagnetic radiation. The complex dielectric function is the response function of the medium corresponding to the electric polarization induced by an applied electric field. Similarly, complex conductivity describes the electric current induced in response to an applied electric field. In general, the system response to a given stimulus is ruled by complex analysis and, in particular, by the so-called Kramers–Kronig relations. A consequence of the Kramers–Kronig analysis is constituted by the sum rules which must be satisfied by several optical parameters.

I.1 Linear Response to External Perturbations Polarization P(t) of a material at time t depends on the electric field E(t) applied to the material from t  = −∞ to t  = t, according to the following equation:  t G(t − t  ) E(t  ) dt  , (I.1) P(t) = −∞

where G(t) is a real Green function of real variable [1]. Assuming that E(t) = E 0 exp[−iωt] ,

(I.2)

it is thus easy to see that 

∞

P(t) = E

G(τ ) exp[iωτ ] dτ .

(I.3)

0

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1

257

258

Appendix I: The Kramers–Kronig Relations and the Sum Rules

As a consequence, the electric susceptibility χ(ω) can be written as  ∞ G(τ ) exp[iωτ ]dτ . χ(ω) =

(I.4)

0

Since εE = E + 4πP = E + 4πχE = (1 + 4πχ) E , we conclude that the dielectric function ε(ω) is given by  ∞ G(τ ) exp[iωτ ]dτ . ε(ω) = 1 + 4πχ(ω) = 1 + 4π

(I.5)

(I.6)

0

Thus, ε∗ (ω) = 1 + 4π



∞

G(τ ) exp[−iωτ ]dτ =

0

= ε(−ω) = ε1 (−ω) + iε2 (−ω) .

(I.7)

Since ε∗ (ω) = [ε1 (ω) + iε2 (ω)]∗ = ε1 (ω) − iε2 (ω) ,

(I.8)

we have demonstrated that the real part of the dielectric function is an even function in frequency ε1 (ω) = ε1 (−ω) ,

(I.9)

while its imaginary part is an odd function in frequency ε2 (ω) = −ε2 (−ω) .

(I.10)

I.2 The Kramers–Kronig Relations Using the Cauchy’s residue theorem we obtain  ∞ χ(ω) = iπχ(ωT ) , P ω − ωT −∞

(I.11)

Appendix I: The Kramers–Kronig Relations and the Sum Rules

259

and, as a consequence, keeping in mind that ε(ω) − 1 , 4π  ∞ 1 ε(ω  ) − 1 P dω  . ε(ω) = 1 +  iπ −∞ ω − ω χ(ω) =

(I.12)

(I.13)

Therefore, the real and the imaginary parts of the dielectric function can be expressed as follows  ∞ ε2 (ω  ) 1 ε1 (ω) = 1 + P dω  , (I.14)  π −∞ ω − ω ε2 (ω) = −

1 P π



∞

ε1 (ω  ) − 1 dω  . ω − ω

−∞

(I.15)

In order to eliminate the negative frequencies, let us rewrite the last equations through straightforward algebraic manipulations (since the real part of the dielectric function is an even function in frequency and the imaginary part of the dielectric function is an odd function in frequency) as follows ε1 (ω) = 1 +

2 P π

2ω P ε2 (ω) = − π



∞

0



∞

0

ω  ε2 (ω  ) dω  , ω 2 − ω 2

(I.16)

ε1 (ω  ) − 1 dω  . ω 2 − ω 2

(I.17)

These are the Kramers–Kronig equations relating the real and imaginary parts of the dielectric function. When the fields are longitudinal, the loss function is the response to the longitudinal displacement. As a consequence, also the components of the loss function obey the Kramers–Kronig relations that, in this case, assume the form:  Re  Im

1 ε(ω)

1 ε(ω)

 = 1+

 = −

1 P π

1 P π 

∞ −∞





∞

Im −∞

 dω  1 , ε(ω  ) ω  − ω

    1 dω  Re − 1 .  ε(ω ) ω − ω

(I.18)

(I.19)

I.3 Sum Rules Physical arguments can be used to establish some sum rules that, taking into account the Kramers–Kronig relations, must be satisfied [2]. Let us consider, for example, the simple case of a high-frequency electric field interacting with a metal.

260

Appendix I: The Kramers–Kronig Relations and the Sum Rules

On the one hand, indicating as usual the plasma frequency with ω p , according to the Drude model on optical properties of metals we can write ε1 (ω) = 1 −

ω 2p ω2

.

(I.20)

On the other hand, according to the Kramers–Kronig theory and taking into account the high-frequency hypothesis as well, we obtain  ∞ 2 P ω  ε2 (ω  ) dω  . (I.21) ε1 (ω) = 1 − πω 2 0 Combining these two equations, we obtain the following sum rule that the imaginary part of the dielectric function must satisfy:  ∞ π ω ε2 (ω) dω = ω 2p . (I.22) 2 0 It is well known from the electromagnetic theory that the following relationship exists between complex dielectric function ε and conductivity σ of a metal: ε = 1+i

4π σ. ω

(I.23)

Simple algebraic manipulations allow us to express the real and imaginary parts of the complex conductivity, σ1 and σ2 , respectively, as follows ω 2 , 4π

(I.24)

ω (1 − ε1 ) . 4π

(I.25)

σ1 = σ2 =

Thus, the real part of the complex conductivity must satisfy the sum rule 

∞

σ1 dω =

0

ω 2p 8

.

(I.26)

Two other very important sum rules are the f-sum rule and the perfect-screening sum rule (ps-sum rule). In order to illustrate these two sum rules, let us consider the case of a metal with the following complex dielectric function ε(ω) = 1 −

ω 2p ω 2 − iΓ ω

.

(I.27)

Appendix I: The Kramers–Kronig Relations and the Sum Rules

261

From this equation it is possible to obtain Re [1/ε(ω)] and Im [1/ε(ω)]:  Re

1 ε(ω) 

Im



(ω 2 − ω 2p )ω 2p

= 1+

1 ε(ω)

(ω 2 − ω 2p )2 + Γ 2 ω 2



Γ ωω 2p

=

(ω 2 − ω 2p )2 + Γ 2 ω 2

.

(I.28)

(I.29)

Since Re [1/ε(ω)] is an even function in ω (Re [1/ε(−ω)] = Re [1/ε(ω)]) and Im [1/ε(ω)] is an odd function in ω (Im [1/ε(−ω)] = − Im [1/ε(ω)]), the Kramers– Kronig relations, (I.18) and (I.19), can be rewritten, in order to eliminate the negative frequencies, as follows  Re 

1 Im ε(ω)

1 ε(ω)



 = 1+

2ω = − P π



2 P π



∞



∞

(I.30)

  1 1 Re −1 dω  . ε(ω  ) ω 2 − ω 2

(I.31)

0



0



ω dω  , − ω2

Im

1 ε(ω  )

ω 2



From (I.28) and (I.30) it follows that (ω 2 − ω 2p )ω 2p (ω 2 − ω 2p )2 + Γ 2 ω 2

2 = P π



∞ 0



1 Im ε(ω  )



ω dω  . ω 2 − ω 2

(I.32)

In the high frequency hypothesis, (I.32) becomes ω 2p ω2

2 ≈ − πω 2

and we obtain the f-sum rule  ∞ 0



∞ 0

 1 ω  dω  , Im ε(ω  ) 

 π 1 dω = − ω 2p . ω Im ε(ω) 2

(I.33)



(I.34)

Equation (I.34) can be also deduced by the use of an exact quantum mechanical approach [3]. Let us now consider the low-frequency limit, ω → 0. In this case, from (I.32) we obtain    −ω 4p 1 2 ∞ 1 ≈ Im dω  . (I.35) ω 4p π 0 ε(ω  ) ω  From this equation the ps-sum rule follows:    ∞ π 1 1 dω = − . Im ω ε(ω) 2 0

(I.36)

262

Appendix I: The Kramers–Kronig Relations and the Sum Rules

Fig. I.1 Plot of Peff versus Wmax = ωmax for Al

Peff

1.0

0.5

0.0

1

10

100

1000

10000

W max (eV)

The ps-sum rule can also be written in the form [4, 5]    1 2 ωmax 1 − Im dω = Peff . π 0 ω ε(ω)

(I.37)

where Peff → 1 when ωmax → ∞. The trend of Peff as a function of ωmax for Al is shown in Fig. I.1. In order to discuss in more detail the f-sum rule, let us consider the plasma frequency expression ωp =

4πne2 m

1/2 .

(I.38)

In the original Drude theory, we are interested in the optical response only due to the outer-shell electrons, so n represents the number of outer-shell electrons per unit volume. If N is the total number of atoms per unit volume, then n = z N where z is the number of outer-shell electrons per atom. On the other hand, when the frequency becomes high enough, core electrons become involved in the excitation as well and the f-sum rule is still valid after including bound electrons in the dielectric function. Let us then introduce the quantity Ωp =

4π N e2 m

1/2 ,

and rewrite the f-sum rule as follow [4, 5]:    ωmax 1 2 ω Im − dω = Z eff πΩ 2p 0 ε(ω)

(I.39)

(I.40)

Appendix I: The Kramers–Kronig Relations and the Sum Rules

263

15

Fig. I.2 Plot of Z eff versus Wmax = ωmax for Al

Zeff

10

5

0

1

10

100

1000

10000

W max (eV)

where Z eff is the effective number of electrons per atom involved in the excitation. With this notation, when ωmax → ∞, Z eff → Z , where Z is the atomic number of the target. The trend of Z eff as a function of ωmax for Al is shown in Fig. I.2.

I.4 Summary In this chapter, we discussed the system response to a given stimulus. In particular, we deduced the Kramers–Kronig relations. We also described the sum rules which must be satisfied by the imaginary part of the complex dielectric function and by the real part of the complex conductivity of a given metal. Furthermore, we studied the so-called f-sum rule and ps-sum rule associated with the energy loss of electrons moving in a medium.

References 1. F. Bassani, U.M. Grassano, Fisica dello Stato Solido (Bollati Boringhieri, Torino, 2000) 2. M. Dressel, G. Grüner, Electrodynamics of Solids. Optical Properties of Electrons in Matter (Cambridge University Press, New York, 2002) 3. G.D. Mahan, Many-Particle Physics, 2nd edn. (Plenum Press, New York, 1990) 4. S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interf. Anal. 11, 577 (1988) 5. J.M. Fernandez-Varea, R. Mayol, D Liljequit, F. Salvat, J. Phys.: Condens. Matter 5, 3593 (1993)

Appendix J

From the Electron Energy Loss Spectrum to the Dielectric Function

The energy loss function (ELF) in the optical limit can be obtained by means of a transformation of the experimental transmitted electron energy loss spectrum (EELS). Such a transformation can be applied to the EELS after erasing the elastic peak and multiple scattering in order to deal with the single-scattering spectrum S(W ).

J.1 From the Single-Scattering Spectrum to the Energy Loss Function The relationship between the single-scattering spectrum S(W ) and the ELF,  1 Im ε(q=0,W ) , is given by [1, 2, 3]

    β 2 I0 t 1 ln 1 + S(W ) = Im (J.1) πa0 mv 2 ε(q = 0, W ) ϑW where I0 is the zero-loss density, t is the sample thickness, a0 is the Bohr radius, m is the electron mass, v is the incident electron velocity, β is the collection semiW2 angle, ϑW = γmv 2 is a characteristic scattering angle for energy loss W , and γ is the relativistic factor. After collecting the transmitted electron energy loss spectrum and applying to it the so-called Fourier-Log transformation—for erasing the elastic peak and multiple scattering [1],—the transformation described by (J.1) allows us to obtain the ELF.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1

265

266

Appendix J: From the Electron Energy Loss Spectrum to the Dielectric Function

J.2 Summary We described a method for calculating the ELF in the optical limit using the experimental transmitted electron energy loss spectrum.

References 1. R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 3rd edn. (Springer, New York, 2011) 2. M. Stöger-Pollach, A. Laister, P. Schattschneider, Ultramicroscopy 108, 439 (2008) 3. D.T.L. Alexander, P.A. Crozier, J.R. Anderson, Science 321, 833 (2008)

Index

A Allotropic form, 7, 68, 69 Amorphous carbon, 7, 68, 69 Angular momentum, 17, 206 Associated Legendre functions, 16, 17 Atomic potential, 195 Auger electron spectroscopy, 7, 8, 54, 151, 153, 157, 159, 170

B Backscattered electrons, 1, 8, 115, 116, 155, 162 Backscattering coefficient, 5, 89, 93, 105, 108, 115, 116, 118–136 Bessel functions, 17, 199, 212, 222 Bethe-Bloch formula, 4, 29, 54, 58, 116 Bethe ridge, 62, 63, 250 Bode quadrature rule, 12, 13 Built-in potential, 178, 179, 183 Bulk plasmon, 29, 76, 77, 79, 80, 103

C C60-fullerite, 7, 68, 69 Central potential, 191, 205 Chen and Kwei theory, 77, 80, 102, 249–251 Computational physics, 11 Continuous-slowing-down approximation, 4, 89, 90, 93, 94, 103, 104, 122–124, 136, 140, 141, 144, 145, 148, 149 Coulomb potential, 192, 193 Critical dimension SEM, 8, 9, 173, 174, 183

Cross-section, 4, 21–26, 29–39, 80, 89, 90, 92, 94, 117, 122–126, 133, 136, 142, 151, 173, 174, 187, 188, 192, 193, 203, 209, 218, 219, 223, 224, 234 D Density-matrix formalism, 40 Depth distribution, 89, 110, 133, 134, 136, 147, 148, 159, 174, 175 Diamond, 7, 68, 69 Dielectric function, 4, 59, 62, 63, 76, 153, 154, 232, 243, 251 Differential elastic scattering cross-section, 203 Dirac equation, 205, 207–211 Dispersion law, 62, 63, 80 Dissociative electron attachment, 2, 180 Doping contrast, 8, 9, 177 Double scattering experiments, 41 E Eigenfunctions, 17 Eigenvalues, 197, 208, 209 Elastic mean free path, 90, 94, 103, 193 Elastic peak electron spectroscopy, 6, 8, 160 Elastic scattering, 4, 5, 21, 29–39, 77, 80, 90– 92, 94–96, 122–126, 133, 136, 151, 152, 187, 188, 192, 193, 205, 209, 218, 219, 223, 224 Electron affinity, 99, 145, 176–178 Electron energy loss spectroscopy, 5, 153 Electron-molecule elastic scattering, 48

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Dapor, Transport of Energetic Electrons in Solids, Springer Tracts in Modern Physics 290, https://doi.org/10.1007/978-3-031-37242-1

267

268 Energy dispersive spectroscopy, 54 Energy distribution, 168, 170 Energy gain, 5, 29, 53, 231, 235, 236, 238, 240, 241 Energy loss, 4–6, 8, 23–26, 29, 53, 59, 63, 64, 76, 77, 79, 80, 89, 90, 93, 95– 98, 101, 102, 104, 136, 140–142, 145, 149, 151, 153, 155–157, 159, 170, 231, 235, 236, 238, 241, 244, 245, 247–249, 251 Energy selective SEM, 173, 176, 178 Energy straggling, 5, 24–26, 89, 93, 94, 103, 104, 140, 142, 144–147, 149, 173 Euler’s method, 15 F F-sum rule, 260–262 Fermi energy, 165 First Born approximation, 3, 4, 30–32, 35, 187, 188, 191, 192 Fourier-Log transformation, 265 Fourth order Runge-Kutta method, 15 Fröhlich theory, 3, 4, 29, 53, 80, 231, 234, 241 Full width at half maximum, 169 G Gaussian quadrature rule, 13 Generalized oscillator strength method, 71, 74, 75, 253, 254 Glassy carbon, 7, 68, 69 Graphite, 7, 68, 69, 153, 154 H Hadron therapy, 1 I Inelastic mean free path, 4, 6, 23, 24, 29, 59, 63, 65, 67, 75, 77, 78, 102, 142, 241, 248, 250 Inelastic scattering, 4–6, 21, 24, 29, 54, 80, 94, 95, 101, 102, 142, 153, 154, 250 Irregular spherical Bessel functions, 18 K Kramers–Kronig relations, 257, 258 L Laplacian operator, 196

Index Legendre polynomials, 16–18, 196, 197, 199, 212–214 Li et al. theory, 249 Linescan, 174–177 Linewidth measurement, 173 M MELF-GOS, 71, 74, 75, 254 Mermin theory, 253 Monte Carlo, 1, 9, 35, 36, 76, 80, 83, 85, 87, 89, 90, 93, 94, 101, 102, 104, 115, 122–136, 140–148, 151, 153– 161, 166–170, 173–179, 183, 231, 241 Most probable energy, 169 Mott theory, 3, 29, 31–34, 36–39, 122–126, 205, 224, 225 Multilayers, 110, 114 Multiple reflection method, 105, 114 N Nanometrology, 8, 9, 173, 183 Numerical analysis, 11 Numerical differentiation, 11 Numerical quadrature, 12 O Operators, 189, 190, 196, 197, 206, 207 Optical data, 63, 64, 154–156 Optical theorem, 203 Ordinary differential equations, 15 P Partial wave expansion method, 195, 203 Partial waves, 196, 199 Pauli matrices, 206, 218 Phase shifts, 198, 199, 212, 222, 223 Phonon, 4–6, 26, 29, 53, 76, 80, 93, 94, 98, 104, 140, 142, 145, 153, 173, 231, 233–236, 238–241 Plasmon, 4, 6–9, 26, 29, 54, 76, 77, 79, 80, 101, 103, 140, 152–155, 159, 173, 251 PMMA, 65–67, 140, 141, 144–148, 160, 161, 173–177 Polarization, 218 Polarization function, 41 Polarization parameters, 40 Polaronic effects, 4, 5, 29, 75, 80, 140 Positron, 65, 133–136, 209, 224 ps-sum rule, 260–262

Index Q Quasi-elastic scattering, 53 R Radial equation, 195, 198, 199 Random numbers, 83–87, 103, 167 Recursion relations, 17 Regular spherical Bessel functions, 19 Relativistic partial wave expansion method, 3, 4, 29, 31, 33, 34, 92, 122, 123, 125, 126, 205, 225 Ritchie theory, 3, 59, 96, 243 Runge-Kutta method, 15 Rutherford cross-section, 3, 4, 29–36, 187, 192, 193, 223, 224 S Scanning electron microscopy, 173, 174, 176, 178, 183 Scattering amplitudes, 201–203 Schrödinger equation, 17, 18, 187, 189, 193, 195 Secondary electrons, 1, 5, 7, 9, 54, 96, 99– 101, 115, 139–145, 147, 149, 151– 153, 160, 162, 164, 166–170, 174, 175, 177–179, 183 Secondary electron yield, 5, 139–141, 143, 144, 146–148, 173, 176

269 Second order Runge-Kutta method, 15 Sherman function, 40, 41 Simpson quadrature rule, 12, 13 Spherical Bessel functions, 18 Spherical harmonics, 214 Spherical Neumann functions, 18, 19 Spin, 208, 210, 216, 217 Spin-polarization, 36, 40 Step-length, 90, 91, 94 Stopping power, 4, 21, 23, 24, 26, 29, 56– 59, 65, 66, 90, 93, 122–124, 127–133, 136 Sum rules, 257 Surface phenomena, 76, 155 Surface plasmon, 76, 77, 101, 103

T Thin films, 105, 106, 108, 110, 114 Total elastic scattering cross-section, 203 Total electron yield, 143, 144 Transmission coefficient, 99–101 Transport cross-section, 121 Trapezoidal quadrature rule, 12 Triple scattering experiments, 43

W Work function, 99