Thermal transport for applications in micro/nanomachining 9783540736059, 3540736050

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microtechnology and mems

microtechnology and mems Series Editor: H. Fujita D. Liepmann The series Microtechnology and MEMS comprises text books, monographs, and state-of-the-art reports in the very active field of microsystems and microtechnology. Written by leading physicists and engineers, the books describe the basic science, device design, and applications. They will appeal to researchers, engineers, and advanced students. Mechanical Microsensors By M. Elwenspoek and R. Wiegerink

Micromechanical Photonics By H. Ukita

CMOS Cantilever Sensor Systems Atomic Force Microscopy and Gas Sensing Applications By D. Lange, O. Brand, and H. Baltes

Fast Simulation of Electro-Thermal MEMS Efficient Dynamic Compact Models By T. Bechtold, E.B. Rudnyi, and J.G. Korvink

Modelling of Microfabrication Systems By R. Nassar and W. Dai Micromachines as Tools for Nanotechnology Editor: H. Fujita Laser Diode Microsystems By H. Zappe Silicon Microchannel Heat Sinks Theories and Phenomena By L. Zhang, K.E. Goodson, and T.W. Kenny Shape Memory Microactuators By M. Kohl Force Sensors for Microelectronic Packaging Applications By J. Schwizer, M. Mayer and O. Brand Integrated Chemical Microsensor Systems in CMOS Technology By A. Hierlemann CCD Image Sensors in Deep-Ultraviolet Degradation Behavior and Damage Mechanisms By F.M. Li and A. Nathan

Piezoelectric Multilayer Beam-Bending Actuators Static and Dynamic Behavior and Aspects of Sensor Integration By R. Ballas CMOS Hotplate Chemical Microsensors By M. Graf, D. Barrettino, A. Hierlemann, and H.P. Baltes Capillary Forces in Microassembly Modeling, Simulation, Experiments, and Case Study By P. Lambert Microfluidics By J.J. Ducr´ee and R. Zengerle Thermal Transport for Applications in Micro / Nanomachining ¨¸ By B.T. Wong and M.P. Menguc

B.T. Wong ¨¸ M.P. Menguc

Thermal Transport for Applications in Micro/Nanomachining With 69 Figures

123

Dr. Basil T. Wong

¨¸ Professor Dr. M. Pinar Menguc

University of Kentucky College of Engineering Department of Mechanical Engineering 318 RGAN Building Lexington, KY 40511, USA E-mail: [email protected]

University of Kentucky College of Engineering Department of Mechanical Engineering 269 RGAN Building Lexington, KY 40506-0503, USA E-mail: [email protected]

Co-Authors of Chapters 8 and 9: Ravi Kumar

Illay “Victor” Kunadian

Co-Author, Chapter 8 University of Kentucky

Co-Author, Chapter 8 University of Kentucky

Jaime A. S´anchez Co-Author, Chapter 9 University of Kentucky

Series Editors: Professor Dr. Hiroyuki Fujita University of Tokyo, Institute of Industrial Science 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

Professor Dr. Dorian Liepmann University of California, Department of Bioengineering 6117 Echteverry Hall, Berkeley, CA 94720-1740, USA

ISSN 1439-6599 ISBN 978-3-540-73605-9 Springer Berlin Heidelberg New York Library of Congress Control Number: 2008921485 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media. springer.com © Springer-Verlag Berlin Heidelberg 2008 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the authors and SPi Publisher Services Cover: eStudio Calamar Steinen Printed on acid-free paper

SPIN 11898191

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Preface

As the advances in nanosciences and nanotechnology open ways to new engineering concepts, understanding and modeling thermal transport at nanoscales are becoming crucial for the success of future applications and discoveries. Traditional transport theories are typically applicable to bulk objects. Among all, the Fourier law of heat transfer and the Ohm law of electricity are valid for relatively large systems, as opposed to the molecular regime where both continuum and local thermodynamic equilibrium approximations become questionable. With the increasing focus on smaller length and shorter time scales, the intricate effects of energy carriers, including electromagnetic waves, electrons, and phonons, need to be considered in the analyses of transport phenomena. The development of new theories requires a knowledge of the true physics concerning these energy carriers and corresponding scattering mechanisms. In addition to theory, future engineers need to be exposed to approximations, models, and solution methodologies to tackle the complexities of these new and challenging applications. The focus of this monograph is on thermal transport modeling from nanoto micro-scale levels, starting from 1 nm. Since one of the primary objectives of nanoscale engineering is material removal within a range of 1–100 nm, we use the term micro/nanomachining and nanomanufacturing throughout the manuscript. The equations and solution methodologies presented here are valid for a number of practical problems, and some for larger bulk systems. We present them coherently and specifically for electron-beam-based applications. The emphasis here is on the particle theories, specifically for electrons and phonons. All the theoretical details are provided, starting from first principles; nevertheless, all the physical concepts introduced need to be further evaluated experimentally, which is left to future studies. The first chapter of the monograph presents a short review of micro/nanomachining and nanomanufacturing within a historical context. Transport theories for different energy carriers are briefly outlined to follow the discussions given in later chapters. A general description of an example problem

VI

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is presented, which is considered as a test case for the numerical simulations conducted in the rest of the monograph. Transport mechanisms at the micro/nanoscale levels are described in Chap. 2, and a summary of the fundamental concepts is given. Properties and behavior of electrons, phonons, and to some extent, photons, are discussed. The Boltzmann transport equation (BTE) is introduced, and its physical meaning is explained. After that, different equations which stem from the BTE are outlined, including the conservation of momentum and energy equations, the moments of the BTE, the electron-beam transport equation (EBTE), the radiative transfer equation (RTE), the phonon radiative transport equation (PRTE), and the electron transport equation (ETE). The RTE, which is quite well-known in the heat transfer community, provides a starting point for understanding the transport equations for electrons and phonons. It should be understood, however, that the RTE is not applicable to radiative transfer at nanoscales; for that the Maxwell equations need to be introduced, which is beyond the scope of this work. The chapter, discusses the governing equations for the modeling of electron-phonon transport, including the Fourier Law, the two-temperature model (TTM) and the dual-phase lag (DPL) model. Finally the details of the electron-phonon hydrodynamic equations (EPHDEs) are presented, which allow the calculation of energy transfer due to the motion of energy carriers within the medium. Two flow charts are provided at the end of the chapter to put all these models in perspective and to help the reader to choose the right set of models for a given scenario. Chapter 3 is devoted to the discussion of Monte Carlo (MC) methods, which are known to be quite versatile and extensively used for the solution of the BTE in different geometries. First, a general introduction is given for MC methods and the probability density functions. After that, two different solution techniques, called continuous slowing-down approach (CSDA) and the discrete inelastic scattering (DIS), are outlined for the solution of the EBTE. Finally, MC methods for the ETE and the PRTE are discussed. In Chap. 4, we focus on the solution of the EBTE with a MC method. Different electron scattering mechanisms inside a participating medium are discussed, and the derivations of the electron properties needed in the MC simulation are given. In Chap. 5, the EBTE is coupled with the Fourier heat transfer formulation to predict micro/nanomachining with a single carbon nanotube (CNT) probe. In addition, extending the concept to an array of probes for sequential machining is discussed. The Fourier law is not necessarily applicable to modeling thermal transport at nanometer scales. In Chap. 6, we determine the range of its validity by comparing the Fourier conduction model with the TTM, where electron and lattice energies are treated separately before coupling of the corresponding energy equations. Comparisons between these two sets of solutions allow the reader to decide under what micro/nanomachining conditions the Fourier law can readily be used.

Preface

VII

We present two advanced electron-phonon modelings in Chap. 7. In the first one, we consider the EPHDEs to describe the electrical and thermal behaviors of the metal semiconductor field effect transistors (MESFETs). In the second, the construction of a MC simulation is described for the electronic thermal conduction to predict the so-called pseudo-temperature profiles of electrons at nanoscale, considering the ballistic nature of electrons. Chapter 8 is devoted to an overview of parallel-computation methodologies used to solve the transport equations. Different parallelization strategies and architectures are discussed. The same problem considered in Chap. 6 is solved using the TTM and the increase in computational speed is discussed for different clusters. With the feature of decreasing size, micro/nanomachining is likely to converge with molecular processes. At such a small scale, the continuum models need to be re-evaluated. Once implemented properly, we can obtain the required insight to thermal transport during a micro/nanomachining process using molecular dynamics (MD) simulations. In effect, MD simulations mimic physical experiments, albeit over a relatively small computation domain, due to the requirements for extensive computational resources. In Chap. 9, a review and discussion of MD simulations are outlined. We present a simple example where the MD simulations are coupled with an MC method used to model electron-beam propagation. Finally, general conclusions are drawn and future works required improving the modeling of micro/nanomachining applications are summarized in Chap. 10. This monograph is intended to serve as a reference rather than a comprehensive textbook. It is written as a handy resource for students and researchers working on the numerical and theoretical aspects of thermal transport phenomena at micro- and nanoscales. Naturally, the subject areas covered here are biased toward our own research efforts, and by no means is it claimed that all the models for all potential applications are discussed. Specific emphasis is on particle theories, and on electrons and phonons as energy carriers. Even though laser-beam based machining can be modeled with the use of similar approaches, no specific attempt is made to discuss the details of laser-matter interactions. In addition, even though “near field radiation transfer” must be considered for nanoscale machining, and even though we have an active research program on the subject, we decided not to include it in this monograph as it requires discussion of “wave” approaches. Readers are encouraged to refer to other textbooks, supplementary materials, and the ever growing body of literature for additional and the most recent information related to these problems. Among all relevant references, we would like mention two relatively new textbooks on the subject, by Chen [30] and Zhang [222], for additional discussion on nanoscale transport phenomena. This book grew out of Basil Wong’s Ph.D. dissertation [208] at the University of Kentucky. However, both the breadth and depth of the discussion provided in the dissertation have significantly been enhanced since then. Additional theoretical approaches, models, comparisons and further practical

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information are included in Chaps. 8 and 9, based on the work of three other students. Ravi Kumar and Illay “Victor” Kunadian contributed to the discussion of the parallelization algorithms and hardware presented in Chap. 8; they worked on nanoscale thermal transport during their respective MS theses. Details of MD simulations in Chap. 9 are provided by Jaime A. S´ anchez, who has recently completed his Ph.D. under M.P. Meng¨ uc¸. During the preparation of this monograph we had extensive discussions with several researchers and students. Professors R. Ryan Vallance (George Washington University), Apparao Rao (Clemson University), Sungho Jin (University of California, San Diego), who are our collaborators in a related NSF NIRT grant, have contributed significantly to our understanding of micro/nanomachining applications. Several graduate students, including Jaime A. S´anchez, King-Fu Hii, Ravi Kumar, Ellie Hawes, Mathieu Francoeur, Illay “Victor” Kunadian and Matt Robinson, helped us to sharpen our views on thermal transport and proofread many parts of the text. Professors A. Rao of Clemson University, Z. Zhang of Georgia Tech, and T. Okutucu of METU, Ankara, provided valuable insight into different aspects of our discussions. We gratefully acknowledge their help and contributions. In the end, however, all omissions and mistakes are solely our responsibility. This material is based upon work supported by the National Science Foundation through a Nanoscale Interdisciplinary Research Team (NIRT) award from the Nano-Manufacturing program in Design, Manufacturing, and Industrial Innovation (DMI-0210559). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Additional support was received from the Kentucky Science and Education Foundation (KSEF-1045-RDE-008) and the University of Kentucky. Lexington, KY, USA, March 2008

B.T. Wong M.P. Meng¨ uc¸

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation for Building Small . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Nanopatterning Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Electron-Beam Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Micro/Nanomachining with Electrons . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Outline of the Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2

Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Energy Carriers: Electrons, Photons and Phonons . . . . . . . . . . . 2.2 Classification of Transport Models . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Time and Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Relaxation-Time Approximation . . . . . . . . . . . . . . . . . . . . 2.5 Intensity Form of Boltzmann Transport Equation . . . . . . . . . . . . 2.5.1 Electron-Beam Transport Equation . . . . . . . . . . . . . . . . . . 2.5.2 Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Phonon Radiative Transport Equation . . . . . . . . . . . . . . . 2.5.4 Electron Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Moments of Boltzmann Transport Equation . . . . . . . . . . . . . . . . . 2.6.1 Continuity or Conservation of Number of Particles . . . . . 2.6.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Conservation of Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Macroscale Thermal Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Micro/Nanoscale Thermal Conduction . . . . . . . . . . . . . . . . . . . . . 2.8.1 Two-Temperature Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Dual-Phase Lag Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 Electron–Phonon Hydrodynamic Equations . . . . . . . . . . . 2.9 Nanoscale Thermal Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Thermal Radiation at Nanoscales . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 16 18 19 20 22 24 24 27 29 30 31 32 34 36 38 39 41 41 43 45 51 52 53

X

3

Contents

Modeling of Transport Equations via MC Methods . . . . . . . . 3.1 Cumulative Probability Distribution Function . . . . . . . . . . . . . . . 3.2 Building a CPDF Table for a MC Method . . . . . . . . . . . . . . . . . . 3.3 Monte Carlo Simulation for Particle-Beam Transport . . . . . . . . 3.3.1 Setting up Computational Grid . . . . . . . . . . . . . . . . . . . . . 3.3.2 Random Number Generator . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Simulation Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Incident Beam Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Direction of Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Distance of Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.7 Attenuation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Monte Carlo Simulation for Thermal Conduction by Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Initial Electron Distributions . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Launching of Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Non-Equilibrium Fermi-Dirac Distribution . . . . . . . . . . . . 3.4.4 Pseudo-Temperature Calculations . . . . . . . . . . . . . . . . . . . 3.4.5 Scattering of Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Monte Carlo Simulation for Phonon Conduction . . . . . . . . . . . . . 3.5.1 Initial Phonon Distributions . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Launching and Tracing Phonons . . . . . . . . . . . . . . . . . . . . 3.5.3 Phonon Pseudo-Temperature Calculations . . . . . . . . . . . . 3.5.4 Scattering of Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Normalization of the Statistical Results . . . . . . . . . . . . . . . . . . . . 3.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 60 61 62 63 63 64 66 67 68 69 70 72 73 73 74 75 75 76 76 78 78 78 79 80 80

4

Modeling of e-Beam Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.1 Electron Scattering Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 Elastic Scattering of an Electron by an Atom . . . . . . . . . . . . . . . 82 4.2.1 Rutherford Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.2 Mott Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3 Continuous Inelastic Scattering Approach: The Bethe Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4 Discrete Inelastic Scattering Treatment: The Dielectric Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5 Electron Reflection and Refraction at a Surface . . . . . . . . . . . . . 100 4.6 Monte Carlo Simulation Results and Verifications . . . . . . . . . . . . 101 4.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5

Thermal Conduction Coupled with e-Beam Transport . . . . . 105 5.1 Modeling Thermal Transport During Micro/Nanomachining . . 105 5.2 Thermal Conduction due to Single Electron-Beam Heating . . . 107 5.2.1 Problem Description and Basic Assumptions . . . . . . . . . . 107

Contents

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5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.2.7

Computational Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Electron-Beam Monte Carlo Simulation . . . . . . . . . . . . . . 111 Auxiliary Heating Using Laser Beam . . . . . . . . . . . . . . . . . 112 Fourier Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Modeling “Melting” and “Evaporation” . . . . . . . . . . . . . . 121 Computational Parameters for Micro/Nanomaching Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.2.8 Electron-Beam Deposition Profiles . . . . . . . . . . . . . . . . . . . 122 5.2.9 Electron-Beam Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.2.10 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.3 Sequential Patterning Using Electron Beam . . . . . . . . . . . . . . . . . 126 5.3.1 Problem Description and Assumptions . . . . . . . . . . . . . . . 126 5.3.2 Computational Methodology . . . . . . . . . . . . . . . . . . . . . . . . 127 5.3.3 Computational Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.3.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.3.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6

Two-Temperature Model Coupled with e-Beam Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.1 Two-Temperature Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.2 Problem Description and Assumptions . . . . . . . . . . . . . . . . . . . . . 137 6.3 Electron-Beam Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . 138 6.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.4.1 Electron-Beam Deposition Profiles . . . . . . . . . . . . . . . . . . . 139 6.4.2 Two-Temperature Model Predictions . . . . . . . . . . . . . . . . . 140 6.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7

Thermal Conduction with Electron Flow/Ballistic Behavior . . . . . . . . . . . . . . . . . . . . . 147 7.1 Electron–Phonon Hydrodynamic Modeling . . . . . . . . . . . . . . . . . . 148 7.1.1 Governing Equations for Electrons and Phonons . . . . . . . 148 7.1.2 Physical Domain and Boundary Conditions . . . . . . . . . . . 150 7.1.3 Thermophysical Properties for the Simulation . . . . . . . . . 152 7.1.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.2 Thermal Conduction by Electrons via Monte Carlo Method . . . 156 7.2.1 Electron Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.2.2 Electron–Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . 159 7.2.3 Electron–Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . 161 7.2.4 Monte Carlo Simulation Results . . . . . . . . . . . . . . . . . . . . . 163 7.2.5 Remarks on Electron Conduction Simulations . . . . . . . . . 165 7.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

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8

Parallel Computations for Two-Temperature Model . . . . . . . 167 8.1 Introduction to Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.2 Parallelization of the TTM for Micro/Nanomachining . . . . . . . . 169 8.3 Parallel Computing Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.4 Implementation of Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.5 Parallel Computing Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 8.6 Parallel Computing Using Parabolic Two Step (PTS) Model . . 180 8.7 Parallel Computing Including an Electron Beam . . . . . . . . . . . . . 181 8.8 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9

Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 185 9.1 Overview of Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.1.1 Interatomic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.1.2 Time Integration of the Equations of Motion . . . . . . . . . . 190 9.1.3 Molecular Dynamics in Different Ensembles . . . . . . . . . . . 191 9.2 Analysis of Atomic Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.2.1 Equipartition Theorem and the Virial . . . . . . . . . . . . . . . . 192 9.2.2 Determination of Phase Change . . . . . . . . . . . . . . . . . . . . . 192 9.2.3 Velocity Autocorrelation Function and Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.2.4 Phonon Spectral Densities . . . . . . . . . . . . . . . . . . . . . . . . . . 194 9.3 Molecular Dynamics Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 9.3.1 Bulk Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 9.3.2 Surface Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.4 Coupling of MD with Monte Carlo Method . . . . . . . . . . . . . . . . . 199 9.5 Simulations of Electron Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 9.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 A

Derivation of Matrix for the Fourier Conduction Law . . . . . . 209

B

Simplified Electron–Phonon Hydrodynamic Equations . . . . . 213

C

Thermophysical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

List of Abbreviations

BCC BTE CNT(s) CPDF CSDA DIS DPLM EBL EBTE EPHDEs ETE FCC FD G-S HCP HDM LHS MC MD MESFET(s) MWCNT(s) NNI NRT ODE(s) PDE(s) PRTE PTS RDF

Body centered cubic Boltzmann transport equation Carbon nanotube(s) Cumulative probability distribution function Continuous slow-down approach Discrete inelastic scattering Dual-phase lag model Electron beam lithography Electron-beam transport equation Electron-phonon hydrodynamic equations Electron transport equation Face centered cubic Finite difference Gauss-Seidel Hexagonal close-packed Hydrodynamic model Left-hand side Monte Carlo Molecular dynamics Metal semiconductor field effect transistor(s) Multi-walled carbon nanotube(s) National nanotechnology initiative Near-field radiative transfer Ordinary differential equation(s) Partial differential equation(s) Phonon Radiative Transport Equation Parabolic two-step Radial distribution function

XIV

List of Abbreviations

RHS RTE SOR STM SWCNT(s) TEM TTM

Right-hand side Radiative transfer equation Successive over relaxation Scanning tunneling microscopy Single-walled carbon nanotube(s) Transmission electron microscopy Two-temperature model

Nomenclatures

A c Cxx xx Cxx d D e E E0 E˙
E f F
ext g h ¯h H I J k kB kT
k n n ˜ nI Na p P qT

Atomic weight (kg mol−1 ) Speed of light (m s−1 ) Specific heat (J m−3 K−1 ); xx denotes the type of the specific heat Scattering cross-section (m2 ); xx depends on the definition Diameter of scatterer (m) Density of states (m−3 ) Electric charge (C) Electron energy (eV) Initial kinetic energy of the electron-beam (eV) Input power of the electron-beam (W) Electric field vector (V m−1 ) Particle distribution function (–) External force vector (N) Average direction cosine in s-direction (–) Planck constant (J s−1 ) Angular Planck constant (J s−1 ) Magnetic field Intensity (W m−2 ) Mean ionization energy (J) Wave number (m−1 ) Boltzmann’s constant (J K−1 ) Thermal conductivity (J m K−1 ) Wave vector (m−3 ) Energy carrier density (m−3 ) Complex index of refraction Imaginary part of the refractive index Avogadro number (–) Momentum (kg m s−1 ) Volumetric momentum (=np) (kg m−2 s−1 ) Thermal heat flux (W m−2 )

XVI

Nomenclatures

Q r
r R Rbeam Rlaser Rp Rx1→x2 Ran s S ST t T v V w W ˜ W Z

Heat (J) Radial location Position vector (m) Cumulative probability distribution function (–) 1/e2 Gaussian radius Radius of laser beam Penetration depth (m) Reflectivity when light propagates from medium x1 to x2 a Random number (–) Axis of propagation (–) Distance of interaction (m) Rate of volumetric energy source (W m−3 ) Time (s) Temperature (K) Velocity (m s−1 ) Potential (V) Average carrier energy Average carrier energy density (eV m−3 ) Scattering rates (s−1 ) Atomic number (–)

Greek Symbols ℓ κ ϕ ρ µ β τr Φ ν Γ σ Θ ω λ λxx Ω θ φ ε Ψ

The characteristics length (m) Absorption coefficient (m−1 ) Azimuthal angle measured from an axis normal to s-axis (degree) Density (kg m−3 ) Direction cosine (–) Extinction coefficient (= κ + σ) (m−1 ) Relaxation time (s) Phase function (sr−1 ) Frequency (s−1 ) Ratio of ℓ to λ (–) Scattering coefficient (m−1 ) Scattering polar angle measured from s-axis (rad) Angular frequency (s−1 ) Wavelength (m) Mean free path (m) Solid angle (sr) Polar angle (degree) Azimuthal angle (degree) Dielectric function (–) Normalized electron-energy deposition profile (m−3 )

Nomenclatures

Subscripts A d e F λ LO ν ph T

Acoustic phonon Drift For electrons Fermi level Wavelength dependent Optical phonon For photons For phonons Thermal component

Superscripts ′

el inel

Scattered Elastic Inelastic

XVII

1 Introduction

In Greek, the word “nano” means “small” or “dwarf.” In the scientific literature, nano is a prefix that refers to “one-billionth,” that is 10−9 of the scale it modifies. One nanometer (nm) is very small, approximately 1/100,000 to 1/50,000 of a single strand of human hair, or about one-half to one-third of the width of DNA, or as wide as a few atoms. Nanotechnology is concerned with this size range, starting from 1 up to 100 nm. Its definition comes from the National Nanotechnology Initiative (NNI; see http://www.nano.gov) as (a) creation and use of structures, devices and systems that have novel properties and functions; (b) ability to pattern and build structures and systems starting at molecular range up to one-hundred nanometer level; and (c) ability to control processes at nanometer range for advanced material processing and manufacturing [167, 168]. Building structures and processes in this nano-realm requires a better understanding of new physics, chemistry and engineering, which maybe significantly different than the established approaches used for macro-systems. Only after that, extensive applications of micro/nanomachining or nanomanufacturing would be realized. Recent advances in the science and engineering have significantly impacted the art of building small things, which in turn has led to many innovations. The very same advances have also helped shrink the size of electronic chips, while allowing them to process a larger amount of data much faster. The famous Moore’s law dictates that the speed of processors doubles every eighteen months, which corresponds to a 1,000 fold increase in the number of transistors crammed in the same area every fifteen years. In 1975, the number of devices integrated on a chip with the area of one quarter of square inch was about 3,000. This number steadily climbed up to one million in 1990 and to more than one billion in 2006 (see Intel website http://www.intel.com/technology/mooreslaw/for the most recent numbers). The Moore law has proven to be valid for the last four decades; yet with decreasing feature size its predictive ability is being challenged [138]. The corollary of Moore’s law (a.k.a. Rock’s law) is about the cost of manufacturing of these devices, which states that the investment into the infrastructure

2

1 Introduction

of fabrication facilities would increase two-fold in every three years. The cost of new facilities and procedures are likely to be prohibitive as we approach nanoscale feature sizes. And, the challenge will always be there to push the limit of building small to the edge, which currently borders about 65 nm for silicon technologies [162], targeted down to 45 nm soon (Intel 2007). The challenges of new science, new technology, and new manufacturing paradigms at the nanoscale are clear cut, which necessitate further experimental and theoretical studies. The realm of nanotechnology is in the so-called mesoscale regime, a regime that lies between the classical domains of engineering where bulk properties and continuum formulations hold, and the molecular regime where quantum mechanics dominate. Only during the last decade or so, physics, chemistry and biology and their interactions have started to be unraveled within this nanoworld. Still, the underlying concepts and the required thermophysical properties are not readily available for successful implementation of nanomanufacturing ideas. As with any other manufacturing process, the success of nanomanufacturing does not only depend on understanding the relevant science, but also the extent of its art, or the successful implementation of engineering principles to make each and every process reliable with precision and repeatability. This requirement goes hand in hand with the development of predictive capabilities to simulate the processes of nanomanufacturing before actually building the systems. These computational models are also crucial to bringing more science to the art of nanotechnology, which is growing in leaps and bounds. Among all, the material processing and material removal requires a better understanding of the energy transfer mechanism by electrons and laser/light beams, and how the energy is dispersed at the nanometer scale to generate small patterns in a predictable way. The concept and promise of nanotechnology are covered in several scientific books and monographs, starting from Eric Drexler’s Engines of Creation [49]. Since then, many other studies have been devoted to nanosciences and nanotechnology, including those recent more technical titles by Kohler and Fritzsche [102], Chen [30], Mansoori [123], Bhushan [18], Roduner [169], Dupas et al. [50], Zhang [222], as well as the easy-to-read accounts for non-scientists by Ratner and Ratner [163] and Booker and Boysen [23]. A recent review of the National Nanotechnology Initiative [206] is also quite useful as it outlines how the NNI has shaped the research and development efforts so far and what the expectations are in future efforts. In this monograph, we mainly focus on conceptual tools to describe thermal transport inside a gold workpiece due to an impinging electron beam. Two other texts, by Chen [30] and Zhang [222] provide extensive coverage of transport phenomena as related to nanoscale devices and processes. Our objective here is more specific. We will outline the underlying physics, governing equations and the numerical models that can be used for understanding and simulations of “top-down” nanomanufacturing processes. The primary

1.1 Motivation for Building Small

3

emphasis here will be on energetic electrons (emitted from nanoscaled tips) and micro/nanomachining based on electron-matter interactions, although many of the principles discussed can readily be used for laser-based machining applications as well. As mentioned before, we focus only on the particle theories and leave the discussion of wave approaches to other references. The detailed experimental validations of these concepts are also left to other accounts.

1.1 Motivation for Building Small There is a consensus among many researchers working in the nanoworld that the potential and challenges of nanotechnology were first heralded by Richard Feynman in his famous 1959 talk titled “There is plenty of room at the bottom” [54]. He wonderfully articulated the concept of building small and inscribing on a piece of dust all the words ever written within all the books published at that time. Feynman’s insight is considered to have given birth to what we now call nanotechnology. Yet, it is well-known that the concepts of atoms and electrons have been discussed by mankind since ancient Greeks, when Lucretius, Democritus, and Epicurus played with the idea and its potential implications back in the first century BC on the shores of Aegean Sea. Nevertheless, the atomic world was simply “imagined” until the beginning of the nineteenth century, when the scientific theories were firmly established by Dalton, Lavoisier, Gay-Lussac, among others. And only in the beginning of the twentieth century were electrons actually “seen” with the help of experiments conducted by Thompson. This was a major breakthrough, as it led to the development of electron microscopy and impacted the atomic scale visualization. That very same discovery was the starting point of Feynman’s articulations, as he suggested the use of electrons as tools and probes. Another major discovery was made in 1982 with the development of the scanning tunneling microscopy (STM) by Binnig and Rohrer [20, 21]. The underlying principle of STM is the quantum mechanical phenomena of tunneling which enables the visualization of subatomic domains in conducting surfaces. After that, the path to innovation was firmly paved, and the introduction of NNI in 2000 simply helped to put the further development on solid ground. Nanomanufacturing is one of the main thrust areas of nanotechnology, as recently stated by the National Research Council on their review of the NNI [206]. In general, nanomanufacturing can be viewed either as a top-down or bottom-up process. In the former, one can manipulate the matter using an external source to create patterns as small as a few nanometers wide. In the latter, the intrinsic and natural properties of molecules themselves can be used to guide them to form new structures with novel properties for different applications. Self assembly is a paradigm of bottom-up processes, and the engine of nanobiotechnology [132], which is another thrust area of NNI. Nanobiotechnology encompasses the building blocks of living organisms

4

1 Introduction

and fundamental of materials and materials processing, and is likely to have major impact on every aspects of our lives [132, 206]. With the advances in metrology and instrumentation, nanomanufacturing and nanomachining are likely to be important engines of the nanotechnology revolution. A recent search in Web-of-Knowledge in March 2008 revealed significant surge of interest in “nanopatterning,” “nanomachining,” and “nanomanufacturing.” A keyword search for “nanomachining” returned 130 papers, whereas 1,140 publications were found for “nanopatterning.” Most of these papers have only been published during the last few years. The earliest accounts, however, trace back to 1988 [147] and 1993 [114]. Still, these numbers of citations are very small compared to more than 1900 hits obtained for “nano-lithography” as a topic keyword and more than 22,500 publications for “self-assembly.”

1.2 Nanopatterning Approaches Nanometer-wide structures can be patterned using external energy sources, including laser- or ion beams, X- or gamma-rays, or electrons. Our focus in this monograph is on electron-based machining. With the recent developments and understanding of field-emission of electrons from short tips and carbon nanotubes (CNTs), there is a great potential for new micro/nanomachining procedures. Given this, we explore the thermal transfer phenomena that take place on a workpiece which is subject to incoming high-energy electron beams. Many of the theories discussed in this monograph may also be applicable to laser-beam based processing. Particularly, with the development of extreme ultraviolet lasers and further understanding of near-field radiative transfer, sub-100 nm applications will be more feasible in the near future. Material processing with high-power and ultra-fast lasers has been explored extensively over the years for numerous industrial applications. Mega-watt lasers, with pulse durations less than 50 fs (10−15 s) are routinely used to produce desired structures or patterns on material surfaces, including micrometer size structures [15, 85, 111, 154, 185, 204]. Resolutions of these structures depend greatly on the incident spatial distribution of the beam. However, a laser can only be focused close to its wavelength due to the diffraction limit. This results in the limiting of the smallest resolution of the material processed by using traditional lasers with wavelengths in the visible spectrum to no less than a few hundred nanometers, or, about one-fourth to one-fifth of the laser wavelength. Therefore, it is not possible to create nanometer-scale indentations using a conventional laser machining approach based on far-field optical arrangements. However, extreme ultraviolet lasers have wavelengths in the order of 100 nm; consequently they can be used to form nanoscale structures. These processes can be analyzed with the solution of the full Maxwell equations to account for the near-field effects. In addition, recent research suggests that near-field effects can be considered using long-wavelength lasers

1.3 Electron-Beam Processing

5

down to nanoscale regime (see for as example [35,65,80,203]). Yet, it is still not possible to achieve molecular level precision with light-based methodologies. Even though X- or gamma-rays may be used for molecular level machining purposes, the cost required for such applications is still prohibitive. An alternative approach for micro/nanomachining is realized by using energized electrons bombarding the target solid and creating localized structural changes. Since electrons have wavelengths much smaller than that of electromagnetic energy involved in laser-beam processes, the diffraction effect does not play any role until a size much less than a nanometer is reached. Focusing the electron beam down to a few nanometers can be achieved by using electromagnetic lenses [51], although this is still one of the main challenges of the technology. Electron-based machining concept is likely to play more important role in the future, although most of the research has focused on electron-beam lithography (EBL) during the last three decades [125, 144, 187]. Nanomanufacturing is likely to continue receiving significant attention within the industry and research institutions due to the ever-growing interest in development and engineering of structures at nanometer-scale [158]. Several patterning techniques have been considered over the years for nanometer scale applications, including scanning probe approaches as summarized by Sotomayor Torres [194] and Garcia et al. [61], and listed by Bhushan [18] and Dupas et al. [50]. Among all, atomic force microscopy techniques are becoming quite attractive [70, 79, 213, 215]. More specific examples of AFM based nanoscale patterning modalities include the dip-pen lithography of Mirkin group [64, 155, 172], and recent surface-wave directed self assembly approach of our group [70, 71]. The extension of these approaches are collectively called scanning probe lithographies [213], which can be extended to applications for nanoscale metrology [158]. These concepts have been discussed extensively in the literature and outlined in review monographs cited above. As we stated above, in this monograph we are limiting ourselves to mostly electron-based micro/nanomachining of metallic (gold) workpiece, although the same set of fundamental equations can be adapted for application to other materials with relative ease. Below, we provide a more detailed discussion of electron-beam processing concept.

1.3 Electron-Beam Processing Electron-beam processing uses electron emission from an electron gun. During this process, a large amount of energized electrons is projected onto a solid workpiece to achieve the desirable machining process. The key requirement is the use of electrons with large kinetic energies, to allow them to penetrate through the lattice of the solid and to transfer their energy via inelastic collisions. If enough energy is deposited to a small volume, then nanoscale machining or material removal (via melting, sublimation, or ablation) can be

6

1 Introduction

achieved depending on the workpiece material. The typical energies of the electrons in an electron-beam process are in the order of tens of kilo-electronvolts (keV). EBL is based on this very same concept and is relatively well established. Yet the cost associated with its infrastructure is well above $1M, making it an exclusive approach afforded by only a few industries [18]. The electron-beam processing, as the name implies, changes the structure and the properties of a workpiece. The physics of the interaction between the impinging energetic electrons and the solid materials is very complex. As the electrons propagate inside the workpiece, they undergo a series of elastic or inelastic scatterings. Elastic scattering refers to the redirection of the propagating electron, while inelastic scattering involves both the redirection and energy attenuation of the electrons. Electrons transfer their energy to the target material by means of inelastic scatterings. Inelastic scattering in this scenario can be classified as an event where an incident electron causes the ionization of the atom by removing an inner-shell electron from its orbit producing a characteristic X-ray or an ejected Auger electron. In addition, inelastic scattering also includes the case where an electron collapses with a valence electron to produce a secondary electron [93]. Due to their small sizes, the accelerated electrons can easily penetrate the solid material through the lattices. Such penetration depends heavily upon the initial energies of electrons and their corresponding scattering patterns. Use of a highly energized electron beam assures large penetration depth for the propagating electrons, which can transfer their energies deep in the workpiece rather than just on the target surface. There are a number of desirable features of the electron-beam processing for application to micro/nanomachining, including: (a) the possibility of finely focused electron beams, (b) the feasibility of generating high-power-density electron beams, (c) the ability to deflect electron beams rapidly and with high accuracy, and (d) the possibility of varying electron energy with acceleration voltage, hence controlling the electron penetration range. The disadvantages include (a) the necessity for high vacuum to achieve field emission (except for non-vacuum welding and electron reactive processing), (b) the generation of harmful X-rays, and (c) the difficulty in processing electrical insulators. With the recent advances in procedures to grow CNTs and manufacturing of sharp metallic tips, field-emitted electrons are more likely to be used in future micro/nanomachining applications. Two practical application areas of electron-beam processing need further discussion. The first one is thermal processing, which includes machining, welding, annealing, and heat treatment. The second application is reactive processing, such as electron-beam processing, polymerization and depolymerization [189, 190]. The machining process is directly related to thermal processing, which is the focus of this monograph. In the following Chapters we will discuss thermal transport phenomena due to nano-tip-based machining and patterning approaches. First, we will summarize the work related to micro/nanomachining with electrons. EBL and the other nanopatterning

1.4 Micro/Nanomachining with Electrons

7

approaches, on the other hand, are beyond the scope of this treatise. Interested readers can refer to many recent articles and texts on the subject, including those by Marrian and Tennant [125], Stomeo et al. [187], Gates [63], Mahalik [118], Mailly [119], Garcia et al. [61], Ocola and Stein [144], Yang et al. [218], Hiemenz [73], and Woodson and Liu [213].

1.4 Micro/Nanomachining with Electrons The concept of micro/nanomachining with electrons requires a clear understanding of electron and solid matter interactions, which have been investigated theoretically and experimentally by different researchers over the years. Some of these historically important theoretical works include those by Whiddington [205], Archard [7], Kanaya and Okayama [96], and Joy [93]. Almost a century ago, Whiddington [205] related the electron penetration range Rp (m) with the electron acceleration voltage V (Volt) and the mass density of the metal ρ (kg m−3 ), given by the following relation: Rp =

a V2 , ρ

(1.1)

where a = 2.2 × 10−11 kg V−2 m−2 . This is the approximate depth that the penetrating electrons can reach after they are incident on a target surface. This implies that most of the energies of the propagating electrons would be absorbed in this range, but not on the surface of the workpiece. As a result, the heating of the surface of the target cannot be achieved via direct electron bombardments, but requires heat to be conducted up from the lower layers of the solid material. Machining of a workpiece is possible only if spatially-resolved phase change within the material can be achieved using an external energy source. Electronbeam melting has been used for practical applications for some time; however, only recently the science behind nanoscale melting is explored more thoroughly. S´ anchez and Meng¨ uc¸ [174, 176] have discussed molecular dynamics (MD) simulations for this purpose and summarized most of the relevant work. (Chap. 9 of this monograph is devoted to the further discussion of MD approaches.) Due to the complicated interactions between propagating electrons and the solid material, devising a physically realistic theoretical analysis of electronbeam machining is quite challenging. Electron propagation can be modeled with the electron-beam transport equation, which is usually solved using statistical approaches, such as Monte Carlo (MC) method, where a large number of electrons inside a solid material are simulated based on probability distribution functions. The desired solution is then generated according to the scorings of these electrons. Taniguchi et al. [189] and Joy [93] argued that even with a finely focused electron beam, material processing at a very small area is difficult to achieve.

8

1 Introduction

MC simulations carried out by Shimizu et al. [181] and Joy [93] suggest that the multiple scattering nature of the electron inside the solid material may cause extensive spread of electrons. However, simulations based on smaller emitters and the use of threshold heating of the workpiece show that the resolution of machining processing at nanoscale level is indeed possible [198]. The numerical approaches presented in the later chapters do not assume any probe type, but rather only the specifications of the electron beam used in the micro/nanomachining process, such as the initial beam energy, spatial beam spread, and the beam power, are important. Hence, any source of electrons that meets the required beam specifications can serve as a micro/nanomachining tool in this application. The concept of machining considered in this monograph is depicted in Fig. 1.1. As discussed above, to effectively remove atoms from the workpiece, a large amount of energy transfer from the probe via electron bombardments may be required. Depending on the probe, this may not be easy to achieve. One way to overcome this setback is to preheat the workpiece to a higher temperature through bulk heating or by the use of a laser beam. This auxiliary heating further increases the temperature of a specified location to nearly the melting point of the workpiece, where the electron bombardments occur, resulting in minimum energy required from the probe to process the material. If any field-emitted electron source is used, the electron-beam processing should be done under vacuum conditions; then the bulk heating can only be achieved by radiative or conductive transfer, but not by convective transfer. A potential candidate for such a micro/nanomachining probe would be a CNT attached to a sharp tungsten needle (see Fig. 1.2). CNT is only one of the recently discovered carbon based nanoscale structures [82,83]. Carbon is a unique element, and takes different textures and morphologies depending on how it forms. Diamond is carbon, so are spherical fullerenes [106], spheroidal hyperfullerenes [40], cylindrical nanotubes [82], conical nanocarbon [29, 179] and toroidal nanorings [126, 178]. Discovered in 1991 by Sumio Iijima, CNTs proved to be very promising for a number of applications, including field emission displays [203] and for nanolithography [191]. Its field emission properties make it a very good candidate for micro/nanomachining [75, 99]. The details of the production and properties of CNTs are well-documented in the literature (see, e.g., [18]); therefore there is no need to give a complete accounting about them. Although experiments regarding the emission property of the CNTs have been performed over the years, the full range of applicability of the CNTs as machining tools remains untested. Above all, the relationships between the maximum current extracted, the maximum voltage applied, the tube diameter, and the magnification of the emitting area on the detector screen need to be addressed. Knowledge of these relationships will be required when choosing the appropriate CNTs, later discussed in this work. The new developments in the field of CNT growth allow very intricate CNT arrays, which can be used for sequential micro/nanomachining applications.

1.4 Micro/Nanomachining with Electrons

9

Fig. 1.1. A schematic of the machining process using an electron source. The figure is not drawn to the scale. The workpiece is considered to be infinite in extent compared to the electron beam. Although a voltage is applied in the figure, this is not a must for facilitating the machining process; any electron beam that can be directed towards the workpiece from an electron source is applicable here

Figure 1.3 depicts one such array [13]. Additional studies on CNT geometries will allow the development of new tools. They are discussed extensively by the Jin group [13, 14, 32–34]. To explore the possibility of micro/nanomachining with field-emitted electrons, S´anchez et al. [177] and S´ anchez and Meng¨ uc¸ [175] presented extensive numerical simulations. It is demonstrated that the energy available for micro/nanomachining with electrons is a function of the geometry of the CNT, i.e., its radius, length, wall thickness and the shape (open or closed), and also

10

1 Introduction (a)

(b)

Fig. 1.2. Structure of probe tips and mounted CNTs for demonstrating the feasibility of nano-scale probes. (a) Tungsten probe fabricated using electro-chemical etching, and (b) Enlarged view of the mounted CNT on the probe [74]

Fig. 1.3. An aligned and patterned periodic CNT array is shown. The array is created by e-beam patterning of Ni or Fe catalyst layer into islands (e.g., 50–200 nm diameters) followed by plasma CVD growth using hydrocarbon gas [13]

1.4 Micro/Nanomachining with Electrons

11

Fig. 1.4. Heating power available to the electrons emitted from a CNT with 4 µm in length, 50 nm in radius, 10 nm of wall thickness and a closed hemispherical tip. The heating power is calculated as a function of the gap distance (Data adapted from S´ anchez et al. [177])

depends on the gap distance between the tip of the CNT and the workpiece. In Fig. 1.4 we show the effect of the gap distance on the heating power available to the emitted electrons from a CNT defined as the emission current times the applied voltage. These results indicate that as the gap distance decreases, a smaller voltage is required to achieve a given heating power. This is because of the enhancement of the field strength on the tip of the CNT as a function of the gap distance which enters in the calculation of the emission current. The available power can be further enhanced by using multiple CNTs and by focusing electrons on the workpiece. At a gap distance of 25 nm, it is ideally possible to obtain power in the fraction of a milli-watt if 100 V is applied, as shown in Fig. 1.4. With the use of multiple CNTs, it is indeed possible to heat a metallic workpiece to its melting temperature. Figure 1.5 depicts the experimental setup demonstrated by Jin group (Zhu et al. [223]) where they used CNTs to heat a molybdenum bar (about 0.5 mm in diameter). The schematic on Fig. 1.5(a) is provided to explain the experimental pictures shown on Fig. 1.5(b) and (c). The red glowing molybdenum is at about 1,500–2,000 K, as its temperature increases with the duration of the experiment. All said, machining with electrons is likely to be one of the promising applications of nanoscale engineering, and is worth further exploration and experimental validation. Our numerical approaches presented in this monograph deal with electron-beam propagation and thermal conduction induced by an electron beam. These approaches are applicable when temperature distribution of the workpiece is desired. In addition, the type of electron

12

1 Introduction

Fig. 1.5. (a) A schematic of the experimental setup by Jin group (Zhu et al. [223]); (b) (c) The actual experiment picture at two different times showing the red-glow of molybdenum bar heated by the field emitting CNTs

source, whether it is from an electron gun or electron field emission, does not alter the numerical approaches given in this context.

1.5 Outline of the Monograph The goal in this monograph is to discuss theoretical and numerical modeling strategies mainly used for electron-beam induced thermal conduction, and hence for simulation of top-down micro- and nanomachining processes. The heat transfer mechanisms involved in these problems are electron-beam transport based on impinging electrons, coupled electron-phonon transport inside the target workpiece, and radiative energy transport when the process is assisted with a laser beam (see Fig. 1.6). The predictive models for these phenomena are necessary and essential for understanding and implementation of future applications. Starting in Chap. 2 we introduce readers to the governing equations of thermal transport for micro/nanomachining. All these derivations originate from the Boltzmann transport equation (BTE), yet based on different assumptions and simplifications. Figures 2.1 and 2.2 outline all the related transport models and their role in micro/nanomachining applications. Modeling of the BTEs for realistic physical conditions is not always straightforward. Among all solution strategies, only MC models can easily be adapted to most practical situations. For this reason, in this monograph we primarily consider the MC methods as the preferred solution techniques. Chapter 3 is devoted to discussion of solution strategies of BTEs using MC methods, including the EBTE, the electron transport equation (ETE), and the phonon radiative transport equation (PRTE). In Chap. 4, the EBTE is further analyzed after giving the theoretical background required for its application to electron-beam based machining. In Chaps. 5 and 6, we introduce additional numerical models, where MC methods for electrons are coupled

1.5 Outline of the Monograph

13

Fig. 1.6. Interactions of electron-beam transport, radiative transfer and electron– phonon transport shown schematically for electron-beam based micro/nanomachining, aided by auxiliary radiative heating

with two different formulations of thermal conduction problem. The Fourier law is employed in Chap. 5 to predict the temperature profile inside the workpiece. The MC method used for this purpose is called the continuous slow-down approach (CSDA), as discussed in Chap. 3. Later, in Chap. 6, the two-temperature model (TTM) is used to study the electron-phonon transport induced by an electron beam. In this case another electron-beam MC method, called the discrete inelastic scattering (DIS) method, is used. The TTM discussed in Chap. 6 can be replaced with the electron–phonon hydrodynamic equations (EPHDEs) to account for the electrical flow and charge accumulation inside a target workpiece. This procedure is outlined in Chap. 7. In addition, a simulation procedure to predict the electronic thermal conduction using a MC method is presented, which accounts for ballistic behavior of electrons. In Chap. 8, we summarize a parallel computation procedure to allow simulation of machining based on the TTM. Different hardware architectures and their relative performances are outlined. In Chap. 9, we present the fundamentals of molecular dynamics (MD) simulations, which is important when working on isolated nano-scale and/or molecular-level patterning processes. Finally, in Chap. 10 we provide an overview of the monograph.

2 Transport Equations

All natural physicochemical and biological phenomena are based on intertwined and complex energy and mass transfer mechanisms. These mechanisms are also the main drivers for all man-inspired manufacturing and machining processes. As humans discovered long time ago, it is almost always necessary to heat an object before forming it into a desired shape and structure. The heat can be supplied via convection, conduction or radiation transfer, with a laser, electron or ion beam, or by X- or γ-rays, among others. As the external energy is transferred to a workpiece material, its temperature increases, eventually making it melt, sublimate, or evaporate. In the process, the total energy is always conserved, as described by the first law of thermodynamics. When heat losses at the boundaries are included, the steady-state condition is achieved at the end of the process. This “large-scale” picture is applicable to bulk materials and stays valid as we go down to sub-micrometer scales. Yet, with the further decrease in size, more attention needs to be paid to the energy carriers themselves rather than treating energy as an average quantity. Particularly at the length scales below 100 nm, the interactions of electrons and electromagnetic waves with the lattice structure of the matter become quite different than those for the bulk systems. Thermodynamic properties including temperature are based on the “continuum” definition, which may not be valid at the molecular level or at the size range of a few nanometers. This means that the traditional laws of energy transfer may not be readily applicable to nanoscale energy and mass transfer problems. At the atomic level, quantum mechanics explain the energy transfer to and within matter quite elegantly. The Schr¨ odinger wave equation provides the nuts and bolts for the formulation which can be used for a comprehensive solution of energy-matter interactions. Yet, for many complex problems of practical interest, particularly at size ranges beyond a few nanometers, the quantum mechanical solutions are not easy to achieve, which necessitates the use of simplified models. A complex problem at nanoscale level can be made tractable by simplifying the underlying physics with the introduction of new

16

2 Transport Equations

governing equations based on either wave or particle concepts. Once such a particle-wave distinction is made, the solution of the corresponding equations to design and improve processes at nanoscale becomes straightforward, yet not trivial. Our objective in this chapter is to discuss the general ideas and formulations for energy transfer in terms of “particles.” We first introduce properties of energy carriers, including electrons, photons and phonons, and then present the governing equations to describe transport phenomena under different approximations. These equations are to be used in later chapters for modeling the electron-beam based micro/nanomachining processes. We have to emphasize here that at nanoscales radiative exchange can only be modeled using the wave approaches, starting either from quantum electrodynamics or the Maxwell equations. In this monograph we will not discuss the wave approaches, but provide references for the interested reader to consult further. To summarize thermal transport models discussed in this chapter, two flowcharts are provided later to outline the applicability of these models according to different length scales of interest and to give a general overview for the strategies required for modeling micro/nanomachining procedures.

2.1 Energy Carriers: Electrons, Photons and Phonons The transfer of energy can be “modeled” using either a wave approach or by considering energy carriers as particles. Particle concept is acceptable if the wavelength of these carriers is much smaller than the characteristic length scales of the physical system under consideration. For thermal transport, we are primarily concerned with the electrons, photons, and phonons. Among these, electrons and phonons (or quanta of energy due to lattice vibrations) can be safely considered as “particles” within the length scales involved in micro/nanomachining. Both electrons and phonons can have wavelengths in the order of sub-nanometers, depending on the energies and temperatures of applications. On the other hand, thermal radiation cannot be modeled using particle approach in micro/nanomachining applications, as interference, polarization, coherence, or tunneling phenomena need to be considered at length scales below wavelength of radiation (usually much longer than 100 nm cut-off). The solution of electromagnetic wave propagation is described by the symmetric Maxwell equations, which can explain all these near-field effects. Yet, explanation of emission and absorption of radiation still needs help from quantum mechanics. In general, the emission of radiation follows the Planck blackbody radiation, at least if the near field effects are omitted. Since we are mostly focused on electron based thermal transport in this monograph, we will not be concerned with the practical implication of near field effects due to electromagnetic waves.

2.1 Energy Carriers: Electrons, Photons and Phonons

17

Electrons exist in all matter; they are quantum particles that orbit the nuclei of atoms. The two main categories of electrons are inner-shell electrons and outer-shell electrons. The inner-shell electrons are bounded tightly to the nuclei, and they do not “wander” around in the matter. This results in both the nuclei and the inner-shell electrons usually being considered together in most transport phenomena. The inner-shell electrons may also be referred to as the bound electrons. Conversely, the outer-shell electrons can propagate nearly effortlessly around the matter. They are primarily responsible for heat or electrical transport in metals and partially in semi-conductors. The outershell electrons are often called the conduction or valence electrons. Among all heat carriers, the wavelength of electrons is generally the shortest, in the order of a few Angstroms (0.1 nm) which makes them the best candidate for micro/nanoscale machining applications. In the heart of all matter, there are atoms which weave into intricate structures called lattices. Energy propagates in a solid medium by diffusion of electrons as well as by lattice vibrations, or waves, which results thermal (heat) conduction. If there is no resistance, the waves are able to propagate freely, resulting in so called “infinite thermal conductivity.” Resistance to propagation of these waves usually takes place because of the interactions of atoms which make up the lattices, resembling to spring-mass systems. Yet, instead of a simple mass-spring system, matter is comprised of many of these tiny systems in all directions. The attenuation of waves within this extensive lattice structure can also be considered in terms of superposition of different vibrations, which mimics scattering of radiation by particles. This superposition gives rise to “beats” and they can be treated as propagating particles, or “phonons.” Phonons are not actual physical particles, but just the beats of coherent vibrations of lattices in matter. Yet, remarkably these lattice vibrations can be explained with the particle concept, including their spectral behavior and absorption and scattering characteristics, which can be either elastic or inelastic. Elastic scattering refers to re-direction of a heat carrier onto a different path without causing any alteration to its energy. An inelastic scattering not only changes the path of the carrier, but also attenuates its energy assuming that the carrier possesses larger energy than its surroundings. With this picture of phonons, diffusion in solid structures follows transport equations similar to those for radiative transfer [30, 120, 222]. The primary heat carriers in solids, electrons and phonons, have different dispersion relations, which relates the angular frequency of waves to their wavelength via the wave speed. For photons, the dispersion relation is simple, as the wavelength and frequency are dependent on each other through the speed of light. For electrons and phonons, however, they can be quite complicated. The dispersion relation for electrons is usually referred to as band structure (Ziman [224]; Ashcroft and Mermin [9]). Band gap refers to a discontinuity in the energy spectrum of the carrier. These band properties dictate the conduction behavior of a material. A material can be a conductor, semi-conductor, or insulator depending on the dispersion relations of the

18

2 Transport Equations

electrons and phonons. For a conductor, the main energy carriers are basically electrons since they propagate at velocities that are several orders larger compared to that of phonons. For a semi-conductor, both the electron transport and the phonon transport could be equally important in conducting heat in the situation where an electric field is being applied. However, there are band gaps in the electronic band structure where no electrons are allowed to have these energy states. Such gaps can often easily be overcome by electrons in the semi-conductors. A material is considered as an insulator if the band gaps are so large that electrons are not able to move freely.

2.2 Classification of Transport Models All transport problems can be formulated and solved, at least in theory, by use of the elegant quantum mechanics. Nevertheless, the solution of many practical problems, including those involving nanomanufacturing, cannot be achieved this way mainly because of the complexity and computational cost involved with the quantum mechanical approach. Instead simplified techniques need to be devised to describe energy propagation in matter in terms of waves or particles. There are two important parameters that need to be considered to correctly simulate different transport phenomena [25, 30, 121, 222]. They are the characteristics length (ℓ) of the object of interest and the wavelength of energy propagation (λ). The parameter that sets the wave theories and the particles transport theories apart is the ratio Γ = ℓ/λ. If Γ is much larger than unity, then the particles transport theories can be used. As Γ approaches unity or less, the wave theories play more important roles in solving the energy transfer. Depending upon the ratio Γ , various assumptions can be made about the nature of the transport to simulate the underlying physics with acceptable accuracy. If the energy propagations are treated as waves, then there will be no need to consider photons to describe radiative transfer. Indeed, the concept of photon is not necessarily comprehensive and descriptive. Discussions provided by Kidd et al. [100] and Mishchenko [133, 134] (and the references there in) indicate the fundamental shortcomings of photon definition, although it provides a convenient simplification to a complex phenomenon. The radiative transfer can be derived rigorously starting from Maxwell’s equations, and include all possible near field effects. When the wavelength of the predominant electromagnetic waves important for radiative energy transfer becomes much smaller than the feature size, then one can refer to phenomological models of radiative transfer, which are effectively particle models. Particle transport theories conveniently replace wave theories when the number of scattering events becomes increasingly significant. Then, only the scattering cross section and the mean free path need to be considered. Under these conditions, the coherence of the waves may not be important in the modeling outcome. Most of the particle transport theories are derived from

2.3 Time and Length Scales

19

the general form of the Louisville equation, which is phenomelogical at its best [30]. This equation is reduced to the well known Boltzmann transport equation (BTE; see Sect. 2.4) with introduction of a series of approximations, both phenomelogical and statistical. Then the BTE can be used to formulate and solve the electron and phonon transport in materials. With same token, BTE can effectively be reduced to radiative transfer equation (RTE) to model radiative energy transfer. Given the success of the BTE for these energy transfer mechanisms, we can safely use it for transport phenomena related to micro/nanomachining.

2.3 Time and Length Scales The time and length scales of a physical system dictate the theoretical assumptions made for the transport equations involving electrons, phonons or light (electromagnetic waves). The thermal conduction inside matter is greatly influenced by these characteristic scales. The mean free time refers to the time interval between two successive collisions suffered by an energy carrier while the mean free path is the distance corresponding to the time interval. They usually depend on the properties of the medium and the wavelength (or energy) of the carriers. Within the mean free time (or the mean free path), the energy carrier travels ballistically without being deflected out of its propagation direction. Depending on these parameters the nature of the thermal wave propagation can vary from purely ballistic to completely diffusion-like, resulting in a temperature distribution that may vary significantly. If one is interested in the transport behavior within the time limit comparable to the mean free time of the heat carriers, then ballistic transport is to be expected [25, 30, 222]. There are two critical length scales that lead to the appropriate simplifications of particle theories. They are the mean free path, lmfp , of the energy carrier and the characteristic size of a volume over which local thermodynamics equilibrium is defined, lr [192, 222]. Generally, lr is greater than lmfp. There are two time scales associated with with these length scales: the mean free time, τmfp and the relaxation time, τr . Another important time scale is the collision time (duration of collision), τc . To be able to reach a local thermodynamics equilibrium, five to 20 collisions of energy carriers are needed. This corresponds to a maximum total collision time of 20 fs for electrons (at Fermi energy) and 2 ps for phonons in a metal [121]. Since τr > τmfp ≫ τc , in metals τr is much greater than 20 fs for electrons and larger than 2 ps for phonons. This yields a value of lr far exceeding 20 nm for electrons and 2 nm for phonons, assuming the velocities of electrons and phonons are 106 m s−1 and 103 m s−1 , respectively. The thickness of the workpiece considered in many nanomachining applications is in the order of tens of nanometers. Consequently, local thermodynamic equilibrium cannot be assumed for electrons, but it might be achievable for phonons. Given that, the time required for

20

2 Transport Equations

a typical machining process is likely to be much greater than any of the time scales stated. This makes the time-averaged assumption possible for the statistical particle transport equation, which is known as the BTE. In order to determine if the nature of the thermal transport is ballistic, semi-ballistic or diffusive, the mean free path (i.e., lmfp) of the energy carrier for the energy range involved needs to be known a priori. The mean free path of a heat carrier varies according to the level of its energy. The lmfp is typically derived using the scattering cross-sections of a given carrier at its particular energy level within a specified material. This requires the derivation of the scattering rate of the carrier in a specific medium. Often, this derivation is quite cumbersome as it requires applications of quantum mechanics, wave theories, and different scattering mechanism. For the case where a high-energy electron-beam propagates inside a solid, the mean free path of penetrating electrons at various beam energies are relatively well known because of intense research activities carried on for applications to electron microscopy (see Chap. 4), where the energy range is in the order of several kilo-electronvolts to mega-electronvolts. For micro/nanomachining applications discussed here, energies of electrons incident from a probe can also be large, and for them the mean-free path information are available (see Chap. 4). However, the electrons and phonons contributing to conduction transport have energy levels less than 10 eV (see Sect. 7.2). For these cases, especially for metals, the available data are scarce, almost non-existent. In the following sections, various transport models are studied for different physical conditions. We will start the discussion from the general BTE and introduce its different forms. Note that the BTE is obtained from a more extensive Louisville equation (LE), which is quite involved for any practical application. The details of the LE are available in the literature [28, 30]; therefore, there is no need to report them here.

2.4 Boltzmann Transport Equation If the wave nature of the particles is not important in the energy transport, the transport phenomena can be modeled using particle transport theory. The governing equation for the particle transport is the BTE. The BTE describes the evolution of a particle probability distribution (denoted as f ) over⇀time ⇀ ⇀ ⇀ ⇀ t, space r , and wave vector k (f may relate to v or momentum p = h ¯ k), as particles experience a series of scattering events and external forces. In principle the BTE is applicable to all energy carriers, including electrons, phonons, or photons as long as the wavelengths of the carriers are small compared to the characteristic length of the object of interest. This means that, for example, the BTE should not be used to model energy transfer of a laser beam with a wavelength of 532 nm striking a 100-nm thick dielectric film, as the physics including the effect of interference, coherence and tunneling would not be accounted for in the analysis. However, the BTE is applicable if the

2.4 Boltzmann Transport Equation

21

laser beam is replaced with an electron-beam, as electrons have wavelengths in the order of nanometers. The rate of change of the particle distribution, f , is balanced by the rate at which the distribution is increased or decreased due to collisions with other particles. The conservation equation follows [9, 66, 225] that:
⇀ 
⇀  ⎛
⇀ ⎞
⇀  ⇀ ⇀ ⇀ ⇀ ⇀ ⇀ ∂f ∂f ∂f r , k, t r , k, t r , k, t ∂f r , k, t ∂r ∂k ⎠ . + · · =⎝ + ⇀ ⇀ ∂t ∂t ∂t ∂t ∂r ∂k col

(2.1)

The terms on the LHS are the change of f in time t, the rate of change of f due ⇀ to advection in space r , and the rate of change of f due to momentum change ⇀ ⇀  while the RHS is the collision term. For the sake of simplicity, f r , k, t is abbreviated as f from this point on, however it is understood that f is a ⇀ ⇀ function of t, r , and k. The conservation of the particle distribution function can then be rewritten using the different notation as:

⇀ ˙ ∂f ∂f ⇀ + v ⇀ · ∇⇀ k · ∇ f = f + . (2.2) ⇀ r k k ∂t ∂t col ⇀

The group velocity of the particles, v ⇀ , is typically a function of the wave k ⇀ ⇀ ˙ vector k. The time evolution of the wave vector k is due to the externally applied fields or forces. This is valid for charged particles such as electrons when they are accelerated through the electric field. Although this term is dropped out for the case where phonons or electromagnetic waves are the energy carriers, it shall be retained in this term for the completeness of the derivation. ⇀ According to semi-classical particle dynamics, the rate of change of k as a ⇀ ⇀ result of the applied electric field E and magnetic field H is given as [9, 224]: ˙ e k =− h ¯

⇀

⇀

⇀ ⇀ 1⇀ F ext . E + v⇀ × H = c k ¯ h

(2.3)

Inserting this expression into the conservation equation yields the general form of the BTE [9, 224]: ⇀

∂f ⇀ F ext + v ⇀ · ∇⇀ · ∇⇀ f = f+ r k k ∂t ¯ h



∂f ∂t



.

(2.4)

col

If the BTE is to be solved rigorously, the collision term shall be evaluated using scattering probabilities of the energy carriers. The collision term refers to the rate at which the distribution is increased or decreased. In terms of the physical sense, the collision term consists of transitions (or scatterings) of

22

2 Transport Equations ⇀

⇀

⇀

quantum particles from the wave vector k ′ into k or from the wave vector k ⇀ into k ′ . This results in the transition probabilities (or scattering probabilities) being properly accounted for in the equations. For phonons and photons the collision term is expressed as:

 
⇀ 
⇀ ⇀ ⇀ ∂f ˜ k → k′ f . ˜ k′ → k f ′ − (2.5) = W W ∂t col ⇀ ⇀ k′

k′

⇀ ⇀ ˜ k ′ → k denotes the rate of scattering On the right-hand side of (2.5), W ⇀ ⇀ from k ′ to k. The first summation is the rate of change of f due to the inscattering of particles from various locations and directions, while the second is the rate of change of f caused by the out-scattering of particles. The collision term is slightly different for electrons since electrons are obliged to Pauli’s exclusion principle, which prevents two electrons with the same state from residing at the same ⇀ location. In order for an electron to undergo a ⇀ transition ⇀ (or scattering) from k ′ to k , there must be an “empty slot” in the k state to accommodate for this scattered electron. If a state is fully occupied by other electrons, then the distribution function f has a value of unity, otherwise, (1 − f ) represents the available probability. The collision term for electrons is then modified accordingly:

 
⇀ 
⇀ ⇀ ⇀ ∂f ˜ k → k ′ f (1 − f ′ ). (2.6) ˜ k ′ → k f ′ (1 − f ) − = W W ∂t col ⇀ ⇀

k′

k′

In its general form, the BTE is virtually intractable using analytical methods owing to the various independent variables and its integro-differential form. This is usually resolved by using the statistical methods such as the Monte Carlo (MC) method which are commonly used to simulate the propagation of quantum particles according to the probability distribution functions.

2.4.1 Relaxation-Time Approximation Solution of the BTE with the general integral collision term as given above is not straightforward and practical for some applications. Instead, it is desirable to approximate the collision term to make the BTE analytically tractable, if not, numerically. To devise such an approximation, however, we need to understand its significance and its functionality. Without any simplifications, the collision term in the BTE is understood to be particle transformations from one particular state to another. If one assumes that a large number of electrons with different energy levels are “poured” into an isolated control volume, the electrons at the higher energy levels would tend to transfer excess energies to those at lower levels through

2.4 Boltzmann Transport Equation

23

collisions. Over time the entire control volume would eventually reach thermal equilibrium. This is where the amount of energy transfer from electrons possessing higher energy to the lower ones and vice versa, are the same, resulting in an equilibrium distribution function eventually being achieved. Consequently, collisions of particles tend to restore equilibrium over a period of time. This suggests that an approximation needs to be introduced using a relaxation-time approach. In the relaxation-time approach, it is assumed that the perturbed distribution function eventually relaxes back to the equilibrium distribution function within a period of time. The collision term is then approximated as:

f − f eq ∂f , (2.7) =− ∂t col τr where f eq is the equilibrium distribution and τr is the relaxation time needed to reach the equilibrium state. The solution of f in the above first-order differential equation (if the partial derivative is replaced with ordinary derivative) can be expressed as:

t eq eq f = f + (f0 − f ) exp − . (2.8) τr The solution for f in the above equation is of exponentially decaying nature, meaning that if f is increased (or decreased) from its equilibrium distribution, it decays (or grows) exponentially to the original state with the relaxation time τr being the time constant. If the relaxation-time approximation is used in the BTE, the specific details of the scattering process are lost. For example, the directional dependence of the transition process is compromised, which may be critical in some applications. Nevertheless, this approximation transforms the BTE into a partial differential equation instead of an integro-differential equation, and hence, its solution can be achieved easily. By solving the BTE we gain the knowledge of how the particles are dispersed or distributed in time, space, and various directions. The particle distribution f by itself may not yield any useful information; however, various physical quantities can be derived from it by performing integrations of the function over the independent variables. The density of particles as a function of time and space can be retrieved by integrating the distribution function ⇀

over the momentum space k: ⇀  n r, t =




⇀  ⇀ ⇀ f r , k , t d k.

(2.9)

Similarly, the momentum density of particles can also be determined using f : ⇀ ⇀ ⇀ ⇀  ⇀ ⇀  m k v k f r , k, t d k , (2.10) p r, t = where m is the mass and v is the velocity of particles.

24

2 Transport Equations

2.5 Intensity Form of Boltzmann Transport Equation In this section, the BTE is reduced to an expression in terms of intensity, which is applicable to modeling electron, phonon or photon transport. By imposing a series of assumptions on the BTE, governing equations for the electron-beam transport equation (EBTE), the RTE, the phonon radiative transfer equation (PRTE), and the electron transport equation (ETE) can be derived. These equations are usually in the integro-differential form where analytical solutions are virtually intractable. For this reason, statistical MC methods are preferable to solve these equations, as discussed in Chap. 3. 2.5.1 Electron-Beam Transport Equation In electron-beam processing, free electrons are the energy carriers. The energy of an electron is characterized by its wave number k (=2π/λ) instead of its frequency or wavelength. Unlike electromagnetic waves, free (propagating) electrons do not get absorbed. This implies that inelastic scatterings change the energy of these carriers but do not attenuate the number of carriers. The wave number of the electrons changes as they undergo inelastic scatterings. From the computational standpoint, the scattering cross section changes once the energy of a propagating electron ensemble is altered. The propagation of energy carriers is described by the BTE; however, the BTE formulation requires the proper implementation of scattering mechanisms. The distribution function f , which is the dependent parameter in BTE as shown in (2.4), may not yield any useful information by itself. Instead, it is better to cast the BTE in terms of intensity. Before attempting to do this, it is informative to state that an electron-beam consists of bundles of free electrons traveling along a given line of sight. In principle these electrons have the same electronic properties as those of the free-electron model. Hence, the electron intensity Ie can be defined as:   ⇀ 1 ⇀ fe r , sˆ, Ee , t Ee De (Ee )ve (Ee ), Ie r , sˆ, Ee , t = 4π

(2.11)

where the electron density of states De and the electron velocity ve are energy (or wave number k) dependent. These properties do not have directional dependence since the electronic band structure is of a spherical parabolic type. If we multiply the BTE (i.e., (2.4)) by the electron energy, density of states, and group velocity (Ee De ve ), we obtain ⇀

F ext ∂ (fe Ee De ve ) ⇀ + v e · ∇⇀ · ∇⇀ fe (fe Ee De ve ) + (Ee De ve ) r k ∂t ¯h

∂ (fe Ee De ve ) = . ∂t col

(2.12)

2.5 Intensity Form of Boltzmann Transport Equation

25

Using the definition of electron intensity, (2.12) can be re-written as: ⇀

F ext ∂Ie ⇀ + v e · ∇⇀ · ∇⇀ fe = I + (Ee De ve ) r e k ∂t h ¯



∂Ie ∂t



,

(2.13)

col

where the intensity Ie is a function of time, energy (or wave number), location, and direction. Note that the third term on the LHS of the equation needs to be expressed in terms of electron intensity. To do so, it is first written as: (Ee De ve )

⇀ ⇀  F ext F ext · ∇⇀ fe = · ∇⇀ (fe Ee De ve ) − fe ∇⇀ (Ee De ve ) . (2.14) k k k h ¯ ¯ h

Then the gradient in wave vector is replaced with partial derivative in energy by using the definition of the group velocity: ⇀  F ext · ∇⇀ (fe Ee De ve ) − fe ∇⇀ (Ee De ve ) k k h ¯

 ∂I
⇀ ∂ ⇀ e − fe (Ee De ve ) . = F ext · v e ∂Ee ∂Ee

(2.15)

It can be shown that the following relation fe

2 2 ∂ (Ee De ve ) = (fe Ee De ve ) = Ie , ∂Ee Ee Ee

(2.16)

holds for the spherical parabolic electronic band structure (or free-electron model). Thus, the electron intensity equation derived from the BTE becomes:

 ∂I
⇀ 2 ∂Ie ⇀ ∂Ie ⇀ e + v e · ∇⇀ − Ie = I + F ext · v e , (2.17) r e ∂t ∂Ee Ee ∂t col where the first two terms on the LHS describe the variation of intensity in time and in space while the third term is the energy gained or lost as a result of the externally applied field on the electron-beam. Note that the electron intensity equation is written along a line of sight, thus, the collision term on the RHS of the equation can be interpreted as intensity lost or gained along the line of sight, due to interactions of the electron-beam with the participating medium. Here the term “interactions” means elastic or inelastic scatterings of the electron beam. Scatterings result in re-directions of electronbeam intensity into and out of a given line of sight. The collision term should consist of in-scattering and out-scattering terms for both elastic and inelastic scatterings. The out-scattering term that describes the intensity lost of electron-beam, both elastically and inelastically, can be expressed as:   inel 1 − σ (E)Φinel (E, Ω → E ′ , Ω ′ ) Ie ve dΩ ′ dE ′ , e 4π E ′ Ω ′ e (2.18)  ′ ′  el 1 ′ ′ − dΩ dE . σe (E)Φel (E, Ω → E , Ω ) I v e e e 4π E ′ Ω ′

26

2 Transport Equations

Here Φe is the scattering phase function, which is analogous to that of electromagnetic waves. It is used to describe the probability of scattering of electron from one energy state in a given direction to another state and direction. The elastic phase function Φel e is completely different from that for the inelastic because of the distinct nature of these two phenomena. phase function Φinel e The factor of 4π in the expression is used for normalization of the phase functions. The out-scattering term can be further simplified by taking the scattering coefficients and intensities outside the integrals such that: σ inel (E) − e ve Ie Φinel (E, Ω → E ′ , Ω ′ ) dΩ ′ dE ′ , e 4π ′ ′ E Ω (2.19) σeel (E) ′ ′ ′ ′ ve Ie Φel (E, Ω → E , Ω ) dΩ dE . − e 4π E′ Ω′ The above expression takes a simpler form as:   − σeinel (E) + σeel (E) ve Ie .

(2.20)

Since

1 Φinel (E, Ω → E ′ , Ω ′ ) dΩ ′ dE ′ = 1, 4π E ′ Ω ′ e 1 Φel (E, Ω → E ′ , Ω ′ ) dΩ ′ dE ′ = 1. 4π E ′ Ω ′ e

(2.21)

Alternatively, it can be written in terms of elastic and inelastic mean free paths as:   1 1 − inel + ve Ie , (2.22) λe (E) λel e (E) based on the fact that the scattering coefficients are inversely proportional to the mean free paths, for instance, σeinel (E) =

1 λinel e (E)

.

(2.23)

Similar to the out-scattering term, the in-scattering of the electron-beam can be expressed as:  inel ′ inel ′ ′  1 σe (E ) Φe (E , Ω → E, Ω) Ie′ ve′ dΩ ′ dE ′ 4π E ′ Ω ′ (2.24)  el ′ el ′ ′  1 σe (E ) Φe (E , Ω → E, Ω) Ie′ ve′ dΩ ′ dE ′ . + 4π E ′ Ω ′

This term is positive since in-scattering means contributions of electron intensity along the line of sight from all other directions. Substituting the

2.5 Intensity Form of Boltzmann Transport Equation

27

out-scattering and in-scattering terms into the electron-beam intensity equation yields:  ∂I
⇀ ∂Ie ⇀ ⇀ e + v e · ∇⇀ F · v I + ext e e r ∂t ∂Ee 
⇀ ⇀ F ext · v e  inel  Ie = − σe (E) + σeel (E) ve Ie + 2 Ee (2.25)  inel ′ inel ′ ′  1 + σ (E ) Φe (E , Ω → E, Ω) Ie′ ve′ dΩ ′ dE ′ 4π E ′ Ω ′ e  el ′ el ′ ′  1 σe (E ) Φe (E , Ω → E, Ω) Ie′ ve′ dΩ ′ dE ′ . + 4π E ′ Ω ′

This equation is called the EBTE, and is very similar to the extensively used RTE. Both equations are of integro-differential type, which is typical in the particle transport theory. The EBTE is the intensity form of the BTE. If there are no external forces applied to the electron-beam, then the EBTE reduces to: ∂Ie ⇀ + v e · ∇⇀ I r e ∂t   = − σeinel (E) + σeel (E) ve Ie    inel ′ inel ′ ′  σe (E ) Φe (E , Ω → E, Ω) Ie′ ve′ dΩ ′ dE ′ E′ Ω′ +     el ′ el ′ 4π ′ ′ ′ ′ ′ E ′ Ω ′ σe (E ) Φe (E , Ω → E, Ω) Ie ve dΩ dE . + 4π

(2.26)

2.5.2 Radiative Transfer Equation BTE is reduced to RTE if the energy carriers are quanta of light, or photons, each having energy of hν for a given frequency ν (or wavelength λ = c/ν). Even though the photon approach is questioned (see [100,133]; and references therein), it describes the phenomenological RTE, i.e., the conservation of radiant energy along a line of sight of a beam propagating within an absorbing, emitting, and scattering medium, quite well for many applications and will be used here for the sake of simplicity of the discussion (although we call caution here, as the photon approach does not explain the radiative transfer in nanoscale structures. The following discussion is important only because it parallels the electron and phonon transport equations to be discussed later). Radiative intensity, i.e., the radiative energy per unit solid angle, time, normal area, within an infinitesimally small frequency interval, is defined in terms of the photon distribution fν , the photon energy hν, the density of states Dν , and the speed of light c:
    Dν (p) ⇀ ⇀ c. (2.27) Iν r , θ, φ, t = fν r , p, θ, φ, t hν 4π p

28

2 Transport Equations

The subscript ν is used to indicate that the equation is for a given specific frequency (or wavelength) which corresponds to a monochromatic laser beam. The summation is to be performed over all polarization branches. This implies that the change of polarization of photons is ignored in the classical phenomenological RTE. Indeed, for this reason, it is desirable to derive the RTE rigorously starting from Maxwell’s equations, as shown by Mishchenko [133]. Multiplying (2.4) by the photon energy, the density of states, and the speed of light, and then performing the summation over all the polarization branches, allows the BTE to be cast in terms of the radiative intensity, which is effectively the RTE:     ∂Iν ⇀ ˜ ν (Ω ′ → Ω)Iν′ (Ω ′ , t) − ˜ ν (Ω → Ω ′ )Iν ⇀ + v ν · ∇Iν = r , Ω, t . W W ∂t Ω′ Ω′ (2.28) To further simplify the BTE, the in-scattering term is rewritten as:  ˜ ν (Ω ′ → Ω)Iν′ = σν Φν (Ω ′ → Ω) Iν′ dΩ ′ , (2.29) W 4π ′ Ω ′ Ω

where σν is the scattering coefficient (in 1/m) and Φν is the phase function of the medium. The in-scattering term contains all the contributions from within the solid angle Ω ′ . The out-scattering term can be given as  ˜ ν (Ω → Ω ′ ) Iν = (κν + σν ) Iν , (2.30) W Ω′

with κν being the absorption coefficient (in 1/m). Therefore, this yields the familiar form of the RTE: ∂Iν ⇀ σν + v ν · ∇Iν = −βν Iν + Φν (Ω ′ → Ω) Iν′ dΩ ′ . (2.31) ∂t 4π Ω ′ Note that βν is simply the sum of κν and σν , and known as the extinction or attenuation constant. This form of RTE assumes that the in-scatterings are elastic which means that a photon scattered with its energy (i.e., frequency) remains unaltered. This is not true in the case of Raman scattering. The outscattering term includes the absorptions and re-emissions of radiation, which are considered as inelastic scatterings. From the quantum mechanics point of view, the RTE asserts that the rate of change of the radiant energy of a propagating ensemble of photons in a given direction is equal to the amount of photons attenuated, emitted along the direction, and in-scattered from all other directions into the given direction. One important thing to note that in thermal radiative transfer the number of photons does not conserve. The ensemble of photons is attenuated in terms of the population of the photons, and the frequency of the photons does not change during scattering events. This in turn reduces the energy of the entire ensemble. The frequency of photons may be altered during Raman scattering,

2.5 Intensity Form of Boltzmann Transport Equation

29

which is inelastic by nature; however, this type of scattering mechanism is not included here. Detailed discussion of the RTE and various solution techniques for different applications can be found in well known texts in the field [136, 182, 201]. 2.5.3 Phonon Radiative Transport Equation In general, lattice vibrations can be described in terms of phonons which can be optical and acoustic types [9,66,224]. Optical phonons have essentially zero group velocity while acoustic phonons have finite group velocity and usually travel at the speed of sound. In this sub-section, we derive a transport equation called the phonon radiative transport equation (PRTE), similar to the RTE, starting from the general BTE. The PRTE describes the propagation of lattice energies in a medium that emits, absorbs, and scatters them along a given line of sight. The PRTE has extensively been used to predict the thermal conductivities of microstructures in dielectric media since its introduction by Majumdar [120]. Similar to the radiative intensity concept, phonon intensity, i.e., phonon energy per unit solid angle, time, normal area, within an infinitesimally small frequency interval, is defined in terms of the phonon distribution fω , the phonon energy h ¯ ω, the density of states Dω , and the group velocity of phonon vg,ω :
    Dω¯ (p) ⇀ ⇀ Iω r , θ, φ, t = vg,ω , fω r , p, θ, φ, t ¯hω 4π p

(2.32)

where the polarization index, p, includes two transverse polarization branches and one longitudinal polarization branch. Following a similar derivation given in Sect. 2.5.2, the PRTE is obtained as [120, 160]: σω ∂Iω ⇀ U 0 + v g,ω · ∇Iω = σω Iω − βω Iω + Φω (Ω ′ → Ω) Iω′ dΩ ′ . (2.33) ∂t 4π Ω ′ In (2.33), it is understood that all the variables are written for phonons. The in-scattering term (i.e., the last term on the RHS) in the PRTE was discussed by Prasher [160], who simplified (2.33) by considering an isotropic scattering approximation in his derivation. He also showed that the “general” version of the PRTE is consistent with the transport theory in acoustically thick regime. Another difference between the RTE and the PRTE is the structure of the emission term. Instead of the Planck function, here we have the product σωU Iω0 , where σωU is the scattering coefficient for Umklapp process while Iω0 is the equilibrium phonon intensity at the angular frequency of ω. The Umklapp process is emission of a phonon when one phonon gives out two phonons. Similarly, absorption means that when two phonons are combined, they yield to a single phonon. A more detailed discussion of Umklapp process is available in Ziman [224, 225].

30

2 Transport Equations

In Chap. 3, we provide a MC procedure for the solution of the PRTE using a MC approach. Other than that, we will not further investigate the PRTE in-depth, which is mostly applicable for semiconductor and dielectric materials. For additional details, the reader is encouraged to refer to papers by Majumdar [120], Mazumder and Majumdar [129], Prasher [160], Lacroix et al. [109], and Zhang [222].

2.5.4 Electron Transport Equation The ETE is also derived from the BTE, and is quite similar to the EBTE discussed in Sect. 2.5.1. The ETE is for low energy electrons within the medium rather than high energy electrons originated from any external electron source, e.g., a field emission probe. These low energy electrons have their energies around the Fermi level, and they are responsible for thermal conduction inside a conducting material. Consequently, the Pauli exclusion principle must be followed. In the following derivation of the ETE, we start from the general BTE which are given by (2.4) and (2.6) and cast it in terms of electron intensity using the definition given in (2.11). In doing so, an equation similar to (2.17) is obtained, although the collision term on the RHS has a different form. Here, the Pauli exclusion principle must be incorporated into the collision term. For out-scattering term, we write: ⎤ ⎡ (E, Ω → E ′ , Ω ′ ) σeinel (E) Φinel e 1

⎥ ⎢ − dΩ ′ dE ′ , ⎣ 1 ′ ′ ′ ′ ⎦ 4π E ′ Ω ′ × Ie ve E D (E ) ve (E ) − Ie 4π (2.34)   el el ′ ′ σe (E) Φe (E, Ω → E , Ω ) 1 −  dΩ ′ dE ′ , 1 ′ ′ ′ ′ 4π E ′ Ω ′ × Ie ve 4π E D (E ) ve (E ) − Ie

where the term (E ′ D (E ′ ) ve (E ′ )/4π − Ie′ ) enforces the Pauli exclusion principle. Similarly, the in-scattering term can be expressed as:  1 σ inel (E ′ ) Φinel (E ′ , Ω ′ → E, Ω) e 4π E ′ Ω ′ e

 1 ED(E)ve (E) − Ie dΩ ′ dE ′ × Ie′ ve′ 4π (2.35)  1 ′ ′ σeel (E ′ ) Φel (E , Ω → E, Ω) + e 4π E ′ Ω ′

 1 ′ ′ ED(E)ve (E) − Ie dΩ ′ dE ′ . × Ie ve 4π

2.6 Moments of Boltzmann Transport Equation

31

Therefore, the general ETE consists of (2.17), (2.34), and (2.35). For the case where inelastic scattering is insignificant, the ETE is given as:

 ∂Ie ⇀ 2 ∂Ie ⇀ ⇀ + v e · ∇⇀ − I I + F · v e ext e r e ∂t ∂Ee Ee  1 ′ ′ σ el (E)Φel =− e (E, Ω → E , Ω ) 4π E ′ Ω ′ e

 1 ′ × Ie ve E D (E ′ ) ve (E ′ ) − Ie′ dΩ ′ dE ′ (2.36) 4π  1 ′ ′ σ el (E ′ ) Φel + e (E , Ω → E, Ω) 4π E ′ Ω ′ e

 1 × Ie′ ve′ ED(E)ve (E) − Ie dΩ ′ dE ′ . 4π Even without consideration of inelastic scattering, the ETE is in much more complicated form than the EBTE and the RTE due to the exclusion principle. Hence, statistical methods like MC methods are needed to obtain a solution, as otherwise the ETE is analytically intractable. The MC method simulation for this equation will be discussed later in Chap. 3.

2.6 Moments of Boltzmann Transport Equation The BTE is an integro-differential equation, which is a function of time, spatial coordinates, directions of energy carriers, and frequency (energy) spectrum; therefore the solution of the BTE is quite involved. To make this complex problem tractable, a set of relatively simple equations can be obtained if the BTE is integrated over all wave vectors. For this, the BTE is multiplied by a function ψ first, followed by an integration over all possible wave vectors. These expressions are called moments of the BTE, and they retain the physics of the problem in terms of directionally and spectrally averaged quantities. In the following sections, we present derivations of such moments equations starting from the general BTE. These are indeed equations for the conservation of number of particles, momentum, and energy, and they are independent of direction (wave vector). Additional details of these derivations are also available elsewhere [184, 193]. ⇀ If the BTE is multiplied by a function ψ and integrated over k, the following is obtained:  ⇀ 

⇀ ⇀ F ext ∂f ⇀ ∂f + v ⇀ · ∇⇀ · ∇ ψ d k = ψ f + d k. (2.37) ⇀f r k k ∂t h ¯ ∂t col

32

2 Transport Equations

This equation can be re-written in the following fashion using the simple differentiation rule as: ⇀  
 ⇀ ∂ψ F ext ∂ (ψf ) ⇀ ⇀ −f + ∇⇀ · ∇⇀ f d k · ψ v ⇀ f − v ⇀ f · ∇⇀ ψ+ψ r r k k k ∂t ∂t ¯h

⇀ ∂ψ ∂ (ψf ) −f dk . = ∂t ∂t col (2.38) Equation (2.38) is the general moment equation for ψ. The following discussions are for specific moments of the BTE where the spherical energy bands for the particle are assumed. 2.6.1 Continuity or Conservation of Number of Particles The continuity equation for particle transport conserves the carrier density. It is obtained by taking the zeroth moment of the BTE with ψ = 1, which reduces (2.38) to: ⇀ 



 ⇀ F
⇀ ⇀ ⇀ ∂f ∂f ⇀ ext · ∇ dk + f d k = v f d k + ∇⇀ · dk . ⇀ ⇀ r k k ∂t h ¯ ∂t col (2.39) The average particle density, denoted as n, is given by:
 ⇀ ⇀  ⇀ ⇀ n r , t = f r , k, t d k . (2.40)

The first term on the LHS of the equation is simply the time derivative of the average particle density:

⇀ ⇀ ∂ ∂n ∂f . (2.41) dk = f dk = ∂t ∂t ∂t

Before proceeding to simplify the other terms in the equation, it is important ⇀ to examine the wave-vector dependent velocity, v ⇀ , which is the velocity of k the particle. Here, it is assumed that:   ⇀ ⇀ ⇀ (2.42) v ⇀ = v ⇀ + v′ ⇀ , k



⇀ ⇀

k

k



is an average velocity or the drift velocity of the particle, to be
 ⇀ ⇀ is the random velocity due to thermal effects denoted as v d , and v ′ ⇀ = v ⇀ where

v

k

⇀

k

k ,T

on the particle. The average velocity is expressed as:
 ⇀ ⇀ v ⇀f dk   k ⇀ ⇀ . vd = v⇀ = ⇀ k f dk

(2.43)

2.6 Moments of Boltzmann Transport Equation

33

Therefore, ⇀ ⇀

v

k

⇀

⇀

= vd + v⇀ .

(2.44)

k ,T

The drift component of the velocity can be viewed as the average velocity that the particles are expected to possess if there are no thermal effects. The thermal effects cause the particle velocity to fluctuate around this drift velocity, which is called thermal velocity. However, the average thermal velocity over the distribution function is assumed to be zero [121, 193], that is:
 ⇀ ⇀ (2.45) v ⇀ f d k = 0. k ,T

Also, note that:

⇀

⇀  ⇀ ⇀ v d f d k = v d n,

(2.46)

since v d is already an average quantity over the wave-vector spectrum. According to these observations, the second term on the LHS of (2.39) can be replaced by the product of the average drift velocity and the average particle density:

  
 ⇀
⇀ ⇀ ⇀ ⇀ ⇀ ∇⇀ f d v n . (2.47) k = ∇ v + v v d k = ∇ · ⇀ · ⇀ ⇀f ⇀ · d d r r r k

k ,T

The third term on the LHS of (2.39) is zero, due to the fact that the divergence theorem can be used to replace the volume integral with surface integral, and the surface integral goes to zero since the integration is to be performed for a surface at a large wave vector where the function f goes to zero (see [184, 193]). Thus,  ⇀ ⇀ ⇀ ⇀ F ext F ext · ∇⇀ f d k = · ∇⇀ f d k = 0. k k h ¯ ¯ h Hence, the zeroth moment of the BTE reads:

⇀  ∂n ∂n + ∇⇀ · v n = , d r ∂t ∂t col

(2.48)

(2.49) ⇀

where it is understood that the gradient is that of space r , and the velocity ⇀ v d is the averaged drift velocity which varies in space and time. The above equation represents the conservation of number of particles. The rate of change of particle concentration or density, which is the first term on the LHS of (2.49), and the in-coming and out-going particle fluxes (i.e., the second term on LHS of (2.49)) are equal to the rate of density generated due to collisions.

34

2 Transport Equations

2.6.2 Conservation of Momentum The momentum conservation equation is obtained by taking the first moment ⇀ ⇀ of the BTE. This time, ψ is the particle momentum, ψ = p ⇀ = m v ⇀ . k k Again, referring to (2.38) and substituting the expression for ψ, the following momentum equation is obtained: ⎤ ⎡
⇀

∂ m⇀
 ⇀ v ⇀f ⇀ ⇀ F ext ⇀ ⇀ ⇀ k ⎣ ⎦d k + · ∇ ∇⇀ · m v d k + m v dk ⇀ v ⇀f ⇀ ⇀f r k k k k ∂t ¯h ⎛  ⇀ ⎞ ∂ m v ⇀f ⇀ k ⎠ dk . (2.50) = ⎝ ∂t col

The first term on the LHS refers to the rate of change of the average momentum density, the second term is the divergence of the flux density of the average momentum of the particles, and the last term is the flux density of average momentum due to externally applied forces. The RHS is the rate of change of momentum due to particle collisions. Equation (2.50) can be simplified starting from the first term on the LHS. Using (2.44) and (2.45), it is reduced to: ⎡ ⎤ ∂ m ⇀ 
  ⇀ v ⇀f ⇀ ⇀ ⇀ k ⎣ ⎦d k = ∂ m v d + v ⇀ f dk k ,T ∂t ∂t (2.51)  ⇀    ⇀ ∂ npd ∂ ⇀ . m v df dk = = ∂t ∂t

The second term on the LHS of (2.50) is rather complicated. It can be rewritten as:
 ⇀ ⇀ ⇀ ∇⇀ · m v ⇀ v ⇀f d k r k k
 ⇀ ⇀ ⇀ = ∇⇀ · m v ⇀ v ⇀f dk r k

k


m vd,i + v⇀ · = ∇⇀ r

k ,T,i


vd,j + v⇀

k ,T,j


= ∇⇀ m vd,i vd,j + vd,i v⇀ · r

k ,T,j


= ∇⇀ m vd,i vd,j + v⇀ · r

v⇀

+ v⇀

k ,T,i k ,T,j


= ∇⇀ · mnv v + mv⇀ d,i d,j r

  ⇀ f dk

k ,T,i

vd,j + v⇀

  ⇀ f dk

v⇀

k ,T,i k ,T,j

 ⇀ f dk .

v⇀

k ,T,i k ,T,j

  ⇀ f dk (2.52)

2.6 Moments of Boltzmann Transport Equation

35

The RHS of (2.53) consists of the kinetic and thermal energy tensors, respectively. For the sake of simplicity the off-diagonal elements of the thermal energy tensor are ignored. Energy is related to the temperature such that:
 ⇀ mv⇀ v⇀ f d k , (2.53) nkB T = k ,T,i k ,T,i

for an ideal gas. Hence, it is observed that: 
 ⇀ · (nvd,i pd,j + nkB T ). · mnvd,i vd,j + mv⇀ v⇀ f d k = ∇⇀ ∇⇀ r r k ,T,i k ,T,j

(2.54)

The third term in (2.50) can be simplified as: ⇀  ⇀ F ext ⇀ mv⇀ · ∇⇀ f d k k k h ¯ ⇀ ⇀ F ext ⇀ · m v ⇀ ∇⇀ fd k = k k h ¯ ⇀ ⇀  ⇀
 ⇀ F
F ext ⇀ ⇀ ext · ∇⇀ · m v ⇀ f d k − · f ∇⇀ · m v ⇀ d k = k k k k h ¯ h ¯ ⇀ ⇀
 ⇀ F
⇀ ⇀ F ext ⇀ ext · ∇⇀ · m v ⇀ f d k − · f ∇⇀ · ¯h k d k. = k k k h ¯ h ¯

(2.55)

Using the divergence theorem on the first term on the RHS of the above equation to transform the volume integral into surface integral, and performing the ⇀ integration, the first term vanishes since f goes to zero for large k. Thus: ⇀  ⇀
⇀ ⇀ ⇀ F ext F ext ⇀ · ∇⇀ f d k = − · f ∇⇀ · ¯h k d k mv⇀ k k k h ¯ h ¯ (2.56) ⇀

= −F ext ·

⇀

⇀

f d k = −nF ext .

By gathering all these terms, the following momentum equation [184, 193] is obtained:   ⇀   ⇀   ⇀ ⇀  ⇀ ∂ npd ∂ npd + ∇⇀ . (2.57) · n v d p d = nF ext − ∇⇀ (nkB T ) + r r ∂t ∂t col

⇀

After the transformation, p d is the average particle drift momentum. It ⇀ should not be confused with p ⇀ , which refers to wave vector dependent k particle momentum. For convenience, the momentum conservation is usually written as:  ⇀  ⇀
⇀  ⇀ ∂P d ∂P d ⇀ + ∇⇀ , (2.58) (nkB T ) + · v d P d = nF ext − ∇⇀ r r ∂t ∂t col

⇀



⇀



where P d = n p d is the momentum density.

36

2 Transport Equations

2.6.3 Conservation of Energy The energy conservation equation is derived by setting ψ to the energy of particles, which is denoted as E⇀ . Note that E⇀ is a function of wave vector ⇀ k k k, therefore all the time and space derivatives of E⇀ are zero. Substituting k E⇀ into (2.38) yields: k ⎤ ⎡
⇀  ∂ E⇀ f
 ⇀ ⇀ ⇀ F ⇀ ext k ⎦d k + ⎣ · ∇⇀ f d k ∇⇀ · E⇀ v ⇀ f d k + E⇀ r k k k k ∂t ¯h 
⎛ ⎞ ∂ E⇀ f ⇀ k ⎠ d k. = ⎝ (2.59) ∂t col

Similar to the derivation of the momentum equation, the first term on the LHS is the rate of change of the average energy density, the second term is the divergence of the average energy density flux, and the third term is the average energy generated due to the external forces. The first term on the LHS of the above equation can be expressed as: ⎤ ⎡

∂ E⇀ f  ⇀ ⇀ ∂ k ⎦d k = ∂ ⎣ (2.60) E⇀ f d k = (nw), k ∂t ∂t ∂t

while the second term can be written as:

 ⇀  ⇀ ⇀ ⇀ ∇⇀ · E d k = ∇ E ⇀ v ⇀f ⇀ · ⇀ v ⇀f dk r r k k k k

!
 ⇀ ⇀ ⇀ ⇀ = ∇⇀ · E⇀ v d f d k + E⇀ v ⇀ f d k r k k k ,T

! ⇀ ⇀ ⇀ E⇀ v ⇀ f d k . · n v dw + = ∇⇀ r k

k ,T

(2.61) The second term on the RHS of (2.61) can be further simplified by assuming that: 1
⇀ ⇀  m v⇀ · v⇀ k 2 k

1 ⇀ ⇀ ⇀ ⇀ · vd + v⇀ = m vd + v⇀ k ,T k ,T 2 "2

" "⇀ "2 1 " "⇀ ⇀ ⇀ = m " v d" + 2 v d · v ⇀ + " v ⇀ " , k ,T k ,T 2

E⇀ = k

(2.62)

2.6 Moments of Boltzmann Transport Equation

37

which yields:

⇀ ⇀ E⇀ v ⇀ f d k k k ,T   ! (2.63) "⇀ "2 ⇀ "2 ⇀ "⇀ ⇀ ⇀ ⇀ ⇀ 1 = m " v d" v ⇀ + 2 v d · v ⇀ v ⇀ + " v ⇀ " v ⇀ f dk . k ,T k ,T k ,T k ,T k ,T 2 The first term vanishes according to the condition given in (2.45). Again, it is assumed that only the diagonal elements of the thermal energy tensor are important and therefore the second term reduces to:

 
 ⇀ ⇀ ⇀ ⇀ ⇀ ⇀ ⇀ 2 m vd · v⇀ f d k = n v d kB T. (2.64) m v d v⇀ v ⇀ f dk = k ,T

k ,T

k ,T

The third term on the RHS of (2.63) is related to the thermal heat flux, and it is expressed as: ! "2 ⇀ 1 ""⇀ " ⇀ ⇀ m" v ⇀ " v ⇀ f d k = q T . (2.65) k ,T k ,T 2

As a result, (2.61) becomes: 

 ⇀ ⇀ ⇀ ⇀ ⇀ ∇⇀ . · E n v w + n v k T + q d k = ∇ ⇀ v ⇀f ⇀ · d d B T r r k

k

(2.66)

Turning our attention to the third term in (2.59), we obtain:

⇀  ⇀
 ⇀ ⇀ F ext F ext · ∇⇀ f d k = · E⇀ ∇⇀ f d k E⇀ k k k k h ¯ ¯ h ⇀ ⇀
 ⇀ F ⇀ F ext ext · ∇⇀ E⇀ f d k − · f ∇⇀ E⇀ d k = k k k k h ¯ ¯ h ⇀ ⇀ ⇀ = −F ext · f v ⇀ d k k

⇀

⇀

= −nF ext · v d .

(2.67)

Gathering all the reduced terms, the energy conservation equation becomes [184, 193]:  ⇀  ∂(nw) + ∇⇀ · n v dw r ∂t

⇀  ⇀ ∂(nw) ⇀ ⇀ · v nk T − ∇ · q + , = nF ext · v d − ∇⇀ ⇀ d B T r r ∂t col

(2.68)

where w is the average energy of the energy carriers and it is a function of ⇀ time and space, and q T is the thermal heat flux. Note that the energy density

38

2 Transport Equations

is given as W = nw, thus one can write:  ⇀  ⇀ ⇀ ∂W ⇀ + ∇⇀ · v d nkB T · v d W = −nF ext · v d − ∇⇀ r r ∂t

∂W ⇀ . · q + − ∇⇀ T r ∂t col

(2.69)

2.6.4 Conservation of Heat Flux Another conservation equation that can be extracted from the BTE is the conservation equation for heat flux, which is obtained after defining ψ = ⇀ ⇀ v ⇀ E⇀ where v ⇀ and E⇀ are the thermal velocity and energy, respeck ,T

k ,T

k ,T

k ,T

tively. Note that the thermal energy is related to the thermal velocity as follows: "2 1 ""⇀ " E⇀ = m" v ⇀ " . (2.70) k ,T k ,T 2 Substituting this into (2.38) yields: ⎡
⎤ ∂ ⇀  v ⇀ E⇀ f
 ⇀ ⎢ ⎥ ⇀ ⇀ ⇀ k ,T k ,T · E ∇⇀ v dk v ⇀ ⇀ ⇀f ⎣ ⎦ dk + r k ,T k ,T k ∂t =



⎛
⎞ ⇀ ∂ v ⇀ E⇀ f ⎜ ⎟ k ,T k ,T ⎝ ⎠ ∂t

(2.71)

⇀

dk . col

The external force term is conveniently omitted in the above expression for the sake of simplicity. The equation shall be simplified term by term, starting from the first in the LHS. Then, the thermal heat flux can be expressed as [193]:
 ⇀ ⇀ ⇀ (2.72) v ⇀ E⇀ f d k. qT = k ,T

k ,T

Using this expression, the first term on the LHS of (2.71) is: 
 ⇀ ∂ ⇀ ∂ qT ⇀ . v ⇀ E⇀ f d k = k ,T k ,T ∂t ∂t

(2.73)

The second term of (2.71) is: 
 ⇀
 ⇀ ⇀ ⇀ ⇀ ⇀ ∇⇀ f dk v v · E · d k = ∇ v f v ⇀ ⇀ E⇀ ⇀ ⇀ ⇀ ⇀ r r k ,T k ,T k k ,T k ,T k

 ⇀ ⇀ ⇀ ⇀ E = ∇⇀ v + v v f d k. · ⇀ ⇀ ⇀ d r k ,T

k ,T

k ,T

(2.74)

2.7 Macroscale Thermal Conduction

39

The divergence of the integral can be reduced to: 

 ⇀   ⇀ ⇀ ⇀ ⇀ ⇀ ⇀ ⇀ ⇀ E E ∇⇀ · vdqT + vTqT , d k = ∇⇀ v f v f + v v · ⇀ ⇀ ⇀ ⇀ ⇀ d r r k ,T

k ,T

k ,T

k ,T

k ,T

(2.75) ⇀ where v T is the average thermal velocity. Since the external force term has already been neglected, it is preferable to ignore the drift velocity term as well. After collecting all the reduced terms for (2.71), the conservation of heat flux becomes:  ⇀  ⇀ 
∂qT ∂qT ⇀ ⇀ + ∇⇀ . (2.76) · vT qT = r ∂t ∂t col

Since the effects imposed by the external forces and by the particle drift on the thermal heat flux are ignored in the derivation, this equation is best suited for describing photon or phonon transports. In the case of electron transport, (2.76) may deviate from the actual physics depending on the order of magnitude of the ignored terms.

2.7 Macroscale Thermal Conduction Thermal transport in bulk objects is considered as macroscale. Governing equations for macroscale transport are the most simplified form of the BTE. Since the characteristic length scale of the bulk object far exceeds the mean free path of the heat carriers, most of the terms in the moments of the BTE can be neglected. Scales of such length determine that electrons and phonons are always in thermal equilibrium, considering that the relaxation time and length are much smaller than the characteristic behavior of the material. Since only a single temperature exists, the electrons and phonons are considered as a whole. The governing equations for the heat flow are then given by (2.68) and (2.76), where the drift velocity and external forces are all negligible. They are:

∂W ∂W ⇀ = −∇⇀ · qT + + ST , (2.77) r ∂t ∂t col  ⇀  ⇀ 
∂qT ∂qT ⇀ ⇀ + ∇⇀ . (2.78) · vTqT = r ∂t ∂t col

The external neutral (i.e., uncharged) heat sources are denoted as ST , which cause the energy of the carriers to increase. The collision term is responsible for restoring thermal equilibrium. Since local thermal equilibrium exists between electrons and phonons, the collision term in the energy conservation equation vanishes. The energy density, which is nw (see (2.68)), refers to the combined energy density of electrons and phonons and it can be expressed as the product of the total specific heat of electrons and phonons, C, and temperature, T .

40

2 Transport Equations

For macro-scale heat transport there is no concern with the detailed scattering mechanism in the collision term for the heat flux. Instead, a relaxation-time approach is used as discussed in Sect. 2.4.1 to obtain the following expression: ∂T ⇀ = −∇⇀ · q T + ST , r ∂t ⇀ ⇀ 
q ∂qT ⇀ ⇀ + ∇⇀ · vTqT = − T . r ∂t τq

(2.79)

C

(2.80)

The divergence of the product of the thermal velocity and the thermal heat flux can be reduced to account for the temperature variation in space. Referring back to Sect. 2.6.4,  
 ⇀
⇀ ⇀ ⇀ ⇀ ∇⇀ · v ⇀ E⇀ v ⇀ f d k · v T q T = ∇⇀ r r k ,T k ,T k ,T  (2.81) 
 ⇀ ⇀ ⇀ f E⇀ v ⇀ d k . v ⇀ · ∇⇀ = r k ,T

k ,T

k ,T

Since the characteristic length at macro-scale transport is much greater than the relaxation length of the heat carriers (i.e., electrons and phonons), the distribution function f should in fact converge to the equilibrium distribution function, denoted as f eq . This depends on the temperature T . The integral can now be re-written as:



 ⇀ ⇀ ∂f eq ⇀ ⇀ ⇀ ⇀ eq dk . d E E v v k = ∇ v f v ⇀ · ∇⇀ ⇀ ⇀ ⇀ ⇀T · ⇀ ⇀ r r k ,T k ,T k ,T ∂T k ,T k ,T k ,T (2.82) Thermal conductivity is defined as [193]:

⇀ ∂f eq ⇀ ⇀ d k. kT = τ⇀ v ⇀ E⇀ v ⇀ (2.83) k k ,T k ,T k ,T ∂T Using this expression and by assuming that the relaxation time for the thermal heat flux is constant, (2.82) can be written as:  k
⇀ ⇀ T ∇⇀ T. (2.84) v q ∇⇀ · T T = r τq r With this simplification, the energy and heat flux conservation equations for macroscopic thermal transport become: ∂T ⇀ = −∇⇀ · q T + ST , r ∂t ⇀ ⇀ kT qT ∂qT + T = − ∇⇀ . r ∂t τq τq

C

(2.85) (2.86)

The above two equations form the so-called hyperbolic heat conduction equation. The hyperbolic heat conduction can be expressed by a single equation.

2.8 Micro/Nanoscale Thermal Conduction

41

This is done by taking the divergence of the heat flux equation and using the energy equation to eliminate some terms. The equation then takes the form:


∂T ∂ST ∂2T . (2.87) = ∇⇀ T + ST + τq · kT ∇⇀ C τq 2 + r r ∂t ∂t ∂t This equation is not necessary for application to micro/nanomachining and will not be considered further. The hyperbolic heat conduction can be reduced to parabolic heat conduction if the time of interest becomes much greater than the relaxation time of the heat flux. Under such a condition, the transient term in (2.86) drops out and the Fourier law is obtained, that is: ⇀

q T = −kT ∇⇀ T. r

(2.88)

Inserting this expression into the energy conservation equation, the parabolic heat equation is obtained: C


 ∂T = ∇⇀ + ST , · k ∇ ⇀T T r r ∂t

(2.89)

where the conduction is purely diffusive both in time and space.

2.8 Micro/Nanoscale Thermal Conduction At the microscopic level, which ranges from micrometer to sub-micrometer length scales, electrons and phonons can exist at different thermal energy levels. Local thermal equilibrium does not exist in this case since electrons and phonons are not at the equilibrium temperatures. If there is energy transfer to the matter, the thermalization process first occurs through electrons and then relaxation takes place between hot electrons and phonons until equilibrium is reached. There are many thermal models available for modeling microscopic heat transport. Derivations of these models will be discussed in the following sections. 2.8.1 Two-Temperature Model External heating of a workpiece first affects the electron gas inside the workpiece, causing the electron energy to elevate substantially compared to the lattice energy. This phenomenon can be modeled by considering two separate temperatures for lattices and electrons. Such a model is referred to as the twotemperature model (TTM) [192, 197]. The TTM neglects the kinetic-energy changes of electrons, and assumes interactions between electrons and phonons through a coupling constant G.

42

2 Transport Equations

To derive the TTM the energy and heat flux conservations for electrons and phonons are written using (2.68) and (2.76), which become:

∂We ∂We ⇀ = −∇⇀ · q T,e + + ST , (2.90) r ∂t ∂t col   ⇀ ⇀ 
∂ q T,e ∂ q T,e ⇀ ⇀ + ∇⇀ , (2.91) · v T,e q T,e = r ∂t ∂t col

∂Wph ∂Wph ⇀ = −∇⇀ · q T,ph + , (2.92) r ∂t ∂t col   ⇀ ⇀ 
∂ q T,ph ∂ q T,ph ⇀ ⇀ + ∇⇀ = , (2.93) v q · T,ph T,ph r ∂t ∂t col

where subscripts “e” and “ph” denote electrons and phonons, respectively. The external heating source, ST , is assumed to be for electrons only. By assuming that the object length far exceeds the relaxation lengths of electrons and phonons, these equations become:

∂We ∂We ⇀ = −∇⇀ · q T,e + + ST , (2.94) r ∂t ∂t col ⇀

⇀

q T,e ∂ q T,e kT,e + ∇⇀ Te = − , ∂t τq,e r τq,e

∂Wph ∂Wph ⇀ = −∇⇀ · q T,ph + , r ∂t ∂t col ⇀

(2.95) (2.96)

⇀

q T,ph ∂ q T,ph kT,ph + ∇⇀ Tph = − . ∂t τq,ph r τq,ph

(2.97)

Since electrons and phonons are transferring energy between each other, the collision term for electrons includes energy lost to phonons while the collision term for phonons involves the energy received from electrons. By using the relaxation time approach, the collision terms for electrons and phonons can be written as:



∂We Tph − Te ∂Wph , (2.98) = − = Ce ∂t col ∂t τe−ph col where τe−ph is the relaxation time between electrons and phonons. By treating the energy density of electrons and phonons as a function of electron temperature and phonon temperature, respectively, the governing equations are transformed into: Ce

Te − Tph ∂Te ⇀ = −∇⇀ + ST , · q T,e − Ce r ∂t τe−ph ⇀

(2.99)

⇀

q T,e ∂ q T,e kT,e + Te = − ∇⇀ , r ∂t τq,e τq,e

(2.100)

2.8 Micro/Nanoscale Thermal Conduction

Cph

∂Tph Tph − Te ⇀ = −∇⇀ , · q T,ph − Ce r ∂t τe−ph

43

(2.101)

⇀

⇀

q T,ph ∂ q T,ph kT,ph + Tph = − ∇⇀ , r ∂t τq,ph τq,ph

(2.102)

where ST is the volumetric heat generation caused by the external heating sources, C is the heat capacity, and k is the thermal conductivity. These four equations are called the general TTM where the ballistic nature of electrons and phonons is maintained in the governing equations. The general TTM can be simplified to produce the so-called hyperbolic TTM, parabolic TTM, and even the Dual-Phase Lag Model (DPLM) [197]. The hyperbolic TTM can be obtained from the general TTM by ignoring the heat propagation of phonons. This is usually justified when considering metals, since the thermal conduction of phonons is weak compared to that of electrons. Thus, the set of four equations reduces to three: Ce

∂Te Te − Tph ⇀ = −∇⇀ + ST , · q T,e − Ce r ∂t τe−ph ⇀

(2.103)

⇀

q T,e ∂ q T,e kT,e + ∇⇀ Te = − , r ∂t τq,e τq,e ∂Tph Tph − Te = −Ce Cph . ∂t τe−ph

(2.104) (2.105)

If the transient behavior of heat flux in the hyperbolic TTM is omitted, the parabolic TTM is formed. 2.8.2 Dual-Phase Lag Model The hyperbolic TTM can be combined easily to yield the so-called dual-phase lag model (DPLM) [197]. This model describes the phonon temperature in a single governing equation, and all the thermal properties of electrons and phonons are considered to be constant. Therefore, two of the three unknowns need to be eliminated from the governing equation. To do so, (2.103) is first differentiated with respect to time to obtain:





⇀ ∂ q T,e ∂ ∂ ∂ST ∂Te Ce Ce − . Ce = −∇⇀ + + T T · e ph r ∂t ∂t ∂t τe−ph ∂t τe−ph ∂t (2.106) The divergence of (2.104) yields: ⇀ 

⇀ ∂ q T,e q T,e kT,e − ∇ = −∇ . (2.107) ∇ · ∇⇀ ⇀ · ⇀ Te ⇀ · r r r ∂t τq,e τq,e r ∂ ∂t

44

2 Transport Equations

Substituting this equation into (2.106) and rearranging it gives: ⇀ 



q T,e ∂ ∂Te kT,e Ce = ∇⇀ + ∇ ∇ · ⇀ · ⇀ Te r r ∂t ∂t τq,e τq,e r



∂ ∂ST ∂ Ce Ce . Te + Tph + − ∂t τe−ph ∂t τe−ph ∂t

(2.108)

If the relaxation time of the electron heat flux, τq,e , is assumed constant, then:


1 1 ∂ ∂Te ⇀ Ce = · q + k ∇ ∇⇀ ∇ ⇀ Te ⇀ · T,e T,e r ∂t ∂t τq,e r τq,e r



(2.109) ∂ ∂ST ∂ Ce Ce . Te + Tph + − ∂t τe−ph ∂t τe−ph ∂t Note that the divergence of the electron heat flux can be eliminated by using the electron-energy equation given in (2.103) where: ⇀

∇⇀ · q T,e = −Ce r

∂Te Te − Tph − Ce + ST . ∂t τe−ph

Thus, (2.109) becomes:





∂ Ce ∂Te ∂ ∂Te Ce Ce ∂ + Ce + Te − Tph ∂t ∂t τq,e ∂t ∂t τe−ph ∂t τe−ph 
Ce Te − Tph ST ∂ST . − · k ∇ ∇⇀ + + = ⇀ Te T,e r r τq,e τq,e τe−ph τq,e ∂t 1

(2.110)

(2.111)

Next, the phonon energy equation (see (2.105)) is solved for the electron temperature:

τe−ph Cph ∂Tph Te = + Tph . (2.112) Ce ∂t Substituting Te into (2.111) to eliminate the electron temperature yields:  



∂ ∂ ∂Tph τe−ph Cph ∂Tph τq,e Ce + Ce ∂t ∂t Ce ∂t ∂t  



∂ ∂Tph ∂ ∂Tph τe−ph Cph ∂Tph + − (Cph − Ce ) Cph + Ce ∂t Ce ∂t ∂t ∂t ∂t 

   ∂ST τe−ph Cph ∂Tph . + ∇⇀ + ST + τq,e · kT,e ∇⇀ T · ke ∇⇀ = ∇⇀ r r ph r r Ce ∂t ∂t (2.113)

2.8 Micro/Nanoscale Thermal Conduction

45

For the case where the electron–phonon relaxation time, specific heats of electrons and phonons and electron thermal conductivity are constant, the equation becomes: τq,e τe−ph Cph

∂ 3 Tph ∂t3

∂Tph ∂ 2 Tph − (Cph − Ce ) + (τq,e Ce + τe−ph Cph + Cph ) 2 ∂t ∂t

 ∂ST kT,e τe−ph Cph ∂
2 . ∇⇀ Tph + kT,e ∇2⇀ Tph + ST + τq,e = r r Ce ∂t ∂t

(2.114)

2.8.3 Electron–Phonon Hydrodynamic Equations By taking the zeroth-, first- and second-order moments of the BTE, which correspond to continuity, momentum conservation and energy conservation equations, respectively, the so-called electron–phonon hydrodynamic equations (EPHDEs) can be derived. The EPHDEs are usually useful for simulating heat transport inside electrical devices in sub-micron scale resolution. However, by doing so, the variables such as distribution, velocity, and energy describing the heat carriers in the equations are replaced by the corresponding average quantities. For instance, the energy-dependent velocity of the carriers is neglected and replaced by an averaged velocity. However, the overall physics of the problem are still preserved. The various conservation equations derived from the moments of the BTE were given in Sect. 2.6 (see (2.49), (2.57), (2.68), and (2.76)). It is important to consider that electrically and thermally conducting material consists of electrons and phonons as energy carriers. Using moments of the BTE, continuity, momentum, and energy conservations for electrons are expressed as:  ∂n


∂ne ⇀ e + ∇⇀ v n · , (2.115) d,e e = r ∂t ∂t col  ⇀  ⇀ 
⇀ ⇀ ∂ P d,e ∂ P d,e ⇀ + ∇⇀ , (2.116) (ne kB Te ) + · v d,e P d,e = −ene E − ∇⇀ r r ∂t ∂t col  

⇀ ∂We ⇀ ⇀ ⇀ + ∇⇀ = −en v n k T v W v · E − ∇ · · ⇀ d,e e B e d,e e e d,e r r ∂t  ∂W


e ˙ e,gen , T + · kT,e ∇⇀ +W + ∇⇀ r e r ∂t col (2.117) ⇀

where it is assumed that an external electric field, E, is applied. The Fourier law of heat conduction is used for the sake of simplicity. Since this law is applied, the assumption is that the characteristic length of the geometry far exceeds the mean free path and the relaxation length of electrons. It is also implied that heat conduction by electrons is not ballistic in time.

46

2 Transport Equations

The first equation, (2.115), enforces the particle conservation for electrons. This is where the rate of change of the electrons in time and space equals the rate of increase of electrons due to collisions. The second equation, (2.116), refers to electron momentum conservation. This is when the rate of change of the electron momentum in time and space is equivalent to the forces exerted by the electric field and by the electron pressure, as well as the rate of momentum gain or loss in collisions. The third equation, (2.117), represents the electronenergy conservation. This is where electron energies are conserved in time and space according to the energy supplied by the electric field, the work performed by the electron pressure, the divergence of heat flux, and the rate of change of energy due to collisions. Similarly, these conservation equations can be written for phonons. The continuity equation does not apply to phonons since they can be created and destroyed. In addition, the phonon thermal velocity is typically much greater than its drift velocity, implying that the phonon momentum conservation is not needed. Hence, all the terms related to the drift velocity of phonons are neglected. As a result, only the energy and heat flux equations remain. Phonons are usually sub-divided into optical phonons and acoustic phonons [9, 66, 224]. Optical phonons exist mostly in semiconductor while acoustic phonons are present in any type of materials. Optical phonons have essentially zero velocity while acoustic phonons propagate and conduct heat. As a result, the spatial diffusion term in the optical-phonon energy equation is typically omitted. Accordingly, the governing equations for optical and acoustic phonons are written as:

∂WLO ∂WLO = , (2.118) ∂t ∂t col

∂WA ∂WA ⇀ = −∇⇀ · q + , (2.119) T,A r ∂t ∂t col   ⇀ ⇀ 
∂ q T,A ∂ q T,A ⇀ ⇀ + ∇⇀ , (2.120) · v T,A q T,A = r ∂t ∂t col

where the subscript “LO” stands for optical phonons and “A” for acoustic phonons. In order to present these equations in a simpler manner, the divergence term in the acoustic phonon heat flux equation is approximated by relating it to the phonon thermal conductivity and temperature gradient, as previously shown in Sect. 2.6. By using the concept of thermal conductivity, it is assumed that the object is “bulk” in size. The internal energy of the phonons can be expressed as a product of the specific heat and temperature. In addition, the ballistic transport of phonons in time shall be ignored. Using these approximations, the above equations for phonon transport are transformed into:

2.8 Micro/Nanoscale Thermal Conduction





∂TLO ∂WLO = , ∂t ∂t col


∂TA ∂WA = ∇⇀ T + · kT,A ∇⇀ . CA r A r ∂t ∂t col CLO

47

(2.121) (2.122)

Next, various collision terms need to be addressed. The relaxation time approximation (see Sect. 2.4.1) can be used to determine these terms. It is important to first discuss the physics of energy transfer mechanisms in this electron–phonon system before attempting to express them in terms of mathematical equations. Due to the externally applied electric field, electrons are accelerated where they gain additional energy from the force field. Since the electric field does not directly affect phonon energy, phonons remain cold while electrons are hot. Electrons start to relax by transferring excess energy to phonons until thermal equilibrium exists between them. Energy is also transferred between optical phonons and acoustic phonons. Following the detailed explanations given by Blotekjaer [22], the various collision terms can be approximated as:

∂ne = n˙ e,gen , (2.123) ∂t col  ⇀  ⇀ me ne v d,e ∂ P d,e =− , (2.124) ∂t τm
col 

2 3k n T /2 + m v /2 − 3k T /2 e B e e B LO d,e ∂We =− ∂t col τe−LO (2.125)
 2 ne 3kB Te /2 + me vd,e /2 − 3kB TA /2 , − τe−A


2 ne 3kB Te /2 + me vd,e /2 − 3kB TLO /2 CLO (TLO − TA ) ∂WLO − , = ∂t τe−LO τLO−A col (2.126)


2 ne 3kB Te /2 + me vd,e /2 − 3kB TA /2 CLO (TLO − TA ) ∂WA + , = ∂t col τe−A τLO−A (2.127) ⇀

where the electron momentum, P d,e , and the average electron energy, We , are expressed as: ⇀

⇀

P d,e = me ne v d,e , 1 3 2 . We = ne kB Te + ne me vd,e 2 2

(2.128) (2.129)

48

2 Transport Equations

According to the above order, the first collision term corresponds to the number of electrons generated due to the incoming electrons from external sources. The second term describes the relaxation of the momentum between electrons, while the remaining three terms account for the energy exchange between electrons and phonons. The momentum relaxation is assumed to be fully isotropic. Collisions between impinging electrons from any external sources and electrons inside the material itself randomize momentum, resulting in no net momentum transfer during scattering. The premise of this assumption is based on the condition that the incoming electrons from the external sources are at the same energy level as the electrons inside the medium. The third term describes the energy exchanges between electrons and optical and acoustic phonons. The fourth term refers to interactions of optical phonons with electrons and acoustic phonons while the last term corresponds to the interactions of acoustic phonons with electrons and with optical phonons. Replacing all the collision terms, electron momentum, and electron energy in the conservation equations with the above expressions, the following set of equations is obtained: 
∂ne ⇀ + ∇⇀ = n˙ e,gen, v n (2.130) · d,e e r ∂t

⇀ kB n˙ e,gen ⇀ e ⇀ ∂ v d,e ⇀ 1 ⇀ + v d,e · ∇⇀ E− ∇⇀ (ne Te ) − + v =− v d,e , r d,e ∂t me me n e r τm ne (2.131) 
∂Te ⇀ + ∇⇀ · v d,e Te r ∂t  T −T
1 2 e LO ⇀ = Te ∇⇀ − k ∇ ∇ · v + ⇀ Te ⇀ · T,e d,e r r 3 3ne kB r τe−LO (2.132)

2 1 1 n˙ e,gen me vd,e Te − TA 2 + − − + − τe−A τm τe−LO τe−A ne 3kB



n˙ e,gen 2 ˙ e,gen , Te + W − ne 3ne kB 2 ne me vd,e ∂TLO 3ne kB (TLO − Te ) TLO − TA = − − , (2.133) ∂t 2CLO τe−LO 2CLO τe−LO τLO−A  n e me v 2
1 ∂TA d,e = + ∇⇀ · k ∇ ⇀ TA T,A r ∂t CA r 2CA τe−A (2.134) 3ne kB (TA − Te ) CLO (TA − TLO ) − . − 2CA τe−A CA τLO−A Details of the derivations are given in Appendix B, which constitute a set of EPHDEs. It is necessary to go over the physical meaning of each of these conservation equations and discuss the implication of each term, which is done below.

2.8 Micro/Nanoscale Thermal Conduction

49

The electron continuity equation given in (2.130) refers to the electron number balance. This is the rate of increase of the electron concentration being equal to the rate of advection and diffusion plus the rate of electrons generated. In the electron momentum conservation (see (2.131)), it is observed that electron velocity tends to increase propagating through decreasing electric field, negative electron concentration gradient, and/or declining electron temperature gradient. The rate of the electron velocity relaxation (or the rate at which the electron reduces its velocity) is dictated by the momentum relaxation time, τm , and the rate of increase/decrease of the externally generated electron concentration, n˙ e,gen . It is normalized by the local electron concentration, ne . Larger momentum relaxation time means longer time for electrons to reduce their velocity. In addition, when there is additional local electron concentration, the rate of momentum relaxation is elevated by those electrons since an increasing electron concentration increases the frequency of electron–electron collisions. The electron energy conservation equation, (2.132), is different than the  ⇀ ⇀ general equation given by (2.117). The Joule heating term −ene v d,e · E does not appear in the equation anymore due to the simplification given in Appendix B, although it is implicitly included. The Joule heating phenomenon is embedded in (2.132) even though it is not expressed explicitly in Thesame goes for the work done by the electron pressure  the equation. ⇀ −∇⇀ v n k · d,e e B Te . It would be rather difficult to point out the explicit r physical meaning of each term in the current form of the electron energy equation. This claim is explored in the next paragraph. The effects of electron concentration generation (or heating by external electrons) on the electron temperature seem to appear in the last three terms on the RHS of (2.132). One can increase the electron temperature locally by imposing a strong localized heating source. This effect is given by the last of the three terms. Judging from the other two terms (i.e., 2 (n˙ e,gen /ne )(me vd,e /3kB )−(n˙ e,gen/ne )Te ), the electron heating appears to have mixed effects on the electron temperature. Increasing the rate of electron heating may increase or decrease the electron temperature depending on the ratio of average kinetic energy to local electron temperature. To clarify this, the average drift velocity of the electrons, vd , should be considered. The vd decreases when the electron heating rate is large as asserted by the electron momentum equation (see (2.131)). Therefore, the combination of the increase in the electron concentration due to electron heating and the decrease in the average electron drift velocity determines whether or not the electron temperature is increased due to increasing average kinetic energy. When electron concentration is increased locally, the electron temperature shall decrease due to the increase in the electronic heat capacity. The effects of the Joule heating and the work done by the electron pressure are lumped together in the divergence of the electron drift velocity and temperature, the two terms which appear on the RHS of (2.132).

50

2 Transport Equations

Next, we discuss the last two phonon energy equations (see (2.133) and (2.134)). Based on the dispersion relation of phonons, optical phonons have essentially zero group velocity, while acoustic phonons are capable of propagating through the medium [224]. Therefore the optical phonon energy equation (i.e., (2.133)) does not have any diffusion term. Optical phonons can generally be viewed as intermediate energy storage between electrons and acoustic phonons. Therefore, the corresponding conservation equation consists of energy exchanges between these heat carriers. Optical phonons gain energy from electrons (both from average electron kinetic energy and thermal energy), as given by the first two terms on the RHS of (2.133). The last term in (2.133) describes energy exchange with acoustic phonons. The acoustic phonon energy equation is similar to the optical phonon energy equation except that there is a diffusion term that describes diffusion of acoustic phonon energy. Until now, the electric field term in the momentum equation has not been addressed. This field can be obtained by using the electric potential and the charge distribution in space. The relationship between these quantities is derived from Maxwell’s equations. The resultant governing equation is the Poisson equation. Snowden [184] has provided a detailed discussion on how the Poisson equation is derived from the Maxwell equations, therefore, the same derivation shall not be repeated here. Instead, it will be assumed that the Poisson equation can readily be used, which is expressed as: ∇2 V =

e (ne − n+ ), εe

(2.135)

where n+ is the density of the positive charges in the material, and the electric field is expressed as ⇀

E = −∇V.

(2.136)

The EPHDEs are now given by (2.130)–(2.135). These equations are to be solved simultaneously in order to predict the electrical and thermal behavior of a system with electrons and phonons as energy carriers. The momentum equation in the EPHDEs can be further simplified to yield the so-called drift-diffusion approximation. This is where the velocity transient and inertia terms are neglected. The drift velocity vector of electrons can be conveniently expressed as: ⇀

v d,e = −

 eτeff ⇀ kB τeff
E− ∇ ne T e , me me n e

(2.137)

where τeff =



1 τe−e

+

n˙ e,gen ne

−1

.

(2.138)

2.9 Nanoscale Thermal Conduction

51

Since the electron drift velocity is explicitly given, it can be substituted into the electron continuity equation to yield:  k
⇀ e ∂ne B − ∇ · τeff ne E − ∇ · (τeff ∇ (ne Te )) = n˙ e,gen, (2.139) ∂t me me where the second term on the left-hand side represents the drift of electrons in and out of the system while the third term includes the gains or losses due to the electron diffusion. During the course of obtaining these governing equations, a number of assumptions have been made, which are by no means unique. If one or more of the assumptions are removed, the EPHDEs will take a rather different form than the derived one.

2.9 Nanoscale Thermal Conduction Nanoscale thermal conduction is fundamentally different from its counterparts at the macro- and microscales. At the nano-regime, the wavelength of heat carriers becomes comparable to the size of the object itself. Under these conditions, the wave nature of the heat carriers cannot be ignored which necessitates the use of wave theories to predict the thermal diffusion. For electromagnetic radiation, for example, photon approach must be replaced by the Maxwell equations to determine the energy propagation in a medium. This means that the BTE (or the RTE) would not be applicable to predict energy transfer at these small scales anymore. For electrons, however, the wavelength is much smaller (or, at least a few times smaller) compared to the size of a nanometer-sized object. Therefore, the BTE is applicable to determine the ballistic and semi-ballistic behaviors of electrons in terms of time and space. In this case, the BTE is reduced to the EBTE, and its solution can be obtained with MC methods. These ballistic and semi-ballistic behaviors are typically described by the collision term in the BTE. The full phase functions or the scattering probabilities of heat carriers need to be included in the collision term rather than simplifications based on the time-relaxation approaches (see Sect. 2.4.1). This makes the nanoscale thermal conduction significantly more challenging than the macroand microscale thermal conduction. The simulation procedures for the nanoscale electronic thermal conduction with MC methods will be discussed in details in Chaps. 3 and 7. These MC procedures are similar to those used for the solution of BTE for electronor laser-beam transport, except that all the statistical ensembles are tracked simultaneously instead of sequentially, one after another. For phonons, i.e., lattice vibrations, the situation is different. Phonon wavelengths typically range from nano/sub-micron to microns depending on their energies. In nanoscale regime, this may dictate the use of wave theories to accurately predict the transport phenomena, particularly for nonmetallic

52

2 Transport Equations

materials. In this case, the BTE is reduced to the PRTE, and its solution can be obtained using MC methods, which is applicable to phonon transport at small wavelength regime. In general, phonons are rather complicated energy carriers compared to both electrons and electromagnetic waves. Modeling of energy transport by phonons is not trivial due to complicated scattering probabilities as they are not well-documented. The details of the simulation procedures for phonon transport are provided by Mazumder and Majumdar [129], where they performed a MC simulation for observing ballistic and semiballistic phonon transport. In Sect. 3.5, we outline a general MC simulation for the solution of the PRTE. In the case where a workpiece size of a few nanometers is encountered, molecular dynamic (MD) simulations are preferred over MC simulations. In Chap. 9, we discuss MD simulations and present a series of simulation results.

2.10 Thermal Radiation at Nanoscales Radiative transfer plays role in many material processing applications either due to an incident external source (laser beam) or because of radiative interaction between different objects. As the object size and the distance between the objects get smaller than the dominant wavelength of emission, the wave nature of radiative exchange has to be considered to account for the underlying physics correctly, including interference, coherence, tunneling and polarization. As a rule of thumb, this cut-off value can be determined using the Wien law, which relates the peak emission wavelength to the temperature of the material via: (λT)max = 2897.6 mm-K. If the temperature of an object is 300 K, its emission wavelength range is centered around 10 mm = 10,000 nm; if the temperature reaches to 1,000 K, the peak wavelength corresponds to 3,000 nm, which is still much larger than 100 nm, the so-called upper limit for nanoscale phenomena. Because of this, for all micro/nanomachining applications, radiative exchange should be modeled starting from the Maxwell equations. The term “near-field radiative transfer” (NRT) is used to define the radiative exchange between two objects within close proximity, less than a wavelength or so. NRT cannot be modeled using the RTE discussed in Section 2.5.2, which is for far-field applications, where the wave nature of radiative transfer is completely omitted. The RTE, which is derived from the BTE, is applicable if the objects are large and/or the distance between them is several wavelengths long. To properly define thermal radiation emission from a nanoscale object, we may have to refer to the quantum electrodynamics, as at length scales smaller than 10 nm or so, the use of continuum and the LTE approaches become questionable. The fluctuational electrodynamics (FED) theory, which is originally derived for macroscopic systems, can be used to calculate the emission from objects with sizes down to 10 or 20 nm. The FED approach is based on the Maxwell equations and relates the strength of the fluctuations

2.11 Comments

53

of the charges inside a body and its local temperature [58, 171, 222]. This theory provides the necessary bridge between the emission and propagation of electromagnetic waves. The fundamentals of NRT are significantly different than the particle theories discussed for calculation of electron and phonon conduction, which cannot be discussed within the constraints of a focused text. In this monograph, we do not provide any discussion on near-field radiation transfer (NRT), which is becoming an important area of research itself. For further details, the reader should refer to the recent relevant literature [58, 92, 140, 142, 152, 222].

2.11 Comments Thermal transport phenomena can be modeled in terms of wave or particle approaches. This choice depends on the nature of the energy carriers, their energies, as well as the size of objects they interact. A wave theory accounts for coherence, polarization, interference, diffraction, refraction, and tunneling according to the amplitude and the phase of the waves while particle theory disregards this information. In this chapter, we discussed the transport phenomena based on particle theories. We considered the BTE, the EBTE, the RTE, the Fourier and hyperbolic heat conduction models, the TTM, the DPLM, and the HDM. These models are applicable to different physical systems depending on time- and length-scales of interest, which are categorized in Fig. 2.1. In our discussions, we did not emphasize the wave behavior of these energy carriers at all. At nanoscales, radiative exchange should always be modeled using the Maxwell equations. On the other hand, particles theories should be sufficient for explaining energy transfer with phonons and electrons. The molecular transport, particularly the molecular dynamics simulation, is preferred if a much finer resolution of modeling of transport phenomena is required. It is covered in Chap. 9. Flow chart in Fig. 2.1 is provided to help in choosing an appropriate transport model for a given application of micro/nanomachining process. One can choose any of the models based on the required level of resolution and complexity of the problem. This choice depends on the time constant of the process, t, and the physical length of the object, L. As expected, the simplest model for thermal conduction is the Fourier law, which is capable of predicting thermal transport for bulk materials (at the coarsest resolution) where details of electrons and phonons are ignored. This is the case when L ≫ lmfp (see definition in Sect. 2.3) and t ≫ τmfp . The TTM can be employed where electrons and phonons can exist at two different temperatures within the same control volume. Following the TTM in complexity would be a model based on EPHDEs, which allows the separation between electron and phonon temperatures and also includes the electrical behavior of electrons as they flow through the material. Although the EPHDEs are quite complete in handling electron and phonon behaviors, the ballistic and semi-ballistic transport of

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Fig. 2.1. Thermal transport phenomena and models are listed for different length scales (in decreasing order, starting from macroscopic levels down to quantum levels at the bottom of the chart). Time constants shown are specific to energy carriers indicated. Thermal radiation and quantum mechanics are not discussed in this monograph. Note that nanoscale radiative transfer cannot be modeled with particle approaches and requires the solutions based on Maxwell equations

2.11 Comments

55

these carriers are neglected, which simplifies the solution. In order to include the ballistic behaviors of the carriers, MC simulations in electron and phonon transports are required. Finally, the use of MD simulation is necessary to predict machining at the molecular level. Coupling between the MD and the ETE needs to be considered if electrons and phonons equally contribute to transport phenomena inside the material and details of nanopatterning to be determined. Analytical solutions of intergo-differential equations such as the EBTE and the RTE are nearly impossible unless various simplifying assumptions are introduced. The true physics of a problem, on the other hand, can be retained if numerical solution techniques are adapted instead of analytical approaches. Among all numerical techniques, the statistical MC methods are the most

Fig. 2.2. Overview of micro/nanomachining modeling approaches is given. The first column lists different models that can be used to determine the thermal conduction in a workpiece. The second column is for the models required to account for the external heating of the workpiece. Final column shows the different output parameters expected from a comprehensive algorithm that will help guiding the experiments

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suitable for the solution of the BTEs due to their flexibility in handling different geometries and scattering behavior of the energy carriers. At the bottom of the flow chart, we show two additional models that need to be used with further decrease in the feature length. Thermal radiation at nanoscales needs to be modeled using Maxwell equations. However, radiation transfer usually treated as a boundary phenomenon and can be model using the RTE. As we approach to atomic and sub-atomic size ranges, none of the models discussed in this monograph will be applicable. Quantum dynamics would be needed to understand the processes at that level [113, 135]. We outline in Fig. 2.2 the transport models that can be used for micro/ nanomachining simulations. For this purpose, first we need to choose a heat conduction model depending on the resolution and the details required. This model is to be coupled with the formulations describing external electron and/or laser-beam heating. This combined procedure constitutes the algorithm to be used for micro/nanomachining applications, which will provide the specific details on spatial and temporal temperature profiles within the workpiece, phase change and material removal rates, as well as the shape and structure of nanopatterns which can be obtained during the process. In the next chapter, we will discuss a series of simulation procedures based on MC solutions of the BTEs.

3 Modeling of Transport Equations via MC Methods

The Boltzmann transport equation (BTE) describes transport phenomenon for an ensemble of energy carriers, such as electrons, phonons, and photons. The BTE is, by definition, an integro-differential equation, representing the conservation of energy of particles along a line-of-sight propagating through a medium. It involves at least seven independent variables, including two directional angles (zenith, θ, and azimuthal, φ, angles), three space coordinates ((x, y, z), or (r, z, θr ), or (r, θr , φr ), depending on the choice of coordinate system), time, and the wavelength or frequency of particles at which the properties are determined. The BTE, as discussed in Chap. 2, is difficult to solve due to the integral term, which describes the in-scattering nature of the transfer process. To overcome this difficulty, several approximations have been introduced over the years, which are either mathematical or physical in nature, and many times used in tandem. For example, the transient nature of the BTE is omitted for many applications. If the medium is assumed non-scattering, then the resulting governing equation is reduced to a simpler differential equation, which is relatively easier to solve. However, for the problems we are interested in here, the scattering cannot be neglected. In addition, scattering nature of phonons and electrons is usually more complicated then that for photons. Consequently, the correct simulation of the transfer problem necessitates the solution of the integro-differential BTE, which is not trivial and requires careful evaluation. In electron-based micro/nanomachining applications, the propagation of electrons inside the workpiece can be modeled via the electron-beam transport equation (EBTE), which is very similar to the radiative transfer equation (RTE). On the other hand, propagation of lattice energies and conduction electrons in materials can be modeled using the phonon radiative transport equation (PRTE) and the electron transport equation (ETE), respectively. All these three equations are simplified versions of the BTE; therefore, the methodologies developed over the years to solve the RTE may come handy in developing solution strategies for the EBTE, the PRTE and the ETE. Yet, in general, the solution of the RTE is not trivial, as it requires extensive and

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3 Modeling of Transport Equations via MC Methods

lengthy derivations and computational power [136, 182]. In addition, these solution methodologies are not readily expandable to arbitrary geometries. This difficulty becomes more obvious if anisotropic scattering and spectral nature of the properties are to be considered, which is necessary for most physically realistic applications of the RTE, EBTE, PRTE, and ETE. Instead of using any approximations, these equations can be solved using statistical Monte Carlo (MC) methods. The derivation of MC methods is relatively straightforward and their implementation can be achieved with little effort. Also, they are quite powerful in accounting for complicated physical situations and geometries. Because of their statistical nature, however, their accuracy may suffer if the number of statistical ensembles used is not large. This problem can be overcome by using a large number of ensembles, which, unfortunately, results a significant increase in required computational time and memory. Nevertheless, with the improvement of computer power each year, MC methods are likely to find more applications in solving the BTE for different energy carriers, and deserves further discussion here. In the context of particle transport phenomena, a MC method refers to the simulation of the propagations of energy carriers such as photons and electrons inside a participating medium, which absorbs and scatters the carriers. A MC method is, by definition, similar to playing the game of roulette at the legendry casinos of Monaco, where the name stems from. To play the game correctly, one needs uniformly distributed dices, a good set of game rules, and a game board. In a MC method, one needs a uniformly distributed random number generator, correct scattering probabilities, and a grid system for storing the particle histories. At first glance, one often wonders how the randomness of the particle simulation can generate a correct solution to the transport equation. Strictly speaking, the simulation is not truly random in the sense that particles or energy carriers travel according to the scattering probabilities. In essence, they are derived based on the laws of physics. The simulation procedures for a MC method starts by releasing a bundle in the medium from its initial location. Each bundle in MC method is comprised of large number of particles, sometimes reaching millions. Then according to a prescribed probability distribution function, the path length and the direction of the bundle are determined. As the bundle propagates in the medium, a fraction of its intensity/energy is absorbed and scattered. Eventually, all of its energy is consumed (attenuated) or the bundle reaches a boundary and contributes to either reflection or transmission distribution. Figure 3.1 depicts sample trajectories of two statistical bundles as they propagate through a scattering medium. It is the contribution of many of these statistical bundles that provides the intensity distributions and solution to the governing equation. It should be understood that a MC method treats propagating waves as discrete packets of quantum particles. The assumption that the wavelength of the propagating bundle is small compared to the characteristic length of the object is always implied. Should this assumption be violated, then the wave

3 Modeling of Transport Equations via MC Methods

59

Fig. 3.1. The zigzag trajectories of two particles in three dimensions generated using a MC simulation are shown. The z-dimension of the medium is set to be 10 units while x- and y-dimensions are infinite in extent

nature of particles needs to be considered and different set of equations such as the Maxwell equations are to be needed to represent the physics. MC methods have many different applications in different fields of sciences. In this monograph, the discussions shall be confined to the simulation procedures applicable to electron-beam transport, radiative heat transfer, and electron–phonon thermal conduction. The simulation procedures for MC methods in electron-beam transport and radiative heat transfer are almost identical with a few exceptions. When it comes to the electron or phonon thermal conduction inside matter, the simulation becomes cumbersome since all the energy carriers must be accounted for simultaneously, meaning that all the statistical ensembles are launched at the same time, in order to determine the scattering properties in time and space. Such a simulation requires a significant amount of computer memory storage and computational time, which needs to be addressed. This chapter is devoted to basic simulation procedures of MC methods. Specific details about the scattering properties for different energy carriers will be discussed in Chaps. 4 and 7.

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3 Modeling of Transport Equations via MC Methods

3.1 Cumulative Probability Distribution Function The basic principle behind a MC technique is the cumulative probability distribution function (CPDF). A CPDF of a variable is how frequently a given value of that variable occurs within a fixed function. As an example, we will consider below a thermal radiation problem where MC methods have extensively employed [182]. Let us assume we would like to model blackbody energy distribution over all wavelengths for a given temperature of 1,000 K. The plot of the spectral emissive power distribution at that temperature is shown in Fig. 3.2. The total emissive power of this object can be expressed as: ∞ E= Eλ dλ, (3.1) 0

and the amount of emissive power if the object emits within the spectral range of 0 − λ (e.g., λ = 2 µm), which is the shaded area in the figure, is given as: Cλ =



λ

Eλ dλ.

(3.2)

0

Therefore, the CPDF of the object which emits at a wavelength smaller than λ can be sampled as: Cλ R(λ) = . (3.3) E

Fig. 3.2. The blackbody emissive power as a function of wavelength at a temperature of T = 1,000 K is plotted

3.2 Building a CPDF Table for a MC Method

61

If R(λ) is replaced with a random number Ran, then the corresponding wavelength λ can be obtained by inverting (3.3), which yields: λ = λ(Ran).

(3.4)

If this process is repeated for N times, a histogram for a range of λ drawn can be obtained. Normalizing this function by N and multiplying by E yields the original spectral emissive power function, Eλ , as N approaches infinity. This example can be viewed as a statistical means of representing the spectral blackbody profile of a given object radiating at 1,000 K temperature. If only one wavelength is drawn from (3.4) using a random number, it will not represent the range of wavelengths that the object emits. It would require a generous amount of statistical ensembles to truly represent the blackbody distribution. That requirement goes in parallel with the demand for computational power, which is the drawback of a statistical approach. Nevertheless, MC method is an approach which easily lends itself to solving the particle transport phenomena in complex geometries, which is described by the BTE. Energy of an ensemble of particles propagating along a line of sight suffers from attenuation (i.e., absorption and scattering) as described by the BTE. Therefore, the statistical approach to solving the BTE can be based on by simply “chopping” the directional energy of particles into a prescribed number of statistical ensembles and allowing each to travel in time and across space independently1 using the “rules” asserted by the BTE and the CPDFs derived from the scattering properties. For example, the intensity form of the BTE for radiative transfer (that is, the RTE) states that intensity should be attenuated exponentially; therefore, the probability of a statistical ensemble being attenuated must increase exponentially accordingly. The CPDFs will then statistically determine the scattering direction or the energy of an ensemble when it encounters scatterings. Similar to the above example for statistically determining the emission spectrum of a blackbody, the results are meaningful only when a large number of statistical ensembles are traced. What a MC method really offers in this context is a statistical approximation to the solution of the BTE in time, space, energy spectrum, and/or direction. Having known the concept behind MC methods, all that is needed to do is to establish a tracking algorithm which simulates the propagations of these statistical ensembles.

3.2 Building a CPDF Table for a MC Method Derivation of an explicit equation for calculation of the scattering direction of energy carriers is always difficult. Instead, a table can be constructed which contains all the scattering data including the CPDF to determine the 1

This means that the propagation of a statistical ensemble will not interfere that of another, so interference effects are ignored.

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scattering angle for a given random number. Once completed, this table can be used for find the argument of a CPDF which is quite difficult to invert analytically. For a given distribution Z(ξ) ranging from ξa to ξb , its CPDF can be expressed as: % ξ ξb ′ ′ Z(ξ ′ )dξ ′ . (3.5) Z(ξ )dξ R(ξ) = ξa

ξa

For the sake simplicity, we define the followings: ξ γ= Z(ξ ′ )dξ ′ , η=

ξa ξb

Z(ξ ′ )dξ ′ .

(3.6) (3.7)

ξa

In preparing the table, R(ξ) is evaluated as a function ξ and stored. Any numerical integration technique can be employed to solve γ and η. For example, to evaluate η using the Composite Simpson rule, [ξa , ξb ] is subdivided into 2M subintervals of equal width: h=

ξb − ξa . 2M

(3.8)

By using ξk = ξa + kh for k = 0, 1, . . . , 2M, η is approximated as: M

η≈

h (Z(ξ2k−2 ) + 4Z(ξ2k−1 ) + Z(ξ2k )). 3

(3.9)

k=1

After a table is populated with both R(ξ)’s and ξ’s, it can be used for any interpolation. For this, a given random number Ranξ is compared against the tabular R(ξ)’s using a bisectional method. A final linear interpolation may be required if Ranξ is in between two tabulated values of R(ξ)’s. After that, the corresponding ξ value is found and a MC simulation is carried out as discussed below.

3.3 Monte Carlo Simulation for Particle-Beam Transport Particle-beam transport concepts discussed in this monograph are for electron and photon beams incident on a workpiece. Both of these energy carriers have been widely used as non-intrusive2 diagnostic tools. For example, the 2

Non-intrusive tools mean that the powers of the beams used are not sufficient to cause structural damage to the specimen. Oppositely, the electron beam for micro/nanomachining is to serve as intrusive tools to melt and subsequently evaporate the target.

3.3 Monte Carlo Simulation for Particle-Beam Transport

63

most commonly applied tool to visualize micro- or nano-scale structures is the scanning electron microscope (SEM), which employs the reflected electrons from a specimen to construct images of the targets [51]. Although there are some minor differences in comparing electron-beam scatterings to photonbeam scatterings, the basic MC simulation procedures discussed below are applicable to both cases. In this chapter, the details of the scattering properties of electrons or photons are not discussed, and they are assumed prescribed. The derivations of these properties are given in Chap. 4. 3.3.1 Setting up Computational Grid In any numerical method, the physical domain is usually divided into grids to facilitate the discretization process. In the case of statistical simulations, grids are considered to collect histories of the propagating ensembles. In other words, they are used to tally the contributions of each statistical ensemble to various locations in space. These contributions, in a sense, can be regarded as raw data. When proper normalizations are utilized, they are transformed into useful statistical distributions corresponding to the expected physical quantities. There is a very distinct property of the grid used in the statistical simulation different than those for the regular numerical simulations. The size of the grid possesses no effects on the accuracy of the statistically simulated results, but it does have an effect on the resolution of the distributions. The ultimate factors that control the accuracy of the simulation are actually the number of ensembles used and the number of runs performed. When either a coarse grid or a fine grid is used in simulation, the resultant distribution is always fluctuating around the statistical mean distribution, provided that the number of statistical ensembles used in each case is sufficient. Using more grid points would produce resultant distributions at higher resolution. Unfortunately, it also means that more statistical ensembles and computational are required to reduce the statistical noises. Since the histories of statistical ensembles are tallied within infinitesimal control volumes for the sake of normalizations, it is better to have grid points centered in each control volume instead of using four grid points to form a control volume. 3.3.2 Random Number Generator Another important requirement for a statistical simulation is a random number generator. A statistical method is like a “rolling dice” method where one randomly draws a number using the dice to decide the outcome. The random number generator needs to have a uniform probability for the range of numbers required, which is typically between 0 and 1. There are a handful number of random number generators given in the literature. Details of these generators will not be discussed here; additional information can be

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found on the various public website, including that of Wikipedia located at http://en.wikipedia.org/wiki/Main Page. The PC version of the Compaq Visual FORTRAN is used in coding all the algorithms in this monograph. It has a built-in random number generator, which goes by the syntax, ran(nseed). The random number generator is based on the work by Park and Miller [149], and requires a large odd integer number, denoted as nseed, to be initialized. This number will change automatically after each time the function is called. However, in order to avoid an exact same series of random numbers being drawn, the initial nseed used in the algorithm should be different for different runs. One way to overcome this problem is to utilize the total number of seconds calculated using year, date and time as the nseed, since it would never be the same at any given moment. 3.3.3 Simulation Steps There are two ways of approaching the simulation procedures. One is using the so-called continuous slowing-down approach (CSDA) and the other is the discrete inelastic scattering (DIS). Just as their names imply, the former treats inelastic scattering events as a continuous phenomenon where energy change corresponds to the distance traveled while the latter assumes the discrete scattering events at which energy change depends on the type of scattering mechanism. The flowcharts of these two MC methods are given in Figs. 3.3 and 3.4. A MC simulation starts by setting up the grid system for storing the particle histories. The simulation proceeds with launching an ensemble of particles one after another, until the last number of ensembles is specified. Each ensemble is treated separately demonstrating that there is no interference factor between any two ensembles. The location and direction of launching the ensembles are defined according to the incident particle beam profile. The most common beam profiles used in the simulation are impulse (or point incident beam), flat (or uniform within a specified area), and Gaussian (i.e., exponential decaying). The location of launching the ensembles needs to be sampled from its own probability distribution, except in the case of the impulse beam profile. The distance of interaction is determined next from another random number. The distance of interaction is equivalent to the mean free path of an ensemble. This indicates the distance the ensemble can travel without being scattered. Once the distance of interaction is set, the ensemble is allowed to propagate from its initial position to the next location following its initial direction. The ensemble is then checked for confinement within the defined control volume. If the condition is met, the ensemble is allowed to scatter, otherwise the contribution of the ensemble to either the transmission or the reflection is tallied. A new ensemble of particles would be initiated according to the incident beam profile and launched from the appropriate location. If the ensemble is still confined within the medium, its weight (or energy) is updated (or attenuated) and its contribution to the absorption is tallied.

3.3 Monte Carlo Simulation for Particle-Beam Transport

65

Fig. 3.3. The flow diagram for the MC simulation using the CSDA

Following the attenuation, the ensemble is tested for its remaining weight. If the energy of the ensemble becomes less than the tolerance, it is disregarded and a new ensemble is launched. If this is not done, then the ensemble of particles is assumed to suffer an elastic scattering and change the direction of propagation. This procedure is continually repeated until a new distance of interaction is determined. Once the simulation is completed (a pre-determined number of ensembles used in simulation is reached), the tallied particle histories are converted to the desired quantities such as the radial or angular scattering, reflectance and transmittance.

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Fig. 3.4. The flow diagram for the MC simulation using the DIS

3.3.4 Incident Beam Profiles MC simulations require a careful treatment of normal incident beam profiles. Here, only the flat and Gaussian beam profiles are considered. Below, the particle launching location for a flat beam is discussed first. For a given source with strength of Λ incident onto a circular area with a radius of r0 , the beam profile can be written as: & ' Λ πr02 ∀ r ≤ r0 , (3.10) B(r) = 0 ∀ r > r0 .

3.3 Monte Carlo Simulation for Particle-Beam Transport

67

If a steady-state simulation is desired, then Λ can be considered as the power of the beam. Otherwise Λ would be amount of energy transferred as a function of time. The strength of the beam is not important in determining the launching of the particles. The CPDF of this beam profile is obtained following the procedure discussed in Sect. 3.1; once it is applied, the following equation is obtained: 2 1 r r B(r′ )2πr′ dr′ = . (3.11) R(r) = Λ 0 r0 Introducing a random number, Ranr , the radius of launching particles for a flat incident beam profile is obtained as: ( (3.12) r = r0 Ranr .

Translating this into the Cartesian coordinates yields [89, 202]: ( x = r0 Ranr cos(2πRanr1 ) ( y = r0 Ranr sin(2πRanr1 ),

(3.13) (3.14)

where Ranr1 is a second random number. For a Gaussian beam profile with a 1/e2 radius of r0 , one can write:   2 2 r . (3.15) B(r) = 2 exp −2 πr0 r0 Using the similar procedure as in the flat incident beam case, the radius for launching particles can be obtained as: ( r = r0 − ln(1 − Ranr )/2, (3.16) which is again translated to Cartesian coordinates [89, 202]: ( x = r0 − ln(1 − Ranr )/2 cos(2π Ranr1 ), ( y = r0 − ln(1 − Ranr )/2 sin(2π Ranr1 ).

(3.17) (3.18)

3.3.5 Direction of Propagation In a MC simulation, both fixed and moving frames of coordinates are employed. The fixed frame of coordinate is simply the coordinate system that is used to define the grid or the geometry. As the name implies, it is stationary and serves as the reference for the moving coordinate frame. The moving frame of the coordinate is based on the direction of the propagation of the ensemble. It is defined in such a way that the direction of propagation coincides with

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the z ′ -axis3 of the moving frame where the scattering polar angle is measured from this axis. The direction cosines of the scattered ensemble must be calculated from the incident directions. The calculations are easy for the isotropically scattering media since the probability of the isotropically-scattered photons is equal in all directions. Immediately after each scattering event, the direction cosines of the scattered particles are given as: µ′x = cos ϕ sin Θ,

(3.19)

µ′y = sin ϕ sin Θ,

(3.20)

µ′z = cos Θ,

(3.21)

where µ′x , µ′y , and µ′z are the scattered direction cosines in x-, y-, and z-directions, respectively. For the anisotropic case, the scattered direction cosines can be expressed in terms of the incident direction cosines (µx , µy , µz ) and the scattering angles (Θ, ϕ): sin Θ (µx µz cos ϕ − µy sin ϕ) + µx cos Θ, µ′x = ( 1 − µ2z sin Θ (µy µz cos ϕ + µx sin ϕ) + µy cos Θ, µ′y = ( 1 − µ2z ( µ′z = − sin Θ cos ϕ 1 − µ2z + µz cos Θ.

(3.22) (3.23) (3.24)

3.3.6 Distance of Interaction The distance of interaction, which is the path length that a bundle can travel without suffering any attenuation by the medium, can be derived from the intensity form of the BTE. The typical intensity form follows the expressions given in Sect. 2.4; however, the transient term is usually neglected. In addition, the in-scattering terms are not required; because only the tracing of ensembles in various directions in space is performed, the in-scattering nature of the transport is automatically built-in in the simulation. When these terms in the equation are dropped, the intensity form of the BTE is obtained as: I dI =− , ds λ

(3.25)

where λ is the mean free path and s is the propagation axis. The solution to this equation can be expressed as: I = I0 exp(−S/λ),

(3.26)

where I = I0 at s = 0 is taken as the initial condition. According to (3.26), the intensity decreases exponentially along the path of propagation. Note that the 3

An orthogonal coordinate system (i.e., x′ y ′ z ′ ) is used to define the moving frame.

3.3 Monte Carlo Simulation for Particle-Beam Transport

69

exponential term is actually the CPDF for the distance of interaction, ranging from 0 for S = ∞ to 1 for S = 0, and can be given as: R(S) = exp(−S/λ).

(3.27)

Continuous Slowing-Down Approach In the CSDA, it is assumed that particles continuously lose energy as they propagate. This means that the inelastic scatterings are implicitly accounted for in simulations, and therefore there is no need to determine the distance of interaction for inelastic scattering separately. Only the elastic scatterings should be considered, and the CPDF is given as: R(Sel ) = exp(−Sel /λel ),

(3.28)

where λel is the elastic mean free path. Therefore, the distance of interaction is obtained when R(Sel ) is replaced with a random number RanS , which yields: Sel = −λel ln(RanS ).

(3.29)

Discrete Inelastic Scatterings The DIS treats inelastic scattering events as discrete; therefore, the energy loss of a particle ensemble occurs at a determined interacting location. This is similar to that of an elastic scattering event. When this method is employed, the CPDF of both elastic and inelastic scatterings can be incorporated into a single equation, given as: R(Seff ) = exp(−Seff /λeff ),

(3.30)

where λeff is the effective mean free path computed from both elastic and inelastic mean free path (i.e., λel and λinel , respectively): −1 −1 λ−1 eff = λel + λinel .

(3.31)

Therefore, the distance of interaction becomes: Seff = −λeff ln(RanS ).

(3.32)

This is the distance a particle can travel before the elastic or inelastic scattering event occurs. 3.3.7 Attenuation of Energy Continuous Slowing-Down Approach The CSDA assumes that energy is absorbed continuously along the path of propagation. The amount of absorption depends on the distance traveled by

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the particles: larger the distance more the absorption. This approach ignores the specific details of the inelastic scattering mechanisms, and greatly simplifies the complexity of the problem. However, one needs to distribute the attenuated energy evenly along the propagation. This means that the coordinates of various locations along the path traveled must be calculated in order to enable the operation if the absorption profile is desired. One important quantity that is required to incorporate this effect into a MC simulation is the amount of energy absorbed per unit distance. For the electron-beam transport, it is quantified as dE/dS, which is the change of energy per unit change in distance (see Sect. 4.3). Except for photons, it is given in terms of the fraction of energy absorbed, and the fraction is expressed as: 1 − exp(−κSel ),

(3.33)

where κ is the absorption coefficient. Discrete Inelastic Scatterings In this approach, elastic and inelastic scattering events are discrete and the distance of interaction is given by (3.32). Upon interaction, a random number will decide an elastic or inelastic scattering event based upon the following equation. The electron will be scattered elastically if: Ranλ
85 keV. Unlike the Rutherford model, the Mott cross section is applicable for both low and high energy scattering. Both models assume that the effect of the

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4 Modeling of e-Beam Transport

vibrational motion of the atom on the kinetic energy of the electron is negligible. Therefore, the Mott model fails when the electron energy falls within the range of the vibrational energy of the atom, which is typically less than 10 eV. 4.2.1 Rutherford Cross Section The differential elastic scattering cross section of the electron by a central potential without considering the spin of the electron and the polarization effect is given as [43, 51]: dCeel (Θ, E) = |f (Θ)|2 , dΩ

(4.2)

where Ceel is the scattering cross section, Ω the solid angle, and f the scattering amplitude. The f is a function of scattering angle Θ and can be expressed as [43]: 2m ∞ f (Θ) = − 2 sin(qr)V (r)rdr. (4.3) h q 0 ¯ The variable q is a function of Θ and refers to the difference between the wave vectors of the electron before and after scattering while V (r) is the potential of the atom as a function of radius r. A Wentzel-like potential for V (r) is given as [43]:

Ze2 r V (r) = − exp − . (4.4) r a Then the differential cross section is expressed as: Z 2 e4 dCeel (Θ, E) 1 = , dΩ 4E 2 (1 − cos Θ + α1 )2

(4.5)

with the screening parameter defined as: α1 =

me4 Z 2/3 . 4¯h2 E

(4.6)

This is the Rutherford cross section where it is explicitly expressed in terms of the atomic number, electron energy, and scattering angle. The differential scattering cross section can be converted into the so-called scattering phase function, which defines the probability of scattering in a given direction per unit solid angle. The phase function Φe for the elastic electron scattering is the normalized scattering cross section, and can be written as: Φe (Θ, E) =

dCeel (Θ, E) , el dΩ Ce,total (E) 4π

(4.7)

4.2 Elastic Scattering of an Electron by an Atom

where the total scattering cross section is dCeel (Θ, E) el dΩ. Ce,total (E) = dΩ

85

(4.8)

Ω=4π

Normalization for the phase function yields: 1 Φe dΩ = 1. 4π

(4.9)

Ω=4π

For the differential elastic cross section given by (4.5), the total elastic cross section is determined to be: el Ce,total (E) =

πZ 2 e4 1 . 2 E α1 (2 + α1 )

(4.10)

Thus, the corresponding phase function becomes: Φe (Θ, E) =

α1 (2 + α1 ) . (1 − cos Θ + α1 )2

(4.11)

Another form of the Rutherford differential cross section can be expressed as [93]: Z 2 e4 dCeel (Θ, E) = dΩ 4E 2



E + 511 E + 1,024

2

−2

Θ + α2 sin2 , 2

(4.12)

where the screening parameter is now given as: α2 =

me4 Z 2/3 . 8¯h2 E

(4.13)

The total elastic cross section in this case is: el Ce,total (E)

Z 2 e4 4π = 4E 2 α2 (1 + α2 )



E + 511 E + 1,024

2

.

(4.14)

The phase function is then expressed as: α2 (1 + α2 ) Φe (Θ, E) =    2 . sin2 Θ2 + α2

(4.15)

It is easily shown that this phase function is exactly the same as that given by (4.11), although the corresponding differential scattering cross sections may have minor differences, which are not important for the present applications to micro/nanomachining. Figure 4.2 depicts the phase functions of electrons scattered by gold atoms as a function of scattering angle at various incident electron energies based on

86

4 Modeling of e-Beam Transport

103 102

Φe (sr−1)

101 0.02 keV

100

0.2 keV

10−1

1 keV 5 keV

10−2

25 keV

10−3

0

20

40

60

80 100 120 140 160 180

θ (degrees) Fig. 4.2. Rutherford scattering cross-section of electron by a gold atom at various scattering angles and different electron energies. Note that an electron tends to be scattered by the gold atom almost equally in the forward and backward directions at low energy. If the electron energy is high, scattering becomes highly forward, meaning that the angular deflection is relatively small

the Rutherford formulation. Here, the reference axis for the scattering angles is the direction of propagation; therefore the zero-degree angle refers to the direction of propagation. The elastic scattering profile becomes highly forward when the electron energy is increased. This implies that the angular deflection of the electron by an atom decreases with increasing electron energies. However, when the electron energy is low, elastic scattering is more evenly distributed between the forward1 and backward directions. Figure 4.3 shows the elastic scattering phase function for copper, silver, and gold, with increasing atomic numbers in the given order. For a material with a higher atomic number, backward scattering can be more pronounced compared to another material with a lower atomic number. The elastic mean free path, which describes the separation distance between two successive elastic scattering events, is also determined from the differential elastic scattering. Since the total elastic scattering cross section refers to the effective scattering area for electrons by an atom, the mean free path can be obtained by taking the inverse of the product of the elastic 1

Forward direction usually refers to angles between 0◦ and 90◦ with respect to the propagating direction while backward direction covers angles from 90◦ to 180◦ assuming that azimuthal symmetry holds.

4.2 Elastic Scattering of an Electron by an Atom

87

103 Materials Z=29; Copper Z=47; Silver Z=79; Gold

2

10

Φe (sr−1)

101 0.02 keV

100

1 keV

10−1 10−2 10−3

25 keV

0

20

40

60

Z

Z

Z

80 100 120 140 160 180

θ (degrees) Fig. 4.3. Rutherford scattering cross-section of electron by different types of atoms (copper, silver, and gold) at various scattering angles and different electron energies. Increasing the atomic number Z tends to enhance back-scattering and weakens forward scattering

scattering cross section and the number of atoms per unit volume. It can be written as: A λel , (4.16) e (E) = el Na ρCe,total (E) where A is the atomic weight, Na the Avogadro number, ρ the density of the material, and (Na ρ/A) the number of atoms per unit volume. The elastic scattering coefficient is the inverse of the elastic mean free path as similar to (2.23). The elastic mean free path, which is a function of the electron energy for different materials, is derived from the two different forms of the Rutherford scattering cross section is plotted in Fig. 4.4. Both models yield the same mean free path profiles for different materials at low electron energies, yet they deviate from each other when the electron energy is high (i.e., E > 50 keV). 4.2.2 Mott Cross Section The Rutherford scattering phase function offers a simple and convenient analytical expression for describing the elastic scattering phenomenon of electrons by an atom. However, due to its basis on high-energy approximation, the phase functions predicted using this method often produce errors when it comes to a relatively low energy electron beam. In order to correctly represent the scattering phase functions for both the low- and high-energy electron beams, the Mott scattering cross section should be employed. The reason that the

88

4 Modeling of e-Beam Transport

10

Mean Free Path (Angstrom)

Z=29; R2

Materials Z=29; copper Z=47; silver Z=79; gold

3

Z=29; R1 Z=47; R1 Z=47; R2

102

Z=79; R1 Z=79; R2

101 100 10−1 10−2

10−1

100

101

102

103

104

Electron energy (keV) Fig. 4.4. The elastic mean free path of electrons derived from the two different forms of the Rutherford scattering cross-section discussed in the text. R1 and R2 refer to the mean free paths derived using (4.10) and (4.14), respectively. Both models yield the same mean free path at low electron energies but deviate from each other at high energies. The elastic mean free paths for different materials are also shown

Rutherford model fails to predict the correct elastic scattering phase function is because in its formulation the spin of the electron is neglected. The spin– orbit coupling between the incident electron and the atom becomes important when the electron energy is low. This is incorporated in the Mott cross section through the Dirac equation. The Mott elastic differential scattering cross section for an electron beam is typically given in the form of [24, 41, 43, 98, 139] dCeel (Θ, ϕ, E) −AB ∗ eiϕ + A∗ Be−iϕ = |f |2 + |g|2 + (f g ∗ − f ∗ g), dΩ |A|2 + |B|2

(4.17)

where the scattering factors f and g are generally functions of a scattering polar angle Θ and wave number of the incident electron beam k, and A and B describe the state of polarization of the beam (e.g., A = 1 and B = 1 yield transverse polarization, A = 1 and B = 0 or A = 0 and B = 1 refer to longitudinal polarization). The third term in (4.17) represents the dependence of the cross section on the azimuthal angle (ϕ), and it vanishes for an unpolarized electron beam. Therefore, scattering of a (partially-) polarized beam generally depends on both Θ and ϕ, as well as the incident energy of the beam.

4.2 Elastic Scattering of an Electron by an Atom

89

The scattering factors are expressed in the following forms [41, 98, 139]: f (Θ, k) =

∞ 1  {(l + 1)[e(2iδ−l−1 ) − 1] + l[e(2iδl ) − 1]}Pl (cos Θ), 2ik

(4.18)

l=0

g(Θ, k) =

∞ 1  [−e(2iδ−l−1 ) + e(2iδl ) ]Pl∗ (cos Θ). 2ik

(4.19)

l=1

Here, δl ’s are the Dirac phase shifts, Pl ’s and Pl∗ ’s are the ordinary Legendre polynomials and the associated Legendre polynomials, respectively. In the above expressions, k represents the wave number of the electron with energy E. They are related according to: k2 =

(E 2 − m2 c4 ) . h 2 c2 ¯

(4.20)

The phase shifts are to be determined from the Dirac equation, which describes the relativistic behavior of an electron including its spin, the magnetic moment of the electron, and the spin–orbit coupling. In order to satisfy the Dirac equation in the relativistic case, the wave function of the electron should have four components leading to a resultant of four simultaneous firstorder partial differential equations (PDEs) as shown by Kessler [98]. These four PDEs can be transformed into two coupled first-order ordinary differential equation (ODEs) [139], which are given as: ! dG± (1 + j) ± 1 l (r) (E − V (r) + mc2 ) Fl± (r) + + Gl (r) = 0, (4.21) hc ¯ dr r ! dFl± (r) (1 − j) ± 1 2 (E − V (r) − mc ) G± + Fl (r) = 0, (4.22) − l (r) + hc ¯ dr r where V (r) is the atomic potential. The + solutions apply to the electron ‘spin up’ case while the – solutions apply to the electron ‘spin down.’ For the + and – cases, j takes the value −(l + 1) and l, respectively. The asymptotic solution of G at large r requires that: ± ± G± l (r) = Jl (kr)cos δl − Yl (kr)sin δl ,

(4.23)

where δl± are the phase shift needed in computing the Mott cross section. The Dirac equations can be further reduced to the following first-order ODE by introducing the change of variables as proposed by Lin et al. [115]: sin φ± l (r) , r cos φ± ± l (r) . G± l (r) = Al (r) r

Fl± (r) = A± l (r)

(4.24) (4.25)

90

4 Modeling of e-Beam Transport

Using the proposed transformation, the Dirac equations now read: j mc2 1 dA± ± l = − cos 2φ sin 2φ± − l l , dr r h ¯ c A± l

(4.26)

dφ± j mc2 1 l = sin 2φ± {E − V (r)} − cos 2φ± l + l . dr r hc ¯ ¯hc

(4.27)

Equation (4.27) can be non-dimensionlized by letting:
m c e r, r˜ = h ¯ hk ¯ K= , me c ˜= E , E m e c2 V (˜ r) , V˜ (˜ r) = m e c2

(4.28) (4.29) (4.30) (4.31)

which results: dφ± j ± l ˜ ˜ = sin 2φ± l + (E − V ) − cos 2φl . d˜ r r˜

(4.32)

In principle, there is no need to solve for A± l as the phase shifts can be ± determined by matching the ratio of the derivative of G± l to Gl between the numerical solution of (4.27) and the required asymptotic solution as given in (4.23) at large r˜. This ratio can be easily obtained for numerical solutions by coupling (4.21), (4.24) and (4.25): 1 dG± (j + 1) l ˜ − V˜ (˜ = −{E r ) + 1} tan φ± . l − d˜ r r˜ G± l

(4.33)

Using the asymptotic solution given in (4.23), we find: KJl′ (K r˜) cos δl± − KYl′ (K r˜) sin δl± 1 dG± l = , ± r Gl d˜ Jl (K r˜) cos δl± − Yl (K r˜) sin δl±

(4.34)

˜ 2 − 1. The derivative of the Bessel in the non-dimensional form where K 2 = E function is given as (Arfken [8]): Jl′ (K r˜) =

l Jl (K r˜) − Jl+1 (K r˜). K r˜

(4.35)

The same relation applies to Yl′ (K r˜) as well. The phase shifts are determined by matching (4.33) and (4.34) at large r˜ where V˜ (˜ r ) ≈ 0 to obtain [41]:

 ˜ + 1) tan φ± + (1 + l + j)/˜ KJl+1 (K r˜) − Jl (K r˜) (E r l

 . (4.36) tan δl± = ˜ KYl+1 (K r˜) − Yl (K r˜) (E + 1) tan φ± + (1 + l + j)/˜ r l

4.2 Elastic Scattering of an Electron by an Atom

91

In summary, the Mott elastic scattering cross section has the form given in (4.17) in which the Dirac phase shifts are to be calculated. These phase ˜ where the asymptotic shifts are determined by solving (4.32) for φ± l at large r solution is reached using (4.36). However, solving (4.32) requires an initial value for φ± l , these can be obtained by simply employing the power series ˜ r ) in r˜ near the origin and inserting them into the expansions for φ± l and V (˜ equation in which all the constant coefficients of the series are unveiled. The procedures for calculating the scattering cross section are outlined 2 below. First, φ± l and V (r) are expanded in terms of r: φ± l =

∞ 

k a± kr ,

k=0 ∞ 

V =−

bm rm−1 ,

(4.37) (4.38)

m=0

where ak ’s and bm ’s are constants. Substituting these approximations into (4.32) and gathering the coefficients of each corresponding power in r, ak ’s and bm ’s are determined as: b0 , j E + b1 − cos 2a± 0 a± , 1 = 1 − 2jcos 2a± 0 sin 2a± 0 =−

± ± 2a± 1 sin 2a0 (1 − ja1 ) + b2 , 2 − 2jcos 2a± 0   ± ± ± ± 2 ± 2 2a± 2 sin 2a0 (1 − 2 ja1 ) + 2(a1 ) cos 2a0 1 − 3 ja1 + b3 ± a3 = , 3 − 2jcos 2a± 0

a± 2 =

(4.39) (4.40) (4.41) (4.42)

with the condition that: π if j < 0, 2 3π π ≤ 2a± if j > 0. 0 ≤ 2

0 ≤ 2a± 0 ≤

(4.43) (4.44)

Using the fourth-order Runge–Kutta method [128], the solution of (4.32) can be approximated as: φi+1 = φi +

2

h(f1 + 2f2 + 2f3 + f4 ) , 6

(4.45)

For the sake simplicity, the symbol “∼” is omitted for the remaining discussion.

92

4 Modeling of e-Beam Transport

where f1 = f (ri , φi ),

(4.46)

f2 = f (ri + h/2, φi + h/2 f1 ),

(4.47)

f3 = f (ri + h/2, φi + h/2 f2 ), f4 = f (ri + h, φi + hf3 ), j f (r, φ) = sin 2φ + [E − V (r)] − cos 2φ. r

(4.48) (4.49) (4.50)

The electron phase functions in gold computed at different electron energies using the Mott cross section are depicted in Fig. 4.5. Notice that there are “humps” over the angular domain for scattering profiles at various electron energies. They are caused by the interferences between the scattered electron waves. The “humps” disappear when the incident electron energy becomes large, since scattering is weak at angles other than the absolute forward direction (i.e., the zero-degree angle). The electron phase functions for copper, silver, and gold atoms are given in Fig. 4.6. In the high-energy regimen, there is a clear trend that backscattering increases when the atomic number increases, or if the atom becomes heavy. In other words, the magnitude of the phase function is higher for scattering angles larger than 90◦ for heavy elements. However, the trend disappears at low electron energies since the phase functions of different elements overlap more frequently.

Material: Gold Mott Phase Function

103 102

Φe (sr−1)

101

0.02 keV 1 keV

0

10

10−1 0.2 keV

10−2

5 keV 25 keV

10−3

0

20

40

60

80 100 120 140 160 180

θ (degrees) Fig. 4.5. Mott scattering cross-section of electrons by a gold atom at various scattering angles and different electron energies

4.2 Elastic Scattering of an Electron by an Atom Materials Z=29; Copper Z=47; Silver Z=79; Gold

103 102

0.2 keV; Z=29 0.2 keV; Z=47 0.2 keV; Z=79

101

Φe (sr−1)

93

100 10−1 10−2 25 keV; Z=29 −3

10

25 keV; Z=47 25 keV; Z=79

0

20

40

60

80 100 120 140 160 180

θ (degrees) Fig. 4.6. The Mott elastic scattering cross section for copper, silver, and gold at various scattering angles and at two different electron energies

103

Material: Gold

102

0.02 keV (Mott)

1

Φe (sr−1)

10

0.02 keV (Rutherford)

100 10−1

25 keV (Mott)

10−2 10−3

25 keV (Rutherford)

-4

10

0

20

40

60

80 100 120 140 160 180

θ (degrees) Fig. 4.7. Scattering cross-sections of electron computed by Rutherford and Mott models at various scattering angles and different electron energies. The Rutherford model deviates significantly from the Mott model at low electron energy levels

Figure 4.7 shows the comparisons of the phase functions computed using the Rutherford scattering model and the Mott cross section. In general, the Rutherford model is considered as a reasonable approximation to the Mott model; however, it is not detailed enough to yield physically accurate

94

4 Modeling of e-Beam Transport

100

Mean Free Path (Angstrom)

Z=47; silver Z=29; copper

10 Z=79; gold

1 10-2

10-1

100

101

Electron energy (keV) Fig. 4.8. Elastic mean free path of electron at various electron energies for the Mott scattering model

results [93]. The predicted electron-energy dependent elastic mean free paths using the Mott total elastic cross section for different metals are illustrated in Fig. 4.8. Notice that the elastic mean free path for gold, a heavier element compared to copper and silver, increases for electron energies smaller than 0.1 keV. This is different from those shown by copper and silver at low electron energies, and implies the importance of the spin–orbit coupling phenomena between the incident electron and the atom for a heavy element such as gold. It also shows that the trend of the elastic mean free path predicted using the Rutherford model is inaccurate for heavy elements, as evident in Fig. 4.9.

4.3 Continuous Inelastic Scattering Approach: The Bethe Theory Inelastic scatterings involve interactions between propagating electrons and electrons inside the matter. Describing the inelastic scattering phenomena accurately for the electron-beam propagation is difficult since electrons can lose their kinetic energies in a number of ways within the matter. Depending on the strength of the electron energy, the physical processes involved must be different. There are a number of theories and formulations available to describe these interactions. In this section, a classical approach or the socalled Bethe theory is discussed to account for inelastic scattering based on continuous slow-down approach (CSDA).

4.3 Continuous Inelastic Scattering Approach: The Bethe Theory

95

Material: Gold

Mean Free Path (Angstrom)

Mott's model

10

1 Rutherfords model

0.1 10−3

10−2

10−1

100

101

Electron energy (keV) Fig. 4.9. Elastic mean free path of electron at various electron energies for the Rutherford scattering model and the Mott scattering model. The Rutherford model always underestimates the elastic mean free path of electrons

The Bethe theory treats the interaction between a propagating electron and electrons at rest based on the classical theories where quantum mechanical effects are not strongly present [51]. Accordingly, a propagating electron continuously loses its kinetic energy along the path of propagation. This loss is quantified by a quantity called the stopping power (usually denoted as dE/dS). This refers to the change of electron kinetic energy per unit length of propagation. To derive the stopping power of an electron, one usually starts by examining the repulsion force between a traveling electron and an electron at rest, and then determines the momentum transfer that is induced by this force when the electron passes near the other. After determining the amount of momentum transferred from the propagating electron to another at rest, the electron stopping power can be obtained by integrating it over a suitable distance at which the momentum transfer is possible between the traveling and orbiting electrons. It is important to also take into consideration the density of atoms and the atomic number. Then, the stopping power, which is known as the Bethe equation, is given as:

dE 2πe4 Na ρZ 1.166E =− ln , (4.51) dS AE J where Z is the atomic number and J is the mean ionization energy. This equation is valid for E > J. In an attempt to extend the applicability of the Bethe equation below the mean ionization energy, Joy and Luo [94] used a

96

4 Modeling of e-Beam Transport

statistical method and modified the original Bethe equation to:   2πe4 Na ρZ 1.166(E + 0.85J) dE =− ln , dS AE J

(4.52)

in order to match the data obtained by Tung et al. [196]. This so-called modified Bethe equation is valid for E > 50 eV. Further discussion of the electron stopping power can be found in the papers by Bethe [17], Luo et al. [116] and the monograph by Dapor [43]. The Bethe equation is a relatively crude approximation for modeling the electron energy losses since propagating electrons are assumed to lose energy continuously along the traveling path and the ionization of electrons is treated as an averaged effect. Consequently, it is less accurate when compared to the case where inelastic scatterings are considered discretely in terms of the inner-shell ionizations, the outer-shell excitations, and the plasma oscillations.

4.4 Discrete Inelastic Scattering Treatment: The Dielectric Theory Another method of treating the inelastic scatterings is the dielectric theory/formulation. In this approach, the energy loss function is directly derived from the experimental optical data to generate the differential inelastic scattering cross sections. The energy loss function is a measure of responses to electrons and atoms in a medium as a whole when exposed to an external disturbance; therefore, this formulation is typically more accurate compared to other independent formulations. This is especially true when the electron energy is low. In this way, the inner-shell ionizations and the outer-shell excitations cannot be distinguished clearly. It is a better approach for low energy electrons, considering that the inelastic electron scatterings are not currently well-understood at the low electron-energy regimen. The double differential inelastic scattering cross section (or the probability for an inelastic scattering event to occur per unit length, energy change, and momentum change) then becomes [156]:   d2 λ−1 1 1 1 inel = Im − , d(¯ hω)dq πa0 E ε(q, ω) q

(4.53)

where a0 is the Bohr radius and λinel is the inelastic mean free path. The imaginary part (denoted as Im[·]) of the negative inverse of the dielectric function ε(q, ω) in the equation describes the probability of energy loss. It is called the energy loss function. The incident energy of the electron is denoted as E while the amount of energy loss is given as h ¯ ω which shall be refered to as ∆E. The energy loss function can be derived from extrapolating the optical dielectric constant as measured by experiments.

4.4 Discrete Inelastic Scattering Treatment: The Dielectric Theory

97

Equation (4.53) can be modified using the variable change to account for the Ω-dependency instead of the q-dependency. This is achieved by using the energy and the momentum conservation, which are given as (E − ∆E = E ′ ) ⇀

⇀

⇀

and (k ′ + q = k), respectively. Using the cosine law and the parabolic freeelectron band structure (i.e., E = (¯ hk)2 /2m0 ), the energy and the momentum conservations give: ( (¯hq)2 = 2E − ∆E − 2 E(E − ∆E) cos Θ. 2m

(4.54)

By taking the derivative of q over Ω in (4.54), dq/dΩ can be obtained. Then multiplying (4.53) by dq/dΩ, we obtain:   d2 λ−1 1 1 1( inel = Im − E(E − ∆E), (4.55) 2 d(∆E)dΩ (πa0 e) E ε(q, ω) q 2 ¯ 2 /me2 ) is used. where the definition of the Bohr radius (i.e., a0 = h If the optical dielectric constant of a given matter is given as ε(ω0 ), then the q-dependent energy loss function becomes:     ω0 1 1 = , (4.56) Im − Im − ε(q, ω) ω ε(ω0 )

where ω0 is the positive solution of the plasmon dispersion equation ωq (q, ω0 ) = ω, which is given as (Penn [153]): ωq2 (q, ωp )

=

ωp2

1 + vF2 ωp q 2 + 3



¯ q2 h 2m0

2

,

(4.57)

or (Ashley [10]; Kwei and Tung [108]; Ritchie and Howie [166]): ωq = ωp +

¯ q2 h . 2m0

(4.58)

where ωp stands for the plasma frequency, vF is the Fermi velocity, and m is the electron mass. Using the quadratic rule, ω0 is obtained as: , 2 2 1 1 4 4 vF2 q 2 ¯hq + v q −4 + 4ω 2 , (4.59) ω0 = − 6 2 9 F 2m0 using (4.57), and: ¯ q2 h , (4.60) 2m0 if the latter plasmon dispersion relation is employed. An example of the energy loss function of gold is shown in Fig. 4.11, which is computed using the complex index of refraction from Fig. 4.10. The corresponding q-dependent energy loss function is depicted in Fig. 4.12. ω0 = ω −

98

4 Modeling of e-Beam Transport

103 102

Material: Gold

1

10

100 n

n, k

10−1 10−2 10−3

k

10−4 10−5 10−6 10−7

10−4

10−3

10−2

10−1

100

101

Wavelength (µm) Fig. 4.10. The complex index of refraction of gold is shown. n is the real part while k is the imaginary part of the index. Data are obtained from Palik [148]

1.0

Material: Gold

Im−[1/ε(∆Ε)]

0.8 0.6 0.4 0.2 0.0 10−1

100

101

102

103

∆E (eV) Fig. 4.11. The energy loss function of gold derived from the complex index of refraction given in Fig. 4.10

4.4 Discrete Inelastic Scattering Treatment: The Dielectric Theory

99

1.00

Im[−1/ε(q,ω)]

0.75 0.50 0.25 0.00

6

4 -1

q (A )

2

0 250

200

150

100

50

0

∆ E (eV)

Fig. 4.12. The probability of energy loss for hot electrons as a function of incident kinetic energy E and momentum transfer in terms of wave number q in gold

The probability of an electron suffering an inelastic scattering event per unit path length and per unit energy change is obtained using (4.53) after integration over all possible wave numbers q of the excited plasmons, and using the energy and momentum transfer conservation. It is given as [48]:   ∞& dλ−1 1 hω p ¯ −1 inel = Im 2 '  d(∆E) πao E 0 ε(ωp ) (∆E)2 − (¯hωp )2 + (¯ hq¯)2 2m0 (4.61)  2 ! h ¯ ×Θ (2k q¯ − q¯2 ) − ∆E d(¯ hωp ), 2m0 when (4.57) is assumed. Using the relation expressed in (4.58), this probability can be simplified to give the following form [48]:   ∞ dλ−1 1 hω p ¯ −1 inel = Im d(∆E) 2πao E∆E 0 ∆E − ¯hωp ε(ωp ) (4.62)  2 ! h ¯ 2 ×Θ (2k q¯ − q¯ ) − ∆E d(¯ hωp ), 2m0 Here, q¯ is the positive solution of the dispersion relation ω = ωq (¯ q , ωp ), E = (¯hk)2 /2m and ∆E = h ¯ ω. The probability for gold is plotted using (4.62) and given in Fig. 4.13. The inelastic mean free path of the electron can then be obtained by integrating the cross section over all energy changes. For this, the Fermi energy

100

4 Modeling of e-Beam Transport

dλ-1/d(∆E) (keV-Α)-1

10 8 6 4 2 0

20

8 40

log E

6 4

60

80

2 100 0

q (A-1)

Fig. 4.13. The probability of energy loss for hot electrons as a function of incident kinetic energy E and momentum transfer in terms of wave number q in gold

EF is taken as the reference level (i.e., the primary electrons lose energies until they reach energy level EF ). The inverse of the mean free path is given as: λ−1 inel

=



E−EF

0

dλ−1 inel d(∆E). d(∆E)

(4.63)

Note that λinel depends on the kinetic energy of the electrons. Therefore, the electron stopping power can be estimated as: dE = − dS



0

E−EF

(∆E)

dλ−1 inel d(∆E). d(∆E)

(4.64)

The inelastic mean free path of hot electrons for gold is depicted in Fig. 4.14, showing that it dips to less than 1 nm at energy levels around 100 eV. Two different electron masses are used in computing the inelastic mean free path: one is the effective electron mass in gold (me = 5.1 × 10−31 kg) and the other is the static electron mass (me = 9.1 × 10−31 kg). However, the result shows that the difference between the two curves is very little.

4.5 Electron Reflection and Refraction at a Surface When an electron encounters a surface between a vacuum and the material, it can be reflected or refracted depending on the energy of the electron and the impinging angle with respect to the surface normal. A quantum mechanical

4.6 Monte Carlo Simulation Results and Verifications

101

103

λin (nm)

102

101

me = 5.1x10-31 kg

100

me = 9.1x10-31 kg

10−1 100

101

102

103

104

E−EF (eV) Fig. 4.14. The inelastic mean free path of hot electrons for gold atoms

expression for describing electron transmission from the material to vacuum is given as [39]: & 4(1−U0 /E cos2 β)1/2 if E cos2 β > U0 , T (E, β) = [1+(1−U0 /E cos2 β)1/2 ]2 (4.65) 0 else. where U0 is the inner potential (which is the sum of the Fermi energy and the work function of the material), and β is the impinging angle with respect to the surface normal. If the electron is transmitted, then β will be altered to β ′ following the expression below while its energy is reduced by U0 amount: ( √ E − U0 sin β ′ = E sin β. (4.66)

Equation (4.65) serves as the criterion for determining transmission or reflection of an electron ensemble in a MC simulation upon encountering the material-air interface. Therefore, re-emission of electrons from the target material may be reduced with the implementation of this restriction in the simulation.

4.6 Monte Carlo Simulation Results and Verifications As discussed in Chap. 3, in this monograph two different MC codes are considered to account for the electron-beam propagation inside a workpiece. The first one is the CSDA and the other is the discrete inelastic scattering (DIS)

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4 Modeling of e-Beam Transport

0.6 0.5

BS yield (-)

0.4 Material: Gold Hunger & Kuchler, 1979 Reimer and Tolkamp, 1980 Bongeler et al., 1993 Bishop, 1963 Heinrich, 1966 Neubert & Rogaschewski, 1980 Drescher et al., 1970 Cosslett & Thomas, 1965 Wittry, 1966 Bronstein & Faiman, 1969 El Gomati & Assad, 1997 SS MC (Joy) PS MC (Joy) SS MC w/ Mott & Exp. SP (this work) SS MC w/ Mott & Bethe SP (this work)

0.3 0.2 0.1 0.0 0.1

1

10

100

Beam Energy (keV) Fig. 4.15. Backscattering yields for various incident electron energies in gold. The medium is assumed to have infinite thickness

approach. The CSDA treats the elastic scatterings using the Rutherford crosssection or the Mott cross-section (see Sect. 4.2). The inelastic scattering is accounted for by using the Bethe Theory as given in Sect. 4.3. As for the DIS method, the Mott scattering cross-section and the dielectric theory are used to account for the elastic and inelastic scatterings, respectively. Further details of the dielectric theory are outlined in Sect. 4.4. Since the MC codes were built specifically for this research, the verifications of the codes were needed in order to ensure the validity of the simulations. For this purpose, backscattering yields defined as the total number of electrons reflected back normalized by the total number of incident electrons can be calculated and compared against the experimental data. In Fig. 4.15, such comparisons are shown. All the data points from different references were compiled by Joy [93].3 Even though there are significant variations of the experimental results, the computational results match most of the data, suggesting that the overall procedure presented here is accurate.

4.7 Comments In this chapter, we have described the scattering properties of electrons to be used in MC methods to simulate the electron-energy deposition profile within a target workpiece. These properties are to be used in conjunction 3

Data are available online at http://web.utk.edu/∼srcutk/htm/interact.htm.

4.7 Comments

103

with the procedures outlined in Chap. 3. Although it is not of interest in this monograph, the surface reflectivity or transmissivity can also be calculated using these simulations. In the next two chapters, we will generate the electron-energy deposition profiles using these formulations and then couple them with the transient heat conduction formulation to predict temperature distribution inside a workpiece. These calculations will be interpreted for different nanoscale machining scenarios.

5 Thermal Conduction Coupled with e-Beam Transport

Modeling thermal transport during an electron-beam based micro/nanomachining process requires understanding of electron-beam propagation and scattering, thermal conduction due to electrons and phonons, and phase change and material removal rates within a workpiece. In Chap. 3, we discussed the solution of the Boltzmann transport equation (BTE) to model electron-beam propagation inside a metallic workpiece using two different Monte Carlo (MC) methods. The details of the electron scattering properties and calculation procedures were outlined in Chap. 4. Following these guidelines, we now have the ability to build a MC model for simulation of the electron-beam propagation inside a scattering and absorbing medium. In this chapter, we will couple a MC method with the traditional Fourier heat conduction equation to predict the evolution of temperature profile inside a workpiece. The results obtained from this coupled analysis will be discussed for potential micro/nanomachining applications.

5.1 Modeling Thermal Transport During Micro/Nanomachining In the “macro” world, thermal transport phenomena as related to material processing are quite well understood. Using available analytical and numerical techniques, we can predict heating, melting, or evaporation within a metallic object with relative ease. Yet, once the dimension of the process is reduced further down to the nanoscale levels, the problem becomes non-trivial and necessitates better understanding of the physics and proper estimation of the required thermophysical properties. The first challenge in modeling a micro/nanomachining process is setting up the physically correct governing equations. In general, transport of energy carriers can be modeled using wave or particle approaches, depending on their energy levels and the size of the medium they interact. It is rather difficult to estimate the nature of the transport without knowing the exact transport

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5 Thermal Conduction Coupled with e-Beam Transport

properties of the material of interest, which is to be determined during modeling. Our emphasis in this monograph is on the modeling of electron-beam based processes. Electrons are highly energized, and much smaller than the lattice features of a workpiece, even at the nanoscale. Because of this, the wave nature of electron transport can be avoided in modeling. Given that, we focus here only on the particle theories and electron-beam transport equation (EBTE). Also, we limit our analyses to metallic workpieces, specifically to gold, although the modeling methods described here can be easily extended to other materials. As we have discussed in Chap. 1, with the use of an electron beam, a small area of workpiece can be micro/nanomachined. The target material undergoes melting, evaporation or sublimation, which yields the material removal, and hence the machining. Modeling of thermal transport during this process requires the formulation of conservation of energy equations. If local thermodynamic equilibrium (LTE) can be assumed, then these equations can be cast in terms of temperature. The temperature range involved in this application may be extensive, starting at room temperature (i.e., 300 K) reaching to the evaporation temperature of metals (i.e., ∼3,000 K). At nanoscale dimensions, the LTE, or the concept of temperature, may not be necessarily applicable; therefore the conservation equations should be used with care. For the most cases we consider in this monograph, the minimum feature size is larger than 20 nm; therefore the LTE can be considered valid, at least as a first approximation. For feature sizes smaller than this, molecular dynamics (MD) simulations are preferable to explore melting and material removal concepts. In principle, energy carriers such as electrons, phonons and photons propagate as waves, defined by their phases. In highly scattering media or hightemperature applications, however, the phase information is usually lost as the propagation becomes highly diffusive. For machining applications, thermal transport inside a workpiece can be treated as diffusive due to the highly scattering of heat carriers. Therefore, the use of particle theories is justified. Particle transport within a medium can be accurately modeled by solving the BTE, which describes the propagation of energy carriers much smaller than the physical dimension of the object of interest. It is, however, difficult to solve the general BTE. MC simulations come in handy to overcome this problem, even though they may require large computational costs and resources. It is relatively easy to construct a MC simulation and to generate the required “rules” for the propagation of heat carrier based on some intuitive thinking. These “rules” may lead to unique statistical distributions, which can be considered as the solution to the problem. In a sense, a MC simulation is very similar to an experimental setup. Without proper calibration and evaluation of the accuracy of the results obtained, it would not be possible to claim its accuracy or applicability. Extra care should be taken to account for the physics of the problem, and the results should be evaluated independently. Below, we will present a systematic and general solution approach that can be used for modeling transport phenomena inside a metallic layer with MC models.

5.2 Thermal Conduction due to Single Electron-Beam Heating

107

5.2 Thermal Conduction due to Single Electron-Beam Heating 5.2.1 Problem Description and Basic Assumptions We consider a workpiece exposed to a single electron beam field-emitted from a nanoprobe. Three modes of thermal transport modes are encountered in this application: (1) overall heating of the workpiece, which is denoted as the bulk heating, (2) local heating using a laser beam to raise the energy of the location of interest, and (3) energy transfer to the workpiece by electron bombardment from field emission. For modeling these heating mechanisms, physically accurate governing equations and the boundary conditions need to be chosen. In principle, this choice strongly depends on the wavelengths of the energy carriers and time and length scales of the system (see Figs. 2.1 and 2.2). The concepts of “melting,” “evaporation,” and “sublimation” at the nanoscale are not well established, as they require further theoretical and experimental investigation. Our specific objectives in this section include: (a) modeling of the electron-beam transport, (b) prediction of the temperature field in the workpiece, and (c) determination of the electron-beam power to effectively remove atoms from a workpiece. The final result of these simulations should be a detailed outline of the conditions required to achieve micro/nanomachining. The models and predictions developed in this monograph will subsequently serve as experimental guidelines for finding suitable nanoprobes for micro/nanomachining. In the following simulations, the workpiece is assumed to be homogeneous and is free of defects and cracks. These assumptions are necessary to model electron, photon, and phonon scatterings from well established theories. After a set of comprehensive models are developed, some of these approximations will be relaxed. Note that if there are impurities and inhomogeneities in the medium, the required properties should be obtained from experiments. A schematic of the modeling problem is depicted in Fig. 5.1. An electron beam with a Gaussian profile in the radial direction is assumed to bombard the workpiece from the top surface. The workpiece is assumed infinitely long in the radial direction. The thickness of the workpiece in the z-direction is denoted as L. The distance between the emitting probe and the workpiece is not as important at this stage. Although this distance changes the incident beam profile, it is not a required parameter in the simulation as we can change the beam profile directly to account for it. (Note that the electron-beam profile on a workpiece as a function of probe material and gap distance was recently explored by S´ anchez et al. [177]) For the sake of simplicity, first a single probe will be used. The machining process involving multi-probes will be analyzed later. Below, two possible micro/nanomachining scenarios are considered. In the first one, we determine the power requirement from the emitting probe, which is illustrated in Fig. 5.1a. In the second scenario, in order to minimize the

108

5 Thermal Conduction Coupled with e-Beam Transport

Fig. 5.1. The micro/nanomachining process is shown. The workpiece is located above a thick substrate, which is assumed to be non-absorbing of the incident laser energy. Case (a) shows only the electron beam used for machining; Case (b) is for both the laser and electron beams. Laser energy is considered to provide the auxiliary heating of the workpiece to achieve the machining process

power input from the electron beam, we consider auxiliary heating using a laser beam at the interface between the workpiece and the substrate. The schematic for this case is depicted in Fig. 5.1b. We assume that there are no electron sources or drains imposed on the workpiece. With additional applied voltages across the workpiece, non-uniform joule heating may occur locally within the workpiece. This phenomenon is best described if the electronphonon hydrodynamic equations (EPHDEs) are used, which we will discuss in Chap. 7. For the results given in this section the workpiece is assumed to be a thin gold film, with thickness of either 200 or 500 nm, deposited on a 10-µm transparent quartz substrate. This thickness of the substrate allows the assumption of “infinitely thick” to electron and phonon propagation, as determined after a series of simulations. Any further increase in the thickness does not change the results within the time domain considered here.

5.2 Thermal Conduction due to Single Electron-Beam Heating

109

To help minimizing the power requirement from the electron beam, we can provide auxiliary heating by a laser beam. For this, we consider a 355-nm wavelength laser to heat the workpiece at the interface between workpiece and the substrate. It is assumed that the laser enters the substrate in the direction opposite to the incident electron beam (see Fig. 5.1b). This configuration is chosen more for the convenience of experimental requirements. The particular laser wavelength is used to have more absorption in gold layer. The substrate is assumed to be transparent to the laser beam. Energetic electrons, emitted from a single electron source with known initial kinetic energies, are incident on the top surface of the workpiece perpendicularly. The point of their origin is arbitrary in the modeling; we set the origin of the coordinate frame at the point where electron bombardment occurs. Since both the electron and the laser beams considered in this simulation are symmetry along the axial direction, a cylindrical coordinate system is employed. Radiation emission by the workpiece during the process may affect its energy balance and should be explored. Indeed, radiation emission is the only transport mechanism that allows the workpiece to lose energy to the surroundings in a vacuum setting. But, after carrying a series of preliminary simulations, the radiation loss term is found to be negligibly small. Radiation exchange may still be important between the tip and the workpiece, especially if they are within a few hundreds of nanometer distance. For these calculations, more elaborate near-field radiation transport calculations are needed, which are beyond the scope of this manuscript (see Francoeur and Meng¨ uc¸ [58] for recent review on NRT). Note that because radiation loss at the surface is neglected, the problem becomes naturally unsteady after a long exposure to external heating sources. However, given that here we only consider relatively short processes, this should not be an issue. 5.2.2 Computational Grid Results from the MC simulations provide the heat source term in the thermal conduction calculations. For this, MC and conduction transfer grids should be compatible. We perform the simulations first in a uniform grid, which is the domain given as (r × z) = (R1 × L1 ) and spans from m = 0 to NR1 − 1 and n = 0 to NL1 −1. The uniform grid is extended from (R1 ×L1 ) to ((R1 +R2 )× (L1 + L2 )) when the conduction is modeled. The laser heating distribution is confined within the uniform grid to avoid significant numerical errors. The cylindrical coordinates considered are defined by the grid index m along the radial direction r and the grid index n along the axial direction z. The grid is stretched in both r- and z-directions to reduce the cost of computations. The grid is then sub-divided into uniform and non-uniform regions due to grid stretching. A schematic of the grid system used is given in Fig. 5.2. The total thickness of the workpiece considered is Lw (= L1 + L2 ) (i.e., n = 0 to NL1 + NL2 − 1) while the thickness of the substrate is Ls (= L3 ) (i.e., n = NL1 + NL2 to NL1 + NL2 + NL3 − 1). The number of grid points

110

5 Thermal Conduction Coupled with e-Beam Transport

Fig. 5.2. The computational grid used in modeling micro/nanomachining. Zones A and B have uniform spacings in both r- and z-directions while zones C and D have non-uniform spacings where the grid is stretched along r- and z-directions with independent factors. Zone A is where the electron-beam MC simulations are performed. Zone B extends zone A uniformly in order to account for the laser heating. The adiabatic boundary condition is applied to all the boundaries except at r = R1 + R2 + R3 where the material is assumed to be at room temperature

required for the calculations, (i.e., NR1 , NR2 , NR3 , NL1 , NL2 , and NL3 ) is determined using the R’s, the L’s, and the ratios of the two adjacent grid spacing (particularly, the latter over the former) for the r- and z-directions. The radial dimension of the workpiece is given by: Robs = R1 + R2 + R3 .

(5.1)

The numbers of grid points required for uniform-grid regions A and B are determined as: L1 , ∆z L2 . = ∆z

Nz1 =

(5.2)

Nz2

(5.3)

5.2 Thermal Conduction due to Single Electron-Beam Heating

111

For region C, the grid spacings are stretched linearly (i.e., multiplied by a constant factor, ξ, consecutively) between adjacent grids along the z-direction, so that: L3 = ∆z

Nz3 

ξl .

(5.4)

l=1

Note that L3 may not be precisely specified by the user because of the rounding error at the last stretched grid spacing. Therefore, if a 1-µm thick workpiece is considered; the actual length of the object will be slightly off depending on the stretching factor used. Once all the required numbers of grid points are determined, the grid spacing and the coordinate of each computational element in the z-direction follows: & ∆z ∀ n = 0, . . . , Nw − 1, ∆zn = (5.5) ξ n+1 ∆z ∀ n = 0, . . . , Ns − 1, ⎧ (n + 0.5) ∆z ⎪ ⎪ ⎨ 1 zn = Nw ∆z + 2 ξ∆z n * ⎪ ⎪ ⎩Nw ∆z − 12 ξ n+1 ∆z + ξ l+1 ∆z l=0

∀ ∀

n = 0, . . . , Nw − 1, n = Nw ,

∀

n = 1, . . . , Ns − 1.

(5.6)

Similarly, the grid spacing and the coordinate of each computational element in the r-direction can be obtained following those in the previous discussion. As a result: & ∆r ∀ m = 0, . . . , Nuni − 1, ∆rm = (5.7) m+1 ζ ∆r ∀ m = 0, . . . , +Nnon − 1,

rm

⎧ (m + 0.5) ∆r ⎪ ⎪ ⎨ 1 = Nuni ∆r + 2 ζ∆r m * ⎪ ⎪ ⎩Nuni ∆r − 12 ζ n+1 ∆r + ζ l+1 ∆r l=0

∀ m = 0, . . . , Nuni − 1, ∀ m = Nuni ,

(5.8)

∀ m = 1, . . . , Nnon − 1,

R2 R1 , Nr2 = ∆r , and R3 = where Nuni = Nr1 + Nr2 , Nnon = Nr3 , Nr1 = ∆r N r3 * ∆r ζ l . For the uniform-grid region, the stretching factor along the rl=1

direction is denoted as ζ.

5.2.3 Electron-Beam Monte Carlo Simulation The MC simulation procedures used to determine the electron-energy distribution in a workpiece and to calculate the required electron-beam properties

112

5 Thermal Conduction Coupled with e-Beam Transport

were already described in details in Chaps. 3 and 4, respectively. The MC method treats an electron beam in terms of discrete electrons where all the electrons have equal energy. Each electron undergoes a series of elastic and inelastic scatterings inside the workpiece. By tallying the histories of many propagating electrons, we can form the resultant profile of absorbed energy. The MC method used in this section is the continuous slow-down approach (CSDA) (see Chap. 3 for more discussions). As the MC simulation traces electron ensembles in space, we can calculate the amount of electron energy deposited within each computational control volume and determine electron distribution. This distribution, denoted as Ψm,n , is typically normalized by the total incident energy and expressed per unit volume. It is mathematically expressed as: Ψm,n =

ψm,n , 2πrm ∆rm ∆zn Nen E0

(5.9)

where ψm,n is the total kinetic energy of electrons deposited at the (m, n) element, Nen is the total number of electron ensembles used (i.e., for the statistical MC simulation, not the actual number of electrons incident on the solid target), E0 is the initial energy of the electrons, and the quantity 2πrm ∆rm ∆zn is the volume of the (m, n) element. The internal heat generelec ation at a given element (m, n), q˙m,n , is then computed with the following expression: elec ˙ m,n , q˙m,n = EΨ

(5.10)

where E˙ is the input power of the electron beam. In general, the electronenergy distribution inside the material is a function of the initial beam energy, the workpiece thickness, and the spatial spread of the beam. 5.2.4 Auxiliary Heating Using Laser Beam Electron-beam based micro/nanomachining can be safeguarded by employing auxiliary heating mechanisms, which will guarantee the transfer of sufficient energy to the workpiece for the necessary machining and material removal process. A collimated laser beam focused on a prescribed area with radius Rlaser can be considered for this purpose (see Fig. 5.3). This area is usually much larger than the domain of interest. We assume that the substrate is transparent to the incident laser beam (i.e., no absorption within the substrate) while the metal layer is absorbing. Since the absorption cross-section in a metal is much larger than the scattering cross-section, the laser heating is analyzed in one dimension along the direction of incidence. The Fresnel reflections need to be considered at the mismatched interfaces where the indices of refraction are different. In the present case, the mismatched conditions are because of the gold workpiece and the quartz substrate. For the normal incident case, the fraction of the incident radiant energy reflected, Ri→t , as the laser propagates from medium i to t, is given as [72]:

5.2 Thermal Conduction due to Single Electron-Beam Heating

113

Fig. 5.3. The laser beam propagation inside the workpiece. The beam is assumed to have circular profile with a radial dimension of Rlaser at wavelength of 355 nm. A 1-D radiation model is used, which yields an exponential decaying of radiant energy in the direction of propagation

Ri→t =



˜i n ˜t − n n ˜t + n ˜i



˜i n ˜t − n n ˜t + n ˜i

∗

,

(5.11)

where n ˜ i and n ˜ t are the complex indices of refraction of the incident and the transmitted media, respectively. In the simulations, the initial heat flux q0′′ of the laser beam propagating through the substrate is prescribed. Once the laser beam hits the quartz-gold interface, a fraction of the heat flux, Rs→w is reflected (which is calculated to be around 0.75), while the remainder is transmitted through the interface. The laser is then absorbed exponentially by gold layer along the direction of propagation. As a result, the radiant heat flux is expressed as a function of depth in the z-direction within a circular area: q ′′ (z) = (1 − Rs→w )q0′′ e−κ(Lw −z) ,

(5.12)

where κ is the absorption coefficient of the workpiece. This absorption coefficient is determined using the imaginary refractive index of the workpiece, nI,w , as [72, 136, 182]: κ=

4πnI,w , λo

(5.13)

where λo is the wavelength of the laser in vacuum. Generally, the incident laser beam is strongly absorbed within the first few tens of nanometers of a metallic layer. For a λo = 355 nm laser, the imaginary index of refraction for gold is nI,W = 1.848 [148]; therefore, κ is 0.0654 nm−1 and 95% of the penetrating photons are absorbed within 46 nm

114

5 Thermal Conduction Coupled with e-Beam Transport

into the gold film. The thickness of the film considered in the following simulations is either 200 or 500 nm which is sufficiently large that the laser beam is never transmitted through the workpiece. To determine the amount of radiant energy per unit volume absorbed by a computational element in the workpiece within a radius of Rlaser , the radiant heat flux is divided by the ∆zn and is expressed as: rad q˙m,n =

q ′′ (z) , ∆zn

(5.14)

for m = 0, 1, . . . , NRlaser − 1 and NRlaser is the radial index at Rlaser . 5.2.5 Fourier Heat Conduction Energy Balance for an Element in the Workpiece or Substrate The next step is to build a thermal conduction model where the electron and laser beams enter as heat sources. As we have already discussed in Chap. 2, there are several conduction models that can be used for modeling the heat diffusion inside a workpiece, including the Fourier law, the TTM, and the EPHDEs. The Fourier heat equation is considered in this chapter. The other two models will be used to have more accurate electron-phonon transport predictions, as discussed in Chaps. 6 and 7. According to the Fourier law of heat conduction, the energy transferred into a computational element is expressed as the product of the thermal conductivity, k, the cross-sectional area, A, and the negative gradient of the temperature perpendicular to the surface, and written as: Q˙ = −kA∇T.

(5.15)

This expression is written for each element (m, n) in the computational domain within the workpiece and the substrate. In addition, the heat generation term is written for each element, and includes the deposited electron energy and absorbed laser energy: rad elec q˙m,n = q˙m,n . + q˙m,n

(5.16)

A non-uniform computational grid is used in simulation (see Fig. 5.4). The energy balance for the uniform grid follows in a similar way except that all the ∆z’s (or ∆r’s) are constant. Thermophysical properties (i.e., conductivity, k, heat capacity, C, and density, ρ) are assumed to be temperature dependent, and may vary from one element to another. Then, heat transferred into the node (m, n) is expressed as:

5.2 Thermal Conduction due to Single Electron-Beam Heating

115

Fig. 5.4. Energy balance is shown for a computational element inside the workpiece or the substrate (except at the interfaces between the two). Non-uniform grid spacings are considered in the formulation. The energy balance for the uniform grid follows similarly except that all the ∆z’s (or ∆r’s) are constant. To be consistent, all the heat is assumed to be transferred into the node of interest, (m, n). The heat generation term as a result of heating by external means (laser or electron beam) is denoted as E˙ sto . The thermal properties (i.e., conductivity, k, heat capacity, C, and density, ρ) are assumed to vary from one element to another due to the transient and spatial temperature variations

 −1

 P +1  ∆r ∆r /2 /2 ∆r m−1 m m P +1 ∆zn + P +1 Tm−1,n − Tm,n , Q˙ 1 = 2π rm − P +1 2 km−1,n km,n (5.17)  −1

 P +1  ∆rm ∆rm /2 ∆rm+1 /2 P +1 ∆zn Q˙ 2 = 2π rm + Tm+1,n − Tm,n , + P +1 P +1 2 km,n km+1,n (5.18) −1   P +1  ∆zn−1 /2 ∆zn /2 P +1 Q˙ 3 = 2πrm ∆rm Tm,n−1 − Tm,n , (5.19) + P +1 P +1 km,n−1 km,n  −1   P +1 ∆z /2 /2 ∆z n n+1 P +1 , (5.20) + P +1 Tm,n+1 − Tm,n Q˙ 4 = 2πrm ∆rm P +1 km,n km,n+1

116

5 Thermal Conduction Coupled with e-Beam Transport

where the superscript ‘P ’ denotes the time step. Accordingly, the heat generation and the unsteady energy terms are given as:

E˙ sto

Q˙ gen = 2πrm ∆rm ∆zn g˙ gen ,   P +1 P T − T m,n m,n P +1 P +1 . = 2πρm,n Cm,n rm ∆rm ∆zn ∆t

(5.21) (5.22)

Note that at each new time step (P + 1) the thermal conductivity is an unknown, as the new temperature value is not known. Because of this the conductivity values need to be extrapolated based on the previous, available, values:      P P P ∂Tm,n ∂km,n ∂km,n P P +1 P km,n = km,n + ∆t = km,n + ∆t. (5.23) ∂t ∂T ∂t The derivative of k in temperature can be easily determined using the data provided in the literature, while the derivative of T in time is to be approximated as: P P P −1 Tm,n ∂Tm,n − Tm,n = . ∂t ∆t

(5.24)

Equation (5.23) applies not only to thermal conductivity but also to the specific heat and density of the material in the finite-difference (FD) equation. Hence, the energy balance for the element (m, n) reads: Q˙ 1 + Q˙ 2 + Q˙ 3 + Q˙ 4 + Q˙ gen = E˙ sto .

(5.25)

Additional simplifications lead to:  P +1  m−1→m P +1 Q˙ 1 = Am−1→m Tm−1,n − Tm,n ; A1 = 1

 P +1  n+1→n P +1 Q˙ 4 = A4n+1→n Tm,n+1 − Tm,n ; A4

gen ; A5 = Q˙ gen = A5 g˙ m,n

P +1 P E˙ sto = Tm,n − Tm,n ,



∆t , +1 P +1 ρP m,n Cm,n



P +1 Km−1→m,n +1 P +1 ρP m,n Cm,n

P +1 Km+1→m,n

 ∆t , rm ∆rm (5.26) 

∆t , rm ∆rm (5.27)   P +1 Km,n−1→n ∆t , (5.28) = +1 P +1 ρP m,n Cm,n ∆zn   P +1 Km,n+1→n ∆t , (5.29) = +1 P +1 ρP m,n Cm,n ∆zn

 P +1  m+1→m P +1 Tm+1,n − Tm,n ; A2 Q˙ 2 = Am+1→m = 2  n−1→n  P +1 P +1 ; A3 Tm,n−1 − Tm,n Q˙ 3 = An−1→n 3



+1 P +1 ρP m,n Cm,n

(5.30) (5.31)

5.2 Thermal Conduction due to Single Electron-Beam Heating

117

where: P +1 Km−1→m,n

P +1 Km+1→m,n

P +1 Km,n−1→n

P +1 Km,n+1→n

=



∆rm rm − 2



P +1 P +1 2km−1,n km,n P +1 P +1 ∆rm−1 km,n + ∆rm km−1,n



,



 P +1 P +1 2km+1,n km,n ∆rm , = rm + P +1 P +1 2 ∆rm+1 km,n + ∆rm km+1,n   P +1 P +1 2km,n−1 km,n , = P +1 P +1 ∆zn−1 km,n + ∆zn km,n−1   P +1 P +1 2km,n+1 km,n . = P +1 P +1 ∆zn+1 km,n + ∆zn km,n+1

(5.32) (5.33) (5.34) (5.35)

The overall FD equation for a node (m, n) is written everywhere inside the workpiece or the substrate except the interfaces between the two, and it is given as: P +1 P +1 − An−1→n Tm,n−1 − Am−1→m Tm−1,n 3 1  P +1  + 1 + Am−1→m + Am+1→m + An−1→n + A4n+1→n Tm,n 1 2 3

(5.36)

P +1 P +1 P gen − Am+1→m Tm+1,n − A4n+1→n Tm,n+1 = Tm,n + A5 g˙ m,n . 2

Depending upon the location of the node, all A’s are to be evaluated using properties of the workpiece or the substrate. Energy Balance for an Element of the Workpiece and the Substrate at the Interface of Two Distinct Materials The energy balance at the interface between the workpiece and the substrate can be obtained in a similar fashion. Here, it is necessary to take into consideration the contact resistance between the two materials and to incorporate the correct thermal properties into the FD equation. A schematic of the energy balance of a computational element of the workpiece on top of the substrate is depicted in Fig. 5.5. The terms given in the energy balance equation are identical to those derived in the previous section except that Q˙ 4 is to be modified to account for the contact resistance and the thermal properties of the substrate. It is expressed as: Q˙ 4 = 2πrm ∆rm



∆zn+1 /2 ∆zn /2 + Rc′′ + P +1 P +1 km,n km,n+1

−1



 P +1 P +1 Tm,n+1 − Tm,n .

(5.37)

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Fig. 5.5. The energy balance of a computational element of the workpiece at the interface between the workpiece and substrate is depicted. The subscript ‘w’ refers to that of the workpiece and ‘s’ refers to the substrate. The contact resistance between two different types of materials is denoted as Rc′′

Following a similar derivation, the energy balance equation for a node in the workpiece adjacent to that in the substrate is written as: P +1 P +1 − An−1→n Tm,n−1 − Am−1→m Tm−1,n 3w 1w

 n+1→n m+1→m n−1→n P +1 Tm,n + 1 + Am−1→m + A + A + A 4w,s 1w 2w 3w  n+1→n

P +1 − Am+1→m Tm+1,n − A4w,s 2w

(5.38)

P +1 P gen Tm,n+1 = Tm,n + A5w g˙ m,n

where  n+1→n

∆t . (5.39) ∆zn  P +1   P +1  km,n w 2 km,n+1  P +1   P +1   P +1 s   P +1  = . R′′ k ∆zn+1 km,n w +∆zn km,n+1 s +2 km,n w m,n+1 s c (5.40) A4w,s


 P +1 K m,n+1→n

w,s

P +1

 K m,n+1→n

= 

w,s  +1 P +1 ρP m,n Cm,n w

The subscripts ‘w’ and/or ‘s’ in the A’s refer to the properties of the workpiece and the substrate, respectively. For the case where the substrate element is of interest at the interface of the two materials, the energy balance equation for the element is similar to those given in Fig. 5.4 with the exception that Q˙ 3 needs to be modified. An

5.2 Thermal Conduction due to Single Electron-Beam Heating

119

Fig. 5.6. The energy balance of a computational element of the substrate at the interface between the workpiece and substrate is depicted

illustration of the notations used is given in Fig. 5.6. Accordingly, the energy balance equation for this node reads:  n−1→n

P +1 P +1 − A3w,s Tm,n−1 − Am−1→m Tm−1,n 1s

 n−1→n m+1→m n+1→n P +1 Tm,n + 1 + Am−1→m + A + A + A 3w,s 1s 2s 4s

(5.41)

P +1 n+1→n P +1 P gen − Am+1→m Tm+1,n − A4s Tm,n+1 = Tm,n + A5s g˙ m,n , 2s

with  n−1→n

A3w,s P +1

 K m,n−1→n w,s

=

P +1

 K m,n−1→n

w,s



+1 P +1 ρP m,n Cm,n



s

∆t , ∆zn

 P +1   P +1  km,n s 2 km,n−1  P +1 w  P +1   P +1   P +1  . = km,n−1 w Rc′′ ∆zn−1 km,n s + ∆zn km,n−1 w + 2 km,n s

(5.42)

(5.43)

The System of Equations and Matrices In order to solve the temperature distribution within the workpiece and the substrate, it is necessary to consider all the FD equations for all the nodes to

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form a system of equations. For the sake of simplicity, the following quantities are first defined: Nr = Nr1 + Nr2 + Nr3 − 1, Nz = Nz1 + Nz2 + Nz3 − 1.

(5.44) (5.45)

The boundary conditions are given as: P +1 P +1 = T0,n T−1,n P +1 P +1 Tm,−1 = Tm,0 P +1 P +1 Tm,N = Tm,N z z −1 P +1 TNr ,n = 2T0 − TNPr+1 −1,n

∀ n = 0, 1, . . . , Nz − 1, ∀ m = 0, 1, . . . , Nr − 1, ∀ m = 0, 1, . . . , Nr − 1,

(5.46)

∀ n = 0, 1, . . . , Nz − 1.

When all the nodal equations are summoned, the corresponding system of ⇀ ⇀ equations can be constructed in the form of AT = B where A is a (Nr × Nz ) × (Nr × Nz ) matrix. Details about matrices A and B are given in Appendix A. If the radiation interaction needs to be incorporated into the system, then the boundary conditions change accordingly. For any boundary exchanging radiant heat with the surroundings, the energy balance at the boundary reads as: −k

  ∂T 4 = εσ T 4 − Tsurr . ∂n

(5.47)

Applying this to the top surface of the workpiece and discretizing the equation yield:   P +1
 

 4 km,n P +1 P +1 4 P +1 Tm,n = εP − Tm,n−1/2 − T T σ 2πrm ∆rm surr . m,n m,n ∆zn /2 (5.48) Note that the non-linearity in the temperature has been conveniently removed by assuming the radiation loss is given by the previous time step P . Next, the temperature at the node (m, n − 1/2) shall be approximated as the mean value between two adjacent nodes, namely nodes (m, n − 1) and (m, n), and it is expressed as: P +1 = Tm,n−1/2

 1  P +1 P +1 T + Tm,n . 2 m,n−1

(5.49)

As a result, the boundary condition at the top of the workpiece becomes:  

P +1  km,n 1  P +1 P +1 P +1 Tm,n Tm,n−1 + Tm,n 2πrm ∆rm − ∆zn /2 2   (5.50)

4   1 +1 P 4 TP . + Tm,n − Tsurr = εP m,n σ 2 m,n−1

5.2 Thermal Conduction due to Single Electron-Beam Heating P +1 P +1 T−1,n = T0,n P +1 P +1 Tm,−1 = Tm,0 P +1 P +1 Tm,N = Tm,N z z −1 P +1 TNr ,n = 2T0 − TNPr+1 −1,n

121

∀ n = 0, 1, . . . , Nz − 1, ∀ m = 0, 1, . . . , Nr − 1, ∀ m = 0, 1, . . . , Nr − 1,

(5.51)

∀ n = 0, 1, . . . , Nz − 1.

After discretizing the entire computational domain, the difference equations for all the nodal points are used to form a system of linear equations, expressed in matrix representation as: − → ⇀ B T = D, (5.52) ⇀

where B is a (NR × NL ) × (NR × NL ) matrix, T is the temperature field, and ⇀

D contains temperature information from the previous time step and heat ⇀ generation terms at various nodes. Equation (5.52) is solved for T using the point successive overrelaxation (SOR) numerical scheme [128] in this study. 5.2.6 Modeling “Melting” and “Evaporation” At nanoscale domains, definitions of “melting” and “evaporation” are not well established, and even the concepts of LTE and continuum are questionable. Molecular dynamics simulations are, to a degree, capable of shedding light on “phase change” and “melting” processes at nanoscales. We will briefly discuss MD calculations in Chap. 9. Further discussions for non-equilibrium thermodynamics as applied to such small domains were provided in [164,200]. There is no question that proper understanding of melting and evaporation at small scales, particularly for applications to electron-beam based micro/nanomachining, needs further research. In our simulations, we assume that melting and evaporation temperatures for a metal workpiece (i.e., gold) are provided to the user a priori. Since the numerical method used in solving the matrix is the point SOR method, the numerical solution is obtained by solving one nodal point after another. This scheme allows us to incorporate melting and evaporation easily. In our earlier work, we proposed a strategy to account for these phenomena [210]. The basic idea here is to solve the system of equations following the normal point SOR procedures until local melting or evaporation temperature is encountered. When temperature at a nodal point within the computational domain exceeds either one of these temperatures, the temperature at that particular location is held fixed at either melting or evaporation temperature. Energy balance is then performed on these locations to determine the amount of energy needed to maintain constant temperatures at these locations. Typically, this requires less energy than that supplied by the external sources, electron beam and/or laser, due to additional energy transfer from the neighboring nodes. The excess energy provided is stored to count towards the local latent heat values. Whenever the latent heats are overcome, then the phase

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change process is assumed to be completed, and temperatures start increasing in those locations. After that, the solution for the next time step is obtained by following a regular SOR scheme once again. 5.2.7 Computational Parameters for Micro/Nanomaching Scenario The numerical models that we discussed in this chapter are valid for any type of electron source that is capable of generating an electron beam. If field emitted electrons are used as the electron source, the entire micro/nanomachining stage needs to be placed under ultra-vacuum condition. Material properties under such condition differ from those under standard pressure and they may be strongly pressure-dependent in addition to temperature dependent. Given this, and to simplify our numerical calculations, we have first used constant properties for gold and quartz. Simulations with varying properties will be discussed later in Sect. 5.3 and Chap. 6. These constant properties required for the conduction code are summarized in Appendix C. In the following sections, we present a series of results. The grid parameters considered are ∆r = 1.25 nm and ∆z = 1.25 nm with 10% increases in the rand 5% increases in the z-grid spacings starting from m = NR1 + NR2 and n = NL1 + NL2 , respectively (see Fig. 5.4). This grid stretching is adapted to maintain the accuracy of temperature predictions in the order of O(1 K). The size of the uniform grid is chosen to be either (r × z) = (400 nm × 200 nm) or (400 nm × 500 nm) when the thickness of the gold film considered is either 200 or 500 nm, as outlined in Wong et al. [210]. 5.2.8 Electron-Beam Deposition Profiles Electron-beam deposition profiles are obtained using the MC simulation discussed in Sect. 5.2.3; they enter into the conduction code as the heat source terms. Figure 5.7 depicts the electron energy deposited within gold for an electron beam with a 1/e2 Gaussian radius of 100 nm (= Relectron ) and an initial kinetic energy of 4 keV (= E0 ). The distribution shown in the figure is an average of five separate MC runs with 10 × 106 statistical ensembles each run to ensure smooth spatial distributions. Results from the MC simulations show that electron penetration depth decreases when initial kinetic energy of electrons decreases. Consequently, electrons are more concentrated near the area of incidence for low-energy electron beams, implying that the deposition amount increases in terms of per unit volume [210]. In addition, the deposition profile is less denses when the incident electron beam has a wider spatial spread. 5.2.9 Electron-Beam Heating Using the MC simulation result given in Fig. 5.7, the temperature distribution inside the workpiece and substrate are computed based on the Fourier

5.2 Thermal Conduction due to Single Electron-Beam Heating

123

Fig. 5.7. Electron-energy deposition profile, Ψ × 109 (nm−3 ), inside gold film. The profiles are normalized by the total incident electron energy. Gaussian beam profiles in the r-direction are used: (a) a 1/e2 radius of Relectron = 100 nm and initial kinetic energy of E0 = 4 keV, (b) Relectron = 50 nm and E0 = 4 keV, and (c) Relectron = 100 nm and E0 = 6 keV. Data are obtained from Wong et al. [267]

conduction formulation. The calculations run until the element at the origin “evaporates” (i.e., latent heat of evaporation is overcome and T > Tevap = 3,129 K). A plot of the temperature distribution at the snapshot when evaporation occurs is shown in Fig. 5.8. For the specific case considered here, where the power of the electron beam used was 0.5 W, the evaporation time was 0.9 ns. The corresponding current required for the electron beam is calculated to be 125 µA. Several other results were obtained by following the same procedures when different voltages and beam specifications were employed. For example, we found that as the spatial Gaussian radius of the electron beam was decreased to 50 nm, the required power for the beam dropped to 0.305 W with evaporation started at 0.7 ns [210]. This suggests that if the spatial spread of the electron beam is decreased, the electric current required from the beam can be further reduced. This is actually one of the important limiting factors in using field emitted electrons for micro/nanomachining purpose. There are no focusing mechanisms used in such application. Therefore, if the electron-beam spread is large, the use of these electrons for machining may not be feasible. We also observed that the electric current required for achieving evaporation was actually reduced when the beam energy decreases. This is due to the fact that electrons are more concentrated within the area of incidence when the initial beam energy is small.

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5 Thermal Conduction Coupled with e-Beam Transport

Fig. 5.8. Temperature distribution inside gold film at t = 0.9 ns. The electron beam impinges z = 0. A 1/e2 radius of Relectron = 100 nm is considered for the Gaussian beam. The electron beam has an initial kinetic energy of 4 keV. The power of the beam is set to E˙ = 0.5 W. The ∆t used in the simulation is 0.005 ps. The gold workpiece has a thickness of 500 nm, and it is deposited on a 10-µm quartz substrate

With the use of the auxiliary laser heating, the electron-beam power requirement can be further reduced. We presented the analysis of radiative energy absorption within the workpiece in Sect. 5.2.4. Using this approach, we can further explore the possibility of using laser to assist micro/nanomachining. In order to avoid experimental complications, we avoid overlapping the electron and laser beams. This is achieved by placing the electron beam and laser at the opposite ends and by making sure that the energy deposition profiles from these beams do not interfere with each other. Using the same configurations given in Fig. 5.8 and adding a laser with heat flux of 5.09 µW nm−2 and beam radius of 300 nm, the time required for evaporation is shortened from t = 0.9 to 0.5 ns (see Fig. 5.9). As one would expect, a higher power laser would decrease this time even further, although the total melt down of the workpiece with the laser heating should be avoided. The numerical experiments show that the laser power used and the thickness of gold layer are correlated. Gold cannot be to thick, as otherwise the laser beam heating from the lower side will not affect the electron-beam machining from the top. By decreasing the workpiece thickness from 500 to 200 nm, we observed that the power requirement of the electron beam to achieve evaporation at approximately 1 ns time was decreased by one half. The temperature distribution corresponding to this case is shown in Fig. 5.10.

5.2 Thermal Conduction due to Single Electron-Beam Heating

125

Fig. 5.9. Temperature distribution inside a 500-nm gold film at t = 0.5 ns. Laser is used in addition to the electron beam. The power of the laser is 1.5 W and it covers a radius of Rlaser = 300 nm incident at the interface between gold and quartz. The input parameters are the same as those given in Fig. 5.8. The time required for the first element at the origin to evaporate is improved from t = 0.9 ns (as in Fig. 5.8) to 0.5 ns. Data are obtained from Wong et al. [210]

5.2.10 Comments In this chapter we have discussed a general methodology to simulate thermal transport within a workpiece during micro/nanomachining. We coupled a MC solution of the electron-beam propagation inside a metallic material with the Fourier heat conduction equation, and obtained the temperature distribution inside the workpiece. A simple radiation model was used to describe the exponentially decaying laser energy within the workpiece. The contributions from the laser and electron beams were treated as heat source terms in the conduction equation. As an example, we considered a 500 nm thick workpiece and a focused electron beam with a Gaussian radius of 100 nm. Results from these simulations show that evaporation of gold film at a nanoscale resolution is possible if combined laser and electron beams are used. The power of the electron beam required to achieve evaporation within 1 ns is found to be around 0.5 W. These values are valid only for the specific case considered; additional simulations need to be carried out to determine the outcome in other possible scenarios. The modeling effort described in this section is rather straightforward due to the various assumptions introduced. For instance, we assumed that the thermophysical properties of the workpiece remained constant regardless of

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Fig. 5.10. Temperature distribution inside a 200-nm gold film at t = 1 ns. The power of the electron-beam is set to 0.25 W. The rest of the input parameters follow those given in Fig. 5.9. Data are extracted from Wong et al. [210]

the temperature change or the vacuum conditions. We relax this assumption in Sect. 5.3, and consider temperature-dependent thermophysical properties in exploring the feasible of sequential patterning. Also, the electrons from the electron beam were assumed to lose energy continuously along the path of propagation. Later in Chap. 6, we model the problem with a different MC method where electrons lose energy in discrete inelastic scattering (DIS) events. The DIS approach is generally considered to be more accurate since electron-energy loss is derived from the refractive indices of the material at different wavelengths. Finally, we expand the simulations to a three-dimensional domain.

5.3 Sequential Patterning Using Electron Beam 5.3.1 Problem Description and Assumptions In this section, we explore a micro/nanomachining scenario using a series of nanoprobes arranged in a rectangular array. The schematic of the problem considered here is depicted in Fig. 5.11. We assume that the pitch between the probes can be as small as 50 nm. This arrangement is based on the discussions with Professor S. Jin of UCSD, whose group is developing such carbon nanotube (CNT) arrays [13] (see Fig. 1.3). Each of these probes is considered to emit electrons in sequential fashion to heat the workpiece. Below, we will

5.3 Sequential Patterning Using Electron Beam

127

Fig. 5.11. The 3-D schematic of material processing using an electron-beam and a laser

first summarize the solution procedure as discussed by Wong et al. [211] and then present a series of results. The physical domain considered here remains similar to that given in the previous section where a thin gold film is deposited on top of quartz. A 3-D Cartesian grid scheme is adapted. In simulations, we do not specify any distances between the probes and the workpiece; rather use a prescribed radial beam spread, which can easily be modified in simulations. Radiation exchange between all surfaces and surroundings are considered in the analysis, even though they have negligible effect. 5.3.2 Computational Methodology In Sect. 5.2, only one electron beam was used and electron energy was supplied continuously. Here, multi-probes (or electron beams) are considered to create the desired pattern at different locations where they fire electrons sequentially. During such a sequential machining process, some parts of the workpiece may cool off while other areas are heated. Therefore, this transient behavior needs to be properly accounted for in the simulations. As expected, the modeling strategy in this transient 3-D case is more complicated and involved. There are three modeling steps to be considered: (1) computation of the electron-energy deposition profiles using the MC simulation, (2) calculation of the radiant-energy deposition due to laser beam heating, and (3) simulation of the temperature field inside the workpiece with the heat sources calculated from 1 and 2. For the solution of the EBTE, we use the same MC simulation procedures given in Sect. 5.2.3. The radiant energy absorbed by the workpiece from the incident laser beam is again determined using the approach discussed in Sect. 5.2.4 except that we assign a circular incident flat beam with a prescribed radius Rlaser (see Fig. 5.11). Heat conduction modeling used

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5 Thermal Conduction Coupled with e-Beam Transport

Fig. 5.12. A typical grid structure used in the simulations. This figure shows the grid in two dimensions at a given time during the simulations. The third dimension follows the same grid stretching method. In the simulation, a three-dimensional grid is used

here follows closely the discretization method described in Sect. 5.2.5 with the exception that 3-D energy balance is performed for each node instead of 2-D. Similarly, we use the same modeling strategy given in Sect. 5.2.6 for simulating melting and evaporation. A sample of the grid scheme used is depicted in Fig. 5.12. We show only the top view of the grid in the figure. The grid spacing in the third dimension follows the same general scheme. The MC simulation for the electron-beam propagation is always performed within the uniform grid area. This applies to the calculation of laser penetration as well. In the conduction simulation, workpiece and substrate change thermophysical properties locally depending on the temperature. To reduce the computational time, we assume that local properties, such as thermal conductivity and specific heat, are evaluated based on the temperature computed in the previous time step. To account for cooling phenomena such as condensation and solidification, we use the same computational strategy given in Sect. 5.2.6 with the exception that the process is reversed. In other words, as a computational node cools to condensation/solidification temperature, it remains at the fixed temperature until all of its latent heat is lost. After conducting extensive numeral simulations, it was determined that evaporation first occurs several nanometers below the top surface. This suggests that explosive evaporation may be possible. This observation is consistent with the electron-energy deposition profile since the maximum amount of volumetric deposition is at layers below the surface as evident

5.3 Sequential Patterning Using Electron Beam

129

in Fig. 5.7. A thorough understanding of the problem can be possible only if an in-depth investigation of thermodynamics, coupled with the transport calculations, is conducted. For this, other numerical approaches can be adapted, including MD simulations. For the sake of simplicity, the explosive evaporation possibility is neglected in the following discussions. We limit our goal to determination of the amount of required electron-beam power for starting evaporation within the workpiece. In the simulations we assume that the nodes at which evaporation takes place have zero thermal conductivity. In other words, they are considered insulated, implying that the nodes act as vacuum after evaporation. This assumption is physically reasonable, since all the material is likely to be removed due to the machining process. Hence, once a computational element at the surface is evaporated, the adjacent nodes are exposed to vacuum and their boundaries become insulated since there are no heat loss mechanisms, assuming that radiative heat loss is negligible. Under these conditions, it is mathematically correct to replace the conductivity of the evaporated elements with a zero value. 5.3.3 Computational Parameters For 3-D simulations, an electron beam with 0.25 W power is considered. This value was chosen after some trial and error, and found to be sufficient for the micro/nanomachining of metallic workpiece. The initial kinetic energy of the beam is 4 keV, and the area of incidence is a spatial Gaussian distribution with a 1/e2 radius of 25 nm. The machining process by the electron beam is assisted by a laser, which has 0.35 W, a wavelength of 355 nm, and a flat area of incidence within a radius of 150 nm. The MC simulations are conducted to solve for the electron-beam propagation using the specified electron energy and beam profile for 10 millions statistical ensembles to achieve sufficiently smooth electron-energy deposition profile. A total of 5 MC runs are used and the average distribution is calculated. The spatial grid spacing in all three directions within the uniform grid is 5 nm and the stretching factor for the non-uniform grid is set to 10% increment with respect to the previous step size. The time step used in the conduction code is 3.125 fs. The material properties of gold and quartz needed in the simulation are depicted in Fig. 5.13. Due to the lack of data beyond the melting temperature, we decided to linearly extrapolate the existing data until the evaporation temperature. 5.3.4 Results and Discussions The effectiveness of 3-D micro/nanomachining procedure can be best evaluated if it is considered for creation of a complicated pattern. For this purpose, we chose the pattern ‘UK’, representing the University of Kentucky. Such a complex patterning process is not trivial to machine, and requires a careful

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5 Thermal Conduction Coupled with e-Beam Transport

Fig. 5.13. The thermal properties of gold and quartz as a function of temperature: (a) Specific heat of gold, (b) thermal conductivity of gold, (c) hemispherical emissivity of gold, (d) specific heat of quartz, and (e) thermal conductivity of quartz. The triangular symbols are the extrapolated values

consideration of a number of issues. For example, whenever a region of the workpiece is evaporated, the boundary of the geometry is changed. Remember, the electron-energy deposition profile was initially obtained from the MC simulations for a prescribed, intact geometry. Therefore, when a computational

5.3 Sequential Patterning Using Electron Beam

131

node becomes null because of evaporation, this geometry is altered. As a result, the deposition profile becomes invalid, and a new MC result needs to be obtained using this evaporated configuration. To avoid this lengthy computation and complication, we choose to stop the electron beam once evaporation is achieved at a given node. However, the procedure can be continued in the same fashion described above beyond the evaporation, provided the thermophysical data are available and enough computational power is devoted to simulations. Figure 5.14 shows all the electron-energy deposition profiles at prescribed locations required to machine the UK pattern, if all the electron beams from

Fig. 5.14. Normalized electron energy-deposition distributions (in units of nm−3 ) predicted using MCM in the electron-beam transport. (a) The intended machining locations to create the UK pattern are shown. Note that there is only one electron-beam used for machining and it is moved from one location to another. This sequential heating can also be achieved with an array of probes firing individually and controllable manner. The figure shows an imaginary heat generation profile as if all of the heat generation profiles generated by the electron beam at various locations are combined together. (b) The internal electron-energy deposition profile, including that by the laser, is depicted. (c) The dimensions of the structures are given. (d) The radius of laser beam is shown

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5 Thermal Conduction Coupled with e-Beam Transport

the array of emitting probes are turned on. However, we sequentially turn on one beam at a time from beam 1 to beam 21. All the beams are assumed to have the same initial kinetic energy and spatial beam spread. Each electron beam is assumed to fire immediately after the previous one has finished machining the surface. The heating sequence is designed such a way that the laser is first used on to preheat the workpiece to a certain “threshold” temperature. Then the machining process starts by turning on the first electron beam to thermally drill a nanohole. Once the first location reaches to the “evaporation” condition, the first electron beam is stopped and the second beam follows immediately. While the machining process is continuing, the first evaporated region suffers condensation and cooling. In this simulation, the workpiece is preheated to 700 K by the laser before an electron beam is initialized, and continues to heat the workpiece throughout the entire process. This temperature is chosen to avoid excessive heating from the laser at the workpiece-substrate interface, which may subsequently melt the entire workpiece. In addition, the heating spots are separated far enough apart that two adjacent sequential electron-beam profiles do not overlap. Therefore, by turning off the electron beam and moving it to the next location, the beam will not interact with the voids created previously. The laser is used alone to heat the workpiece for the first 0.12 ns; after that electrons start bombarding. The sequential electron machining can also be obtained by using an array of nanoprobes, which can be turned on and off in a controllable manner. Figure 1.6 depicts one possible configuration of CNT arrays which can be used for this purpose. Figure 5.15 shows the temperature distribution of the workpiece and the substrate at 0.42 ns when the probe at location #21 (see Fig. 5.14) is turned off after evaporation occurs. The dark blue dots on top of the surface are the evaporated regions or empty voids created by the electron beams. It is interesting to note that this machining scenario does not necessarily produce uniform nanoholes, as evident in Fig. 5.15. One of the reasons is because the non-uniform spatial heating by the laser. However, this effect seems to be minor, compared to the non-uniform heating and cooling at various locations due to electron beams. The second nanohole created is actually larger than the first one since the electron beam heating on the first location further raises the temperature at the second location by conduction and by laser heating. As a result, the second electron beam, which has the same power as the first beam, removes wider and deeper area when evaporation starts. In order to create uniform nanoholes, one possible way is to switch off all the heating sources after the first evaporation occurs and wait for the workpiece to reach thermal equilibrium with the ambient condition before proceeding to machine the next hole. The transient behavior of temperature at various locations on the surface of the workpiece, specifically for points 1–5, 17, 19, and 21, are depicted in Fig. 5.16. For the first 0.12 ns, the laser is on to preheat the top center of

5.3 Sequential Patterning Using Electron Beam

133

Fig. 5.15. Temperature profiles (K) of the workpiece on top of the substrate. These snap shots are for 0.42 ns after the laser beam heating is started. The electronbeam is turned on after the laser heats the workpiece for about 0.12 ns. (a) The 3-D temperature distribution. (b) The top view of the geometry showing the heated areas. (c) The A–A cross-section indicating the temperature profile from the side. (d) The same as (c), but for the B–B cross-section

the workpiece to approximately 700 K. The first electron beam is activated following that. A drastic increase of temperature is observed in all the specified locations as a result of energy transfer from electrons to the material, as seen in Fig. 5.16. 5.3.5 Comments We have presented a detailed methodology for forming nanostructures in a sequential fashion using an array of electron beams. A MC method based on CSDA was used to model the EBTE to determine the electron-energy deposition distribution rates within a thin gold film. A laser beam was employed in parallel to provide additional heat source to help the machining process. The dimensions of the nano-pattern created on the workpiece were about 200 nm × 300 nm. An electron beam with power of 0.25 W was shown to be capable of micro/nanomachining within 0.5 ns of actual time if assisted by a 0.35 W laser at a wavelength of 355 nm. Such an intricate pattern would not be obtained if a laser-beam was used alone since the lateral spread of photons

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Fig. 5.16. Transient temperature (K) of (a) Points 1–5, and (b) Points 17–21 (see Fig. 5.15 for positions of these points). Data are obtained from Wong et al. [211]

at the given wavelength would cover roughly the entire pattern. It should be understood, however, that the values we report are limited to the cases and the assumptions made here, including the electron beam profile, gold layer uniformity, gap distance between the probe and the workpiece. Additional simulations need to be carried on if a more specific case is to be considered. For the simulations given in this section, we used the Fourier heat conduction model, implying that electrons and phonons always exist at the same temperature. This approach, which provides a simple and easily tractable methodology, can be relaxed by considering a TTM. The energies of electrons incident on the workpiece can be assumed to be transferred to the electrons inside the workpiece first, and consequently increase the electron temperature. Equilibrium is then established between electrons and phonons when electrons transfer excessive energy to phonons. The TTM is more realistic for micro/nanomachining applications, especially for short time scales. We will explore such a TTM approach in the next chapter.

6 Two-Temperature Model Coupled with e-Beam Transport

We have so far outlined a methodology to determine thermal transport within a metallic workpiece subject to incident electrons from an electron beam. Electron-beam propagation in the workpiece was analyzed by solving the Boltzmann transport equation (BTE) with a Monte Carlo (MC) method. MC simulations provided the electron-energy deposition profile, which then served as the heat source term in the Fourier heat conduction equation. The Fourier law, however, does not distinguish electron and lattice temperatures. This means that the initial transient where electrons interact with matter cannot be resolved. To remedy this, a different and physically more realistic methodology, dubbed “two-temperature model” (TTM), can be used. The TTM approach is the focus of this chapter. The TTM is coupled with an alternative MC simulation based on discrete inelastic scattering (DIS) approach and include secondary electron generations within the workpiece after the first scattering event takes place. This new MC method replaces the continuous slow-down approach used for the simulation of the electron-beam propagation, and allows a more flexible and accurate modeling of micro/nanomachining.

6.1 Two-Temperature Model An energy-beam machining process is based on the interactions between a workpiece and external heat sources. Photons from a laser beam or electrons from an electron beam incident on a target workpiece transfer their energies to the electrons within the lattice first. This process causes substantial change in the electron temperature within the material compared to the lattice temperature. Eventually electrons and phonons reach to equilibrium by electron–phonon collisions; however, the difference in phonon and electron temperatures within the first few nanoseconds of machining may have significant impact on robustness of the material removal at nanoscale. Modeling of this phenomenon may require separation of the energies of phonons and electrons. This means that the conservation of energy equations are to be written

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6 Two-Temperature Model Coupled with e-Beam Transport

separately in terms of electron and phonon temperatures. Such a model is usually referred to as the TTM [30, 76, 197, 222]. The TTM is expressed in two coupled energy conservation equations: Ce

∂Te = −∇ · (ke ∇Te ) − Ge−ph (Te − Tph ) + ST , ∂t ∂Tph = −Ge−ph (Tph − Te ) . Cph ∂t

(6.1) (6.2)

The subscripts “e” and “ph” denote that of electrons and phonons, respectively. The C’s are the heat capacities of the energy carriers, and ke is the thermal conductivity of electrons. Equation (6.1) conserves the electron energy, which is coupled with the phonon-energy equation (see (6.2)) through the electron–phonon coupling constant, denoted as Ge−ph . The external heat generation term, ST , is included in (6.1). It is the electron-energy deposition distribution obtained from the solution of the BTE via MC simulation for the electron beam. The TTM can be solved numerically using a finite-difference method, where the first-order time and the second-order space discretization are employed. We discretize the TTM in our simulations using a centered-difference approximation in space and a forward Euler approximation in time. The discretized (6.1) and (6.2) are then written as: +1 P (Te )P m,n,o − (Te )m,n,o ∆t 1  2 +1 P +1 P +1 (∆x) = − ke (Te )P m+1,n,o − 2(Te )m,n,o + (Te )m−1,n,o  1 +1 P +1 P +1 (∆y)2 − ke (Te )P m,n+1,o − 2(Te )m,n,o + (Te )m,n−1,o  ' +1 P +1 P +1 2 − ke (Te )P m,n,o+1 − 2(Te )m,n,o + (Te )m,n,o−1 (∆z)   +1 P P +1 − Ge−ph (Te )P m,n,o − (Tph )m,n,o + Sm,n,o ,

Ce

Cph

(6.3)

+1 P   (Tph )P m,n,o − (Tph )m,n,o +1 P +1 = −Ge−ph (Tph )P m,n,o − (Te )m,n,o . (6.4) ∆t

These two equations are for a node (m, n, o) where m, n, and o correspond to the x-, y-, and z-directions indices, respectively. Following the procedure given in Sect. 5.2.5, two separate systems of equations can be formed from (6.3) and (6.4). They are to be solved iteratively in order to obtain the solution to P +1 is the heat generation due to the electron-energy the TTM. In (6.3), Sm,n,o deposition from the electron beam. This quantity is determined from the MC simulation.

6.2 Problem Description and Assumptions

137

6.2 Problem Description and Assumptions We will consider a simple example to discuss the TTM coupled with the MC solution of electron-beam propagation. Figure 6.1 depicts the schematic of the machining process within a 3-D rectangular workpiece, although only 2-D workpiece is shown in the figure. The electron beam is due to an electron gun or field emission from a nanoprobe and incident on the surface of the workpiece perpendicularly. These electrons penetrate into the workpiece, and transfer their energies to the material electrons first. This process elevates the energies of electrons, while phonons remain “cold,” since the electron beam does not directly interact with them. This initial process, which typically takes place within picoseconds (ps) or less after the incidence of the electron beam, creates a thermal non-equilibrium between electrons and phonons. Thermal equilibrium between electrons and phonons is then restored by energy transfer between these heat carriers. For capturing this phenomenon, the TTM is used. In the simulations we assume that the workpiece has perfect lattice structure and finite dimensions. Similar to the modeling strategy used in the previous chapter, the electron beam is assumed to have a spatial Gaussian distribution with a 1/e2 radius. During the process, local material properties may change as a function of the transient temperature; therefore we consider the temperature-dependent properties. The heating process is considered to take place inside a vacuum chamber. The radiation loss is neglected based on

Fig. 6.1. The simple schematic of the case study of a workpiece as it is exposed to the electron-beam heating

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6 Two-Temperature Model Coupled with e-Beam Transport

several a priori calculations, where we did not observe any significant impact of radiation on heat transfer during the machining process. Detailed investigation of radiative heat transfer may be required to study this effect in-depth for longer machining applications. The MC simulations for the electron-beam transport used here is the DIS method, which is different than that outlined in Chap. 5. A brief summary of the specific MC simulation based on the DIS approach is given next. The numerical strategy used in this chapter is similar to that given in Chap. 5. The major difference between is the replacement of the Fourier heat conduction equation with the TTM. A typical grid scheme considered in these simulations is depicted in Fig. 5.12 where it consists of uniform and nonuniform grid spacings. Results from the MC simulations for the electron-beam transport are obtained within the uniform grid. Electron deposition within this uniform grid is mapped onto flexible grid for lattice transport calculations.

6.3 Electron-Beam Monte Carlo Simulation The DIS method, as the name implies, treats inelastic scattering as discrete events. The amount of electron energy lost is independent of the distance traveled. In the DIS formulation, the energy loss function is derived from the experimental optical data for the workpiece, and the differential inelastic scattering cross sections are generated accordingly. Therefore, the energy loss function describes the responses of electrons and atoms in a medium as a whole when the medium is exposed to an external energy source. Since the inelastic scattering cross-sections are derived from the measured optical data, this approach is typically more accurate compared to the continuous slow-down approach (CSDA), especially when the electron energy is low. Simulation procedures for the DIS method are given in Chap. 3 and the required properties for the simulation are discussed in Chap. 4. We can easily implement the generation of secondary electrons and their propagations in the DIS MC simulation [48, 93, 105]. A secondary electron is “born” if the amount of energy transferred from the electron beam to an electron inside the material is greater than the Fermi energy level. Therefore, secondary electrons are generated everywhere within the workpiece when machining process starts. This creates a cascade effect since a secondary electron with high energy can produce another secondary electron along its path of propagation. We can easily account for this effect within the MC simulation since secondary electrons follow the same scattering probabilities as primary electrons (i.e., electrons originated from the electron beam). In the simulation, we assume that all the statistical ensembles, including secondary electrons, propagate until they either exit the medium or their energies fall below that of the surface barrier. The following discussions are based on the recent simulations obtained by Wong et al. [212].

6.4 Results and Discussions

139

6.4 Results and Discussions Physical dimensions and thermophysical properties considered for the 3-D TTM simulation are listed in Table 6.1 and Appendix C. The workpiece (anode) is 1 µm thick and 8.5 µm wide. The workpiece material is assumed to be gold under a vacuum condition of 10−8 torr. The melting temperature for gold is around 1,336 K [84]; however, the sublimation temperature is around 1,080 K for a vacuum pressure of 10−8 torr [77]. Under these conditions, gold tends to sublimate before melting is actually achieved. With enough high-energy electrons from the electron beam, it is possible to sublimate the workpiece. This can be achieved during the period that thermal equilibrium between electrons and phonons does not exist. Detailed experimental and theoretical studies on how electron and phonon temperatures change are required to correctly account for sublimation under this situation. To avoid this complication, the maximum temperature in the simulations is limited within 1,080 K. 6.4.1 Electron-Beam Deposition Profiles The electron beam incident on the workpiece is considered to have the energy of 500 eV with a 500 nm radius Gaussian profile. The energy distribution is expressed per unit volume, which is normalized by the total incident electron energy. A typical 3-D electron-energy deposition profile computed using the DIS method is shown in Fig. 6.2. Additional deposition profiles (given in terms of the 2-D cross-section in the x–z plane at y = 0) under different computational parameters are depicted in Fig. 6.3. Similar to the MC results obtained in Sect. 5.2.8, we observe that the deposition amount is usually maximum at a point below the top surface of the workpiece, as is evident in Figs. 6.2 and 6.3. Table 6.1. Expressions for thermophysical properties (see Appendix C for additional details) Property

References

Electronic heat capacity Ce = 70Te (J m−3 K)

[9, 87, 197, 225]

Electronic thermal conductivity  2 1.25  2  ϑe + 0.16 ϑe + 0.44 ke = Cϑe ; (ϑ2e + 0.092)0.5 (ϑ2e + 0.16ϑl ) ϑe = Te /TF ; ϑl = Tl /TF

[87]

Phonon heat capacity −1 Cl = 8.33 × 103 Tl (J m−3 K )

[9, 197, 225]

Electron–phonon coupling constant Ge−ph = 2.1 × 1016 (W m−3 K−3 )

[87]

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6 Two-Temperature Model Coupled with e-Beam Transport

Fig. 6.2. Electron-energy deposition distribution for Rbeam = 500 nm and E0 = 500 eV is shown. The numbers given in the figure are in terms of normalized quantities, per unit nm3 . Data are obtained from Wong et al. [212]

For the same incident beam profile and initial kinetic energy of the beam, the deposition profile computed using the DIS method is in general more diffusive than that computed using the CSDA. This is mainly because of the presence of the secondary electron generation mechanism considered in the DIS method, which further diffuses the electron energy deposited in the workpiece. Even though the overall trends observed for electron deposition rates are similar to those outlined in Chap. 5, there are a few key differences. First, if the electron beam has a wider spatial spread, the amount of electron energy deposition drops. This is easily observed by comparing the magnitude of the values depicted in Fig. 6.3a, b (i.e., Rbeam = 100 nm → 500 nm). Second, the maximum amount of the deposition increases when the beam energy is lowered, although the penetration depth of the beam is reduced as well. Figure 6.3b–d also depicts this effect. Overall, DIS is a more realistic approach than the CSDA for determining the electron-beam penetration during the specific applications considered here. 6.4.2 Two-Temperature Model Predictions Using the electron-energy deposition distributions obtained from the MC simulations, we can now compute the temperature distribution inside the workpiece. The electron and phonon energy equations in the TTM are solved simultaneously to obtain the temperature distribution, assuming that the deposition profile remains unaltered throughout the entire heating process.

6.4 Results and Discussions

141

Fig. 6.3. Electron-energy deposition distributions for four different cases are depicted: (a) Rbeam = 100 nm and E0 = 500 eV, (b) Rbeam = 500 nm and E0 = 500 eV, (c) Rbeam = 500 nm and E0 = 420 eV, and (d) Rbeam = 500 nm and E0 = 350 eV. The numbers given in the figure are in terms of normalized quantity, per unit nm3 . Data are obtained from Wong et al. [212]

The use of the TTM was deemed to be computationally expensive for determining the entire transient temperature profiles ranging from femtoseconds (fs) to nanoseconds (ns) with fs time steps. Instead, a different methodology was adapted, where the TTM simulation was repeated using several different time steps (i.e., from fs to ps). A series of transient curves can be generated using this methodology. Later, these results can be superposed to determine the entire transient temperature distribution. For instance, a time step of 1 fs (= ∆t) is used to obtain the transient result until 0.1 ps and then the same simulation is repeated with ∆t = 10 fs until 2 ps. Superpositioning these two transient curves yields a transient curve ranging from 1 fs to 2 ps. The time evolution of temperature in the workpiece is desired in order to observe the machining process. For this purpose, the transient maximum

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6 Two-Temperature Model Coupled with e-Beam Transport

Fig. 6.4. The maximum transient electron- and phonon-temperature curves of the target workpiece in unit of Kelvin for the case where Rbeam = 500 nm and E0 = 500 eV are shown. The current of the electron-beam used is 0.5 mA. Data are obtained from Wong et al. [212]

temperature within the workpiece is tracked. Figure 6.4 shows an example of the transient maximum temperature profile in gold with dimensions of (8.5 µm × 8.5 µm × 1 µm). The entire transient curve is constructed using multiple time steps as described previously. During the heating process, electrons have higher temperature than phonons, which is as expected since the energies of the incident electrons are directly transferred to the material electrons. Although there is a possibility that the electrons transfer their energy to the phonons directly, this is not considered in the conduction algorithm, which is consistent with the basis of DIS formulation of the MC method. In the MC simulations, we assumed that electrons from the electron beam can lose their energies via an inelastic electron-electron scattering process (i.e., collisions between the electron beam and electrons orbiting atoms). Atoms are responsible for deflecting the electron beam elastically, and hence they do not gain any energy from the beam. Therefore, the electron beam deposits energy onto electrons only in this analysis. In order to determine the amount of energy directly absorbed by phonons from the beam, we need to include the electron–phonon scattering in the electron-beam MC simulation. This will be left to the future studies, when different workpiece materials need to be considered in addition to metals, such as gold in the present case. The electron and phonon temperatures converge into a single temperature after 10 ns into the heating process, as seen in Fig. 6.5. This implies that the temperature profiles predicted using the TTM should in fact be identical

6.4 Results and Discussions

143

Fig. 6.5. A close-up view of the transient temperature profiles given in Fig. 6.4

to that simulated using the Fourier law beyond 10 ns. This is consistent in our various simulations using different electron-beam energies, currents, and workpiece thicknesses. It is important to note, however, the time at which electron and phonon temperatures collapse into single temperature depends greatly on the coupling constant between electrons and phonons (i.e., Ge−ph ). Should this quantity change, the above condition may differ. The precise value of this coupling factor depends on the actual experiments, and need to be considered in tandem with the experimental efforts for a specific problem. Figure 6.6 shows electron temperature distribution inside the workpiece after 0.1 µs into the heating process using the electron beam. At this particular instance, the electron and phonon temperatures are identical; therefore, only the electron temperature distribution is shown. In general, the electron temperature distribution differs from that of phonons when the time of interest is less 10 ns as evident in Fig. 6.5. In order to verify that solutions of the TTM and the classical heat equation are identical after 10 ns, the Fourier heat conduction equation is used to predict temperature profile for the same case given in Fig. 6.4. The simulation results predicted using this approach are plotted against that of the TTM, which is shown in Fig. 6.7. The representative simulations suggest that temperatures between electrons and phonons in the gold layer reach equilibrium after about 10 ns, implying that the use of Fourier law of heat conduction is acceptable for simulations beyond 10 ns. This conclusion is, however, for the specific set of parameters used here; for different materials with different electron–phonon coupling constants, the time at which the Fourier law is applicable may be different.

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6 Two-Temperature Model Coupled with e-Beam Transport

Fig. 6.6. The electron-temperature distribution (K) of the target workpiece for Rbeam = 500 nm and E0 = 500 eV is shown. The current of the electron-beam used is 0.5 mA, and the temperature profile shown is the snapshot at t = 0.1 µs

Fig. 6.7. The comparison between maximum temperatures predicted using the Fourier heat conduction equation and the TTM is shown. The parameters used are Rbeam = 500 nm, E0 = 500 eV, Ie = 0.5 mA, and Lz = 1,000 nm. Data are obtained from Wong et al. [212]

6.5 Comments

145

6.5 Comments In this chapter, we have discussed the details of the TTM to simulate micro/nanomachining based on an electron beam incident on a gold workpiece. The propagation of electrons in the workpiece and their interaction with the lattice were considered separately by coupling the DIS electron-beam MC simulation with the TTM. The representative simulations suggest that temperatures between electrons and phonons in the gold layer reach equilibrium after 10 ns. Therefore, the Fourier law can be used if the process duration is in the order 10 ns or larger, as beyond that value there would be little difference between the electron and phonon temperatures. Note that for other geometrical configurations, this cut-off value maybe slightly different. Although the TTM separates electron and phonon temperatures, it neglects the possibility of electron flow and charge accumulation inside the workpiece. In order to include the effects of electron flow, we need to use more general electron–phonon hydrodynamic equations (EPHDEs), which are discussed next, in Chap. 7.

7 Thermal Conduction with Electron Flow/Ballistic Behavior

In Chaps. 5 and 6, we outlined two different numerical procedures to model thermal transport during electron-beam based micro/nanomachining processes. We used the Fourier law and the two-temperature model (TTM), considered both electron and laser beams, and determined the temperature profile inside a gold workpiece. Also, we predicted the times required to melt and evaporate the gold layer for a number of design parameters. The solution of the electron-beam transport equation (EBTE) obtained using a Monte Carlo (MC) simulation was coupled with the conduction equations. These predictions suggested that micro/nanomachining is achievable with an electron beam or with field emitted electrons if the assumed simulation parameters can be realized in the experiments, as outlined in Chap. 5. In the simulations, however, we have always assumed that the effect of charge accumulation inside the workpiece was negligible. If the workpiece is electrically grounded, then electrical flow inside workpiece may change the temperature distribution. In order to study these effects, we need to use electron–phonon hydrodynamic equations (EPHDEs). In this chapter, we first introduce EPHDEs and then use them to describe electrical and thermal behaviors of metal semiconductor field effect transistors (MESFETs). The computational methodology discussed below can easily be adapted to study the effect of electrical flow during the micro/nanomachining process. This solution can also be coupled with the EBTE, similar to coupling process described in Chaps. 5 and 6. In the second part of the chapter, a new MC methodology is introduced for modeling electronic thermal conduction. This new approach allows us to observe the degree of deviation of a semi-ballistic temperature profile from that assumed by a diffusion approach (i.e., linear temperature profile). Results from this study can be of importance in determining the limit at which the diffusion approach for heat carriers fails. When a MC simulation for thermal conduction replaces the heat conduction equation, “effective transport properties” are not required anymore since the MC method directly uses scattering properties of heat carriers instead. These scattering properties are usually

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7 Thermal Conduction with Electron Flow/Ballistic Behavior

derived from wave theories by closely examining wave interactions between heat carriers and scatterers. Thus, nanoscale thermal conduction can be studied properly without using any bulk properties. This new MC approach will be beneficial to future micro/nanomachining modeling efforts as it can be easily modified to include heating by an electron beam.

7.1 Electron–Phonon Hydrodynamic Modeling Electrons emitted from a nanoprobe usually have significant amount of momentum and energy. A typical micro/nanomachining process involves propagations of these electrons inside a target material. The collisions of these incoming energetic electrons with electrons circulating atomic nuclei (i.e., outer-shell electrons and conduction electrons) and lattices (i.e., phonons) require detailed consideration of momentum and energy transfer. Modeling of this phenomenon requires a separate analysis of electron and phonon temperatures as electrons do not reach equilibrium with phonons instantaneously. Although the TTM and the dual-phase lag model (DPLM) allow differentiation of electron temperature from phonon temperature, the momenta of electrons and the electrical behavior of the material are not accounted for in their formulations. In addition, electrons are negatively charged; therefore, the incoming electrons (i.e., originated from a probe) induce an electric field inside the material due to locally unbalanced positive and negative charges. This does not only affect the propagation of electrons in the medium, but it also alters the energy transfer between electrons and phonons. The physics of the problem is correctly accounted for only by using the EPHDEs where all the thermal and electrical characteristics of the material are included in the governing equations. In the following sub-sections, the EPHDEs are used to model the electrical and thermal behaviors of the MESFETs. Although this application is not directly related to the micro/nanomachining process, the computational code built for this purpose can be served as a foundation for our future machining modeling. We will first perform numerical verifications of the code on results published for MESFETs in the literature. Then the code will be coupled with electron-energy deposition profiles obtained in the electron-beam MC simulation, although this will be left as a future studies. 7.1.1 Governing Equations for Electrons and Phonons Electrical and thermal behaviors of a MESFET can be described by using the EPHDEs (see Sect. 2.8.3 for the derivation). The MESFET is typically a semiconductor. In Chaps. 5 and 6, the workpiece considered was gold, and therefore only phonons in the acoustic mode were considered. Only one phonon energy equation was used for modeling the machining process. However, phonons in a MESFET can be both optical (LO) and acoustic (A) phonons. As a result, the

7.1 Electron–Phonon Hydrodynamic Modeling

149

phonon energy equation is divided into two separate conservation equations for LO and A modes, respectively. Detailed discussions on these equations are given in Chap. 2 and we shall not repeat them here. Only the governing equations are presented. Referring to Sect. 2.8.3, the complete EPHDEs consist of the following governing equations: ⇀ e (ne − n+ ); E = −∇ V, (7.1) εe 
∂ne ⇀ + ∇⇀ = n˙ e,gen , v n (7.2) · d,e e r ∂t

kB n˙ e,gen ⇀ e ⇀ 1 ⇀ ⇀ + v d,e · ∇⇀ (n T ) − E − ∇ + v = − v d,e , ⇀ e e r d,e me me n e r τm ne (7.3) 
∂Te ⇀ + ∇⇀ · v d,e Te r ∂t  T −T
1 2 ⇀ e LO = Te ∇⇀ − ∇ k ∇ · v + ⇀ · ⇀ Te T,e d,e r r 3 3ne kB r τe−LO

(7.4) 2 1 1 n˙ e,gen me vd,e Te − TA 2 + − − + − τe−A τm τe−LO τe−A ne 3kB



2 n˙ e,gen ˙ e,gen , Te + W − ne 3ne kB

∇2 V =

⇀

∂ v d,e ∂t

2 ne me vd,e 3ne kB (TLO − Te ) TLO − TA ∂TLO = − − , ∂t 2CLO τe−LO 2CLO τe−LO τLO−A

 n e me v 2
1 ∂TA d,e = + · k ∇ ∇⇀ ⇀ TA T,A r ∂t CA r 2CA τe−A

3ne kB (TA − Te ) CLO (TA − TLO ) − . − 2CA τe−A CA τLO−A

(7.5)

(7.6)

The first equation in the EPHDEs is the Poisson equation, which is for the electrical potential distribution, hence the electric field, inside the material. The second, third, and fourth conservation equations describes electron continuity, momentum, and energy conservation in time and space. The final two equations are the LO-phonon and the A-phonon energy conservation equations, respectively. Physical meaning of each term appeared in these governing equations are discussed in details in Chap. 2. The EPHDEs above are derived based on the assumption that external electrons, for example those from an electron beam, are being injected into the system. The density of electrons generated is denoted as n˙ e,gen and the ˙ e,gen . For micro/nanomachining volumetric energy generated is given as W applications, these terms need to be evaluated carefully. In the present case, we

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7 Thermal Conduction with Electron Flow/Ballistic Behavior

are only interested in modeling electrical and thermal responses of MESFETs; therefore, these variables are not required in the simulation. Eliminating these terms from the governing equations for electrons yields: 
∂ne ⇀ + ∇⇀ = 0, (7.7) · v n d,e e r ∂t ⇀ ⇀ e ⇀ kB v d,e ∂ v d,e ⇀ ⇀ + v d,e · ∇⇀ ∇⇀ (ne Te ) − , (7.8) v d,e = − E− r ∂t me me n e r τm 
∂Te ⇀ + ∇⇀ · v T d,e e r ∂t  T −T
1 Te − TA 2 ⇀ e LO = Te ∇⇀ − k ∇ ∇ − · v + ⇀ Te ⇀ · T,e d,e (7.9) r r r 3 3ne kB τe−LO τe−A

2 me vd,e 1 1 2 − − , + τm τe−LO τe−A 3 kB The EPHDEs for MESFETs now comprise of (7.1), (7.5)–(7.9), which need to be solved simultaneously to predict electrical and thermal of electrons and phonons in the material. The numerical discretization of these equations can be obtained using an upwind method for any of the advection terms and a central difference for the heat diffusion terms. Either implicit or explicit discretization schemes can be used. These numerical schemes are readily available elsewhere [11, 12, 90, 151, 170, 184]. Below we will outline a series of predictions based on our preliminary EPHDEs solver. 7.1.2 Physical Domain and Boundary Conditions The physical domain considered for modeling a MESFET is depicted in Fig. 7.1. A MESFET, made of GaAs, consists of two layers; the first, top one, is an active layer and the second is semi-insulating. The active layer has an electron concentration in the order of 1017 cm−3 , whereas the lower semiinsulating layer carries three orders of smaller number of electrons. There are three terminals on top of the active layer, including the source, the drain, and the gate terminals. In this study, each terminal is assumed to be 500 nm wide. The active layer has a thickness of 150 nm while the semi-insulating layer is 250 nm thick. When voltages in the terminals are applied correctly, electrons flow from the source terminal to the drain terminal. While electrons are propagating through the material, a negative voltage can be applied at the gate terminal. Since the gate terminal is insulated electrically and thermally, the negative voltage drives electrons under the gate area away, and hence a depletion region is created where the density of electrons is much smaller compared to the rest of the layer. As a result, the gate terminal acts as a means for controlling the electrical current in MESFETs. The electric field applied to a MESFET is typically high where electrons become energetic in very short amount time. As these electrons propagate

7.1 Electron–Phonon Hydrodynamic Modeling

151

Fig. 7.1. (a) A two-dimensional view of a MESFET with an active layer deposited on top of a semi-insulating layer and (b) the corresponding boundary conditions given in the figure

through the channel, energy transfer occurs between electrons and lattices, and hence lattice vibrations are induced (or, phonons are created). Heat generation occurs in the MESFETs under such operating condition where hot spot is usually observed between the gate and the drain terminals. When temperature of a MESFET increases, thermophysical properties are altered and therefore electrical flow is affected. This poses an interesting challenge for engineering design of these devices. Additional details about the operating conditions and physics of MESFETs will not be discussed here as they are readily available elsewhere [60, 110, 122, 184, 193]. The boundary conditions required to solve the EPHDEs presented in Sect. 7.1.1 for a MESFET are given in Fig. 7.1. All the boundaries in the domain are insulated except those under the source, gate, and drain terminals. The source, gate, and drain terminals have known applied voltages. The temperature at the source and drain terminals are assumed to be the ambient temperature, i.e., 300 K, at all time, and the electron densities at both terminals are constant.

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7 Thermal Conduction with Electron Flow/Ballistic Behavior

7.1.3 Thermophysical Properties for the Simulation Thermal conductivities and relaxation times for electrons and phonons need to be known a priori for the solution of the EPHDEs. The thermal conductivity of electrons can be expressed as [122]: ke = (r + 2.5)

2 ne kB τm Te , m∗e

(7.10)

where r is a parameter which varies depending on the material of interest. The thermal conductivity of electrons is a function of the electron density, the electron momentum relaxation rate, and the temperature. Since all these quantities vary locally, so does the electronic thermal conductivity. This necessitates consideration of temperature-dependent properties in solving the EPHDEs. The electron-momentum relaxation rate, τm , for GaAs as a function of the electron energy, we , can be determined as outlined by Tomizawa [193]: τm (we ) =

1 + (we /wc )6 , Cimp + C1 + D1 (we + w0 )(we /wc )6

Cimp = 1.6



NI 1019

0.4

× 1013 (s−1 ).

(7.11)

(7.12)

Here, NI is the impurity concentration in the layer in units of cm−3 . For constant lattice temperature of 300 K, the values for C1 , D1 , wc and w0 , are 2.9 × 1012 , 1.7 × 1014 , 0.3 eV, and 0.039 eV, respectively. The electron energy can be obtained from its thermal and velocity components: we =

3 1 kB Te + me ve2 . 2 2

(7.13)

For non-constant lattice temperature, the relaxation rate is inversely proportional to the lattice temperature such that:   300 1 + (we /wc )6 τm (we ) = . (7.14) Tph Cimp + C1 + D1 (we + w0 )(we /wc )6 Tomizawa [193] also provided a similar expression for the electron–phonon relaxation rate, given as: τe−ph (we ) =

(we + 0.2)[1 + (we /wc )8 ] , F1 + G1 (we /wc )8

(7.15)

where F1 and G1 are given as 0.4 × 1012 and 2.8 × 1012 , respectively. Additional data for the electron-momentum and the electron–phonon relaxation times are given by Carnez et al. [27]. They provided three

7.1 Electron–Phonon Hydrodynamic Modeling

153

curves/figures for we versus Ess , vss versus Ess , and m∗ versus Ess . In order to determine the required relaxation times, first the electron energy, we should be know, which is obtained from (7.13). Using their first figure for we versus Ess , the quantity Ess can be determined. With Ess known, vss and m∗ can be obtained from the other two curves. Unfortunately, the source data for all those curves cannot be readily obtained; therefore, we did not reproduce them here. Instead, the required data were directly interpolated from these figures as needed to obtain the relaxation times. Accordingly, both relaxation times are computed using the following formula: m∗ (we )vss (we ) , qEss (we ) we − w0 . τe−ph (we ) = qEss (we )vss (we ) τm (we ) =

(7.16) (7.17)

The results for the relaxation times according to the two sources are plotted in Figs. 7.2 and 7.3. Note that there are large discrepancies between the calculated relaxation times as a function of electron energy. The reasons for this difference still remain unknown and an in-depth investigation would be highly desirable.

Fig. 7.2. The electron momentum relaxation rates as a function of electron energy for various electron concentrations are depicted. The relaxation rates are derived from the MC simulation of the electron propagation. The material is GaAs and the effective mass used in the simulation is 0.067 m0 , unless otherwise specified

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7 Thermal Conduction with Electron Flow/Ballistic Behavior

Fig. 7.3. The electron–phonon relaxation rates as a function of electron energy obtained from two different sources are depicted. The relaxation rates are derived from the MC simulation of the electron propagation. Note that there is a large discrepancy between the results of the two simulations

The expressions for the heat capacities of optical and acoustic phonons of GaAs are [122]:

1.94 θLO (J m−3 K−1 ), TLO 1.948 θD 5 4 CA = 9.17 × 10 − 4.40 × 10 (J m−3 K−1 ), TA

CLO = 3.06 × 105 − 2.40 × 104



(7.18) (7.19)

where the values for θLO and θA are assumed to be 429 and 344 K, respectively. 7.1.4 Results and Discussions In this section, we present a series of simulation results to describe the electron–phonon coupling in a GaAs MESFET. For the sake of simplicity, we assume that electrons are first transferring energy to LO-phonons, and the A-phonons subsequently gain energy through the hot optical phonons. Hence, the electron–phonon relaxation time (i.e., τe−ph ) given above is assumed to be the electron-optical phonon relaxation time (i.e., τe−LO ).

7.1 Electron–Phonon Hydrodynamic Modeling

155

There are a total of four parameters involving relaxation times. They are τm , τe−LO , τe−A , and τA−LO for the momentum, the electron-optical phonons, the electron-acoustic phonons, and the acoustic-optical phonons, respectively. In deriving the governing equations, all these four parameters need to be included. However, τe−A is omitted in the following simulation since there is no information regarding the magnitude of this parameter. For τA−LO , 8 ps is chosen, as suggested by Fushinobu et al. [60]. Systems of equations are obtained and solved accordingly using the point successive-overrelaxation (SOR) numerical scheme. Solving equations of this kind may not be efficient using a SOR method and better solution schemes can be adapted. Increasing the speed of the convergence of the computational code may be achieved using various alternative approaches, for example, the Douglas–Gunn time splitting method [131]. In order to verify the validity of the numerical solution, the drain currents are first computed for various drain voltages and gate voltages. The source voltage is always fixed at 0 V. For all the cases, the simulation is run until a steady-state condition is reached, which occurs when the source and the drain currents reach the same magnitude. Figure 7.4 depicts the drain current as a function of the drain voltage and the gate voltage. The lattice temperature (i.e., both the optical phonons and the acoustics phonons) is set to room temperature throughout the simulation and the entire MESFET. We

Fig. 7.4. The drain current versus the drain voltage characteristic of a MESFET is shown. The lattice temperature is assumed to be constant, which is at 300 K

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7 Thermal Conduction with Electron Flow/Ballistic Behavior

made this assumption to compare our results with those of Yoganathan and Banerjee [220]. Results given in Fig. 7.4 closely represent those obtained by Yoganathan and Banerjee [220]. Sample figures for the electron temperature, the electron density, the potential, the current density, the optical phonon temperature, and the acoustic phonon temperature of the case where the drain voltage and the gate voltage are set to 3 and 0 V, respectively, are given in Figs. 7.5 and 7.6. These are typical simulation results for a MESFET operating at a given set of conditions. The results obtained from the solution of the EPHDEs are in agreement with the published data. This suggests that the approach discussed here is accurate and self-consistent, and can be adapted for the simulation of electron-based micro/nanomachining processes. However, the model needs to be modified to include electron-beam heating. Coupling the electron-beam MC simulation with the EPHDEs would require, as input, the electron concentration distribution in the medium and the corresponding energy density generated from an incident electron beam. These quantities can be easily obtained from a MC simulation by implementing minor modifications.

7.2 Thermal Conduction by Electrons via Monte Carlo Method All the transport equations that we have discussed so far are based on the assumption that the heat is conducted diffusively within a workpiece. This assumption is usually acceptable when the temperature of the workpiece is sufficiently high, implying that scattering of heat carriers is so strong that these carriers are not capable of conducting heat ballistically. However, when the scattering rate is low or the workpiece is at low temperature or the workpiece is very thin, heat carriers can propagate long distances within the material without being scattered. As a result, heat transport is no longer diffusive but ballistic or semi-ballistic in nature. Under this condition, temperature profile may not be linear, and may deviate significantly from the diffusion approximation. In this section, a MC simulation is outlined to describe thermal conduction by electrons inside thin gold films. With the use of this new MC method, we will be able to investigate the application limit of a typical diffusion approach. The basic MC simulation procedures have already been outlined and discussed in Sect. 3.4. We will not repeat them here, but focus on the scattering properties required for the simulations. After that, we will present the temperature profiles obtained for ballistic electron transport scenarios. The scattering properties required in the MC method for electronic thermal transport are obtained from derivations of electron-impurity scattering, electron–electron scattering, and electron–phonon scattering. In the case of using a semiconductor as the workpiece, these properties are readily available [55, 88, 193] as the electron transport phenomena inside semiconductors

7.2 Thermal Conduction by Electrons via Monte Carlo Method 0V

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4486 4271 4057 3842 3627 3412 3198 2983 2768 2553 2338 2124 1909 1694 1479 1265 1050 835 620 406

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Fig. 7.5. The electron temperature field, the electron concentration, and the potential distribution inside the MESFET after 40 ns of the simulations

have extensively been studied due to the rapid development of electronic chips at the micro- or nanoscale size. On the other hand, electron transport in noble metals at the nanoscale level is still under extensive investigation owing to the peculiar overlapping of the electronic band structures. Determining

7 Thermal Conduction with Electron Flow/Ballistic Behavior 0V

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Fig. 7.6. The electron current density, the optical (LO) phonon temperature and the acoustic (A) phonon temperature inside the MESFET after 40 ns of the simulations

the scattering rates for electrons as well as for phonons in metals is still a challenging research area. For the micro/nanomachining scenario discussed in this monograph, the workpiece is heated beyond melting temperature, which makes this problem different than those in the semiconductor industry. Thus, we need these

7.2 Thermal Conduction by Electrons via Monte Carlo Method

159

scattering properties to be valid within the temperature range considered. In the following discussions, the parabolic band structure of gold is taken into consideration. Also, we accounted for the electron–electron and the electron– phonon scattering rates while the electron-impurity scattering is neglected for the sake of simplicity. The scattering nature of electrons in gold is based on the selection rules presented according to the time-dependent perturbation theory, which is discussed by Ziman [224]. Using these scattering rates in the new MC method, we investigate the electron transport phenomena inside a gold film at the nanoscale. 7.2.1 Electron Band Structure The electronic band structure of electrons in any material is obtained by solving the Schr¨ odinger equation. The electronic band structures of transitional elements such as silver, copper, and gold are generally complicated owing to the peculiar overlapping and the anisotropic characteristics of the d-bands [36, 37]. Incorporating these band structures into a MC simulation requires extensive computing resources and power. Here, we will introduce a series of approximations in determining these properties. More accurate derivations can be carried out later to improve the accuracy of the MC simulation using more complex band structures. First, a parabolic band structure with an effective electron mass is assumed. This band is assumed to be independent of the wave number, but is a function of the electron wave vector, and is expressed as: E(k) =

¯ 2 k2 h , 2m∗

(7.20)

where ¯h is the Planck constant, k is the wave number, and m∗ is the effective mass of electron. In this case, m∗ is the effective mass of electron in gold, which is 0.6515 mo, mo being the electron rest mass (i.e., 9.1095 × 10−31 kg). In the simulation, electrons with energies larger than the Fermi energy, EF , are allowed to propagate; otherwise, they are assumed to be bonding strongly with atoms and not moving. 7.2.2 Electron–Electron Scattering There are two different types of electrons that need to be considered in predicting electron–electron scatterings: conduction1 electron-inner shell2 electrons 1

2

Conduction electrons or outer-shell electrons are electrons with energies larger than the Fermi energy. They behave like free electrons, as discussed by Ziman [224]. These are the electrons that are being simulated using the MC approach. Inner-shell electrons are electrons packed closely with nuclei, combinations which are referred to as atoms and they are usually not able to propagate freely.

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7 Thermal Conduction with Electron Flow/Ballistic Behavior

and conduction electron-conduction electrons. In general, coulomb forces of electrons pose long range effects, and consequently an electron may influence other electrons far from its present location. Thus, derivation of the electron–electron scattering rates is quite complicated, which is still an active research area. In our next set of simulations, we will only consider the electron– electron scattering for the conduction electrons, as they involve the short range interactions. The rate of scattering for these electrons given by Ziman [224] as: Pe−e



⇀

⇀

′

⇀

⇀

k , k ; k 1, k

′

1



"

"2 ⇀

⇀ ⇀ " " ⇀ ⇀ ⇀ ⇀ ⇀ "M k , k ′ ; k 1 , k ′ 1 " δ k ′ − k + k ′ 1 − k 1 + ⇀ g " " 
(7.21) × δ E⇀′ − E⇀ + E⇀′ − E⇀ ,

2π = h ¯

k

k

k

k1

1

where M is the transition matrix element. The delta functions are for the momentum and energy conservations corresponding to the scattering process ⇀ ⇀ between two electrons with wave vectors k and k 1 , and to energies E⇀ and k E⇀ . The wave vectors and energy of an electron after scattering is denoted k1

by the prime sign. ⇀ When the reciprocal lattice vector, g , is not considered in the momentum conservation, the scattering process is of Normal (N ) type; otherwise, it is a Umklapp (U ) process [9, 66, 224]. The N -processes conserve total electron momentum while the U -processes changes total electron momentum. If there were only N -processes present in the scattering mechanism, then the thermal conductivity of a material would be infinite since the total momentum of the carriers is unaltered. When U -processes are included, the material has finite thermal conductivity because the overall electron momentum is modified by those scattering processes. Electron–electron scattering is to be determined as ⇀

a function of the wave vector or the energy (i.e., k or E⇀ ) of a propagating k electron, and that requires the integrations of (7.21) over all other possible ⇀

⇀

⇀

wave vectors k 1 , and scattered wave vectors k ′ and k ′ 1 :



⇀  ⇀ ⇀ ⇀ 
⇀ ⇀ ⇀ ⇀ ′ ′ Pe−e k = Pe−e k , k ; k 1 , k 1 f⇀ 1 − f⇀′ 1 − f⇀′ dk ′ dk ′ 1 dk ′ 1 . k1

k

k

1

(7.22) The Pauli exclusion principle for the electrons is accounted by the terms
 
1 − f⇀′ 1 − f⇀′ . These integrations can be performed by assuming that k

k

1

the spherical parabolic electronic band structure prevails, as given by Ziman [224]. Thus, the scattering rate for the electron–electron N -processes becomes: N Pe−e (k)

= 2πℵ k

2



0

2k

1 dq, (q 2 + qs2 )2

(7.23)

7.2 Thermal Conduction by Electrons via Monte Carlo Method

161

where ℵ=−

Ξ2 e4 (kB T )2 π3 ¯ h4 vk3



∞

−∞

z 

e−(Ek −EF )/kB T e−z qs2 =

4πne2 . kB T

 dz, + 1 (ez − 1)

(7.24) (7.25)

Similarly, the U -processes for electron–electron scattering is derived as:  2k  ′ 2k 1 z U 2 Pe−e (k) = 2πℵ k dK dq. (7.26) 2 2 2 g−2k 2g g−q (K + qs ) In (7.24), Ξ 2 ≈ 1 for N -processes while Ξ 2 ≈ 0.01 for U -processes. In the formulation electron scattering mechanism is assumed to be elastic. This may not be entirely justified from the physical point of view for the electron–electron interactions. However, derivation of a more realistic electron–electron scattering rate is quite a challenging task, and cannot be justified without having a way to evaluate its accuracy with the experimental data, which are not available. Using these expressions in the simulations, we can predict the electron– electron scattering rates as a function of temperature for gold, as depicted in Fig. 7.7. The rates at different temperatures converge into a single curve for electron energies exceeding 6.0 eV, as the integral given in (7.24) converges to a single value regardless of the magnitude of Ek . Compared to the electron– phonon scattering rate near the Fermi energy, the electron–electron scattering rate is several orders smaller in magnitude, showing that the electron–electron interactions are not of importance for electron transport near the Fermi energy level. For the MC simulation discussed in this section, we did not consider any external sources, such as a short-pulsed laser or an electron probe, which causes an increase in the energy level of electrons to several electron-volts. The present simulations are for electrons with energies in the order of kB T above the Fermi level following the Fermi-Dirac statistics. 7.2.3 Electron–Phonon Scattering The electron–phonon interaction involves the collision of a propagating electron with an atom within a lattice which leads to the creation or destruction of a phonon. During these collisions, an electron can gain or lose energy. Accurate calculation of the electron–phonon scattering rate requires the knowledge of the actual electronic band structure, which is not a trivial task. However, with the assumption of an isotropic, parabolic band structure, the electron– phonon scattering rate can be obtained easily, as discussed by Ziman [224]. Here, we will follow this simple approach.

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7 Thermal Conduction with Electron Flow/Ballistic Behavior

Fig. 7.7. The electron–electron scattering rate as a function temperature for gold as computed using (7.23) and (7.26). Data are obtained from Wong and Meng¨ uc¸ [209]

Generally, the probability of electron–phonon scattering per unit time can be obtained from the time-dependent perturbation theory as [224]: Pe−ph



⇀

⇀

k, k ′ ; q , p ⇀



=

"

"2
 " ⇀ ⇀ 2π "" ′ " M k, k h ω δ E ⇀ ⇀ − E⇀ ± ¯ " q ,p k k′ h " ¯

⇀ ⇀ ⇀ ⇀ × δ k′ − k ± q − g .

(7.27)

The Bardeen’s self-consistent form for the scattering matrix element M is employed in the simulations. Note that there are two delta functions appearing in (7.27). The first implies the energy conservation of the electron–phonon collision, while the second indicates the momentum conservation in the process. In a MC simulation, the scattering rate as a function of incident electron ⇀ wave vector k is required. This can be achieved by performing two integrations ⇀′

over the scattered electron wave vector k and the entire phonon wave vector ⇀ q and then summing up the results for all three phonon polarization branches (i.e., two transverse and one longitudinal polarizations). The electron–phonon scattering rate then becomes: ⇀


 ⇀
⇀  ⇀ ⇀ ′ ⇀ Pe−ph k = Pe−ph k, k ; q , p 1 − f⇀′ dk ′ d q . (7.28) p

k

⇀ ⇀ k′ , q

7.2 Thermal Conduction by Electrons via Monte Carlo Method

163

Assuming isotropic, parabolic band structure, the expression for the scattering rate accounting for the electron–phonon N -scattering can be derived as [224]: Q

(1 − fk ) 1 N (2 nq + 1)|P˜q (K)|2 qdq. (7.29) (k) = Pe−ph 4πmN ¯ hvk 0 ωq For the electron–phonon U -processes, it is derived as: Q 2k

z ′ (1 − fk ) q U Pe−ph (2 nq + 1)|P˜q (K)|2 dK dq, (7.30) (k) = 8πgmN ¯ hvk g−2k g−q ωq ⇀′

⇀

where P˜q (K) is the overlap factor and K = | k − k|. It is assumed here that electrons are scattered only by the phonons in the longitudinal branch and ⇀′

⇀

the scattering is elastic (i.e., | k | = | k|). This is justified as the energy of a phonon in a metal is generally small compared to that of an electron [224]. Equations (7.29) and (7.30) are evaluated numerically using Bardeen’s form of overlap factor and Debye’s model for the phonon dispersion relation (i.e., ωq = vg q where vg is the group velocity of phonon) with a cutoff wave vector Q. The longitudinal phonon group velocity can be calculated from the slope of the dispersion curve for the longitudinal branch, as given by Singh and Prakash [183]. The electron–phonon scattering rate for gold is depicted in Fig. 7.8. Generally, the scattering rate increases as the electron energy increases. Although it is not shown here, the scattering rate for N -processes increases rapidly for electron energy near the Fermi level, and then decreases gradually as the energy increases. The further increases in all the curves beyond 6 eV (as portrayed by Fig. 7.8) are because the possibility of U -processes occurring increases when the energy of electron becomes high. Additional details about the N - and U -processes are given by Ziman [53, 224, 225]. 7.2.4 Monte Carlo Simulation Results Using the electron–electron and electron–phonon scattering rates derived Sects. 7.2.2 and 7.2.3, a series of MC simulations were performed. This type of MC simulation is rather different from the electron-beam MC simulation. It requires tracing of all electron ensembles simultaneously as opposed to launching electron ensembles one after another independently, which is more computational intensive. The numerical procedures used for this purpose are given in Sect. 3.4, therefore there is no need to repeat them. In the following simulations, we consider a gold layer between two parallel boundaries and determine the thermal conduction between them. MC simulations are performed until steady-state is reached within the medium. After that, the temperature profiles between the two boundaries are computed. In the simulations, we divide the number of electron energy intervals

164

7 Thermal Conduction with Electron Flow/Ballistic Behavior

Fig. 7.8. The electron–phonon scattering rate as a function temperature for gold as computed using (7.29) and (7.30). Data are obtained from Wong and Meng¨ uc¸ [209]

into 20 discrete levels and each has 50 electron ensembles. Each computational element possesses a total of 1,000 electron ensembles initially. The dimensions of the geometry are expressed as L × L × L where L is the thickness of the film that is divided into 1,000 computational cubic elements. As a result, the total number of ensembles in the entire computational domain adds up to 1,000,000 [209]. Figure 7.9 shows 1-D temperature profiles for gold thicknesses of 1 m and 100 nm. The straight lines indicate the temperature profile predicted by the Fourier law while symbols are results from the MC simulations. The temperature range used between the two boundaries is 300–400 K. We arbitrarily choose 1 m for gold thickness in order to determine if the temperature profile computed by the MC code converges to that in the diffusion limit where the Fourier law is valid. As seen in Fig. 7.9a, the model predicts the results for bulk cases correctly. After this, the thickness of gold layer is decreased to 100 nm to observe if there is any change in the profile. The agreement of the MC results for 100 nm film with the bulk results indicates that the Fourier law is still valid (see Fig. 7.9b). This is not surprising since the total electron collision rate is about 0.2 fs−1 near 300–400 K, implying that the mean free time of electrons is approximately 5 fs. In the simulation, the average velocity of electrons is about 1 nm fs−1 . Hence, the mean free path of electrons is roughly 5 nm. In order to observe semi-ballistic behavior of these electrons, the thickness of gold

7.2 Thermal Conduction by Electrons via Monte Carlo Method

165

Fig. 7.9. Temperature profiles for a film thickness of (a) 1 m, and (b) 100 nm demonstrating diffusion-like behavior and obeying the Fourier law of conduction (given as straight lines). The top boundary is at 400 K while the bottom boundary is at 300 K. Results are obtained from Wong and Meng¨ u¸c [209]

needs to be in the same order as the mean free path. Consequently, a film with thickness of 100 nm should demonstrate diffusion-like heat transport. Next we consider a temperature range of 300 K to 1,300 K and gold film thicknesses of 1 and 10 nm. These results are plotted in Fig. 7.10. It is clearly evident from these results that the temperature profiles no longer follow the Fourier law. There are sharp discontinuities in the temperature profiles near the boundaries. This effect emerges since electrons are capable of transferring energy ballistically from one end to the other, causing the temperature near the upper boundary to drop, and the temperature near the lower surface to increase. In general, the diffusive behavior of the film starts fading as the mean free path of electrons becomes comparable to the medium thickness. The temperature profiles shown in Fig. 7.10 are within the semi-ballistic regime. In the case where the transport is completely ballistic, the temperature profile will be flat as that reported by Mazumder and Majumdar [129] for semiconductors, where only phonons need to be considered. 7.2.5 Remarks on Electron Conduction Simulations When thickness of a metallic workpiece is 10 nm or less, the diffusion approach fails to predict thermal conduction by electrons. To overcome this problem, a new MC simulation is introduced, which allows us to observe the change in the temperature profiles when semi-ballistic electron transport is encountered. This type of MC simulation can be coupled with the electron-beam MC simulation to study the impact of the semi-ballistic nature of electron transport on the micro/nanomachining process.

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7 Thermal Conduction with Electron Flow/Ballistic Behavior

Fig. 7.10. Temperature profiles for a film thickness of (a) 5 nm and (b) 10 nm demonstrating semi-ballistic-like behavior at which the linearity of the Fourier law breaks down. The top boundary is at 1,300 K while the bottom boundary is at 300 K. Data are obtained from Wong and Meng¨ u¸c [209]

It is important to note that the electron transport is one of two transport mechanisms encountered in an electron–phonon system. We presented the results only for the electron transport, which is the dominant mechanism for metallic workpieces. For phonon transport, a different MC simulation can be constructed following the same basic procedures described in Sect. 3.5; however, it will require the use of the phonon properties specific to the material. More details of MC simulations for semiconducors where only the phonons are considered are given by Mazumder and Majumdar [129] and by Lacroix et al. [109].

7.3 Comments In this chapter, we have discussed two advanced applications for modeling thermal transport at nanoscales. In the first one, we used the EPHDEs for understanding of electron–phonon transport in MESFETs. The second application was for modeling electronic thermal conduction at nanoscale using a MC method. These two methods can be coupled with the MC methods for the EBTE to provide insight into electron-based micro/nanomachining simulations. The EPHDEs enable us to study the electron flow and accumulation inside the workpiece due to the penetration of an electron beam, which may influence the heat transfer within the material. On the other hand, the spatial propagation of heat within the workpiece is still treated as diffusive in the EPHDEs. This assumption can be remedied using a MC simulation of electronic thermal conduction, as the ballistic or semi-ballistic behavior of heat carriers can greatly alter the heat transfer.

8 Parallel Computations for Two-Temperature Model∗

In previous chapters, we introduced several theoretical and numerical models for transport phenomena for application to micro/nanomachining. They included the Boltzmann transport equation (BTE) for electrons and phonons, statistical solution techniques based on Monte Carlo (MC) methods, and the Fourier law and the two-temperature model (TTM) for conduction heat transfer. These equations need to be considered simultaneously as all the underlying phenomena involving electrons, phonons and photons are coupled. As expected, the solution of the final set of equations would be quite involved and would unavoidably require extensive computer power. This bottleneck can be overcome by introducing novel computational schemes. Among those, the use of parallel computation approaches is the most attractive of all. In this chapter, our goal is to outline the procedures required to parallelize computer codes. The physical problem we consider is the same as the one discussed in Chap. 6 where the TTM was used to determine the temperature profiles inside a workpiece bombarded by field-emitted electrons from a nanoprobe. Several tabular comparisons are provided to show the impact of parallelization strategies on the overall computation time required in simulations.

8.1 Introduction to Parallelization Parallel computing has become a very attractive approach for enhancing computational power in many scientific and engineering applications. The primary objective of parallel computing is to gain computational speed. As the cost of parallel hardware decreased relative to fast workstations, trend shifted toward clusters of complete computers using a standard communication interface; this provided efficient and very cheap alternative parallel machines. And finally, with better algorithms and parallel programming tools, parallel environment has become researchers’ most important tools for computationally intensive ∗

This chapter is co-authored with Ravi Kumar and Illay “Victor” Kunadian.

168

8 Parallel Computations for Two-Temperature Model

tasks. With the advent of myriad technologies at software as well as hardware ends of parallel computing field, it is very important to understand them in order to choose the right parallel environment suitable for solving a certain problem. Our objective in this chapter is to outline the pros and cons of parallel computation modalities for micro/nanomachining applications. Three most important aspects for implementing parallelization are listed as: (I) A parallelization algorithm to break the tasks; (II) Parallel software environment; (III) Parallel hardware environment. The first of these (I) is the foremost step, as an effective parallelization is heavily based on developing an algorithm to break the task into smaller tasks which can be solved simultaneously. The extent to which a problem can be parallelized depends on numbers of factors. If these factors are not taken care of properly, the resulting computational burden may exceed that corresponding to serial applications. However, adhering to good algorithm design, with suitable selection of hardware and software architecture one can always come up with an efficient parallel implementation. Steps involved in developing a good algorithm can be summarized as: (i) Task decomposition, (ii) Interprocessor Communication, and (iii) Fine Tuning. Task Decomposition (Step i) can be further divided into two viz. data decomposition and functional decomposition. Undoubtedly, the first step in parallelization is to understand the problem that has to be solved in parallel. Some types of problems can be decomposed and executed in parallel with virtually no need for tasks to share data. As an example we can consider a task of MC simulation of energized electrons penetrating through the work piece and getting absorbed. The total number of electrons to be considered for simulation can easily be divided among multiple processors that can act independently of each other to do their portion of the work. These types of problems are very straight forward and are highly scalable. Very little or no inter-task communication is required, consequently they are listed under computation driven problem category. However, most parallel applications are not quite so simple, and do require tasks to share data with each other. These types of the problems are called communication driven and require massive interprocessor communication (Step ii). For example, a 3-D heat diffusion problem requires the knowledge of all temperatures around a node, as they all enter into the conservation equation solved at each node. On the other hand, functional decomposition involves identification of functional parallelism, which comes from the programming algorithm. The entire task can be divided to different processors on the basis of similar functionality. Finally, fine tuning (Step iii) is another way of implementing parallelism into the parallel program. The entire wall-clock time consumed in parallel computing can be categorized into either computation time or communication time. Using message passing subroutines, it is possible to overlap computation time with communication time thus reducing the entire wall-clock time. The other requirements for implementing parallelization schemes include the selections of the right software (II) and suitable hardware architecture

8.2 Parallelization of the TTM for Micro/Nanomachining

169

(III). These areas are quite vast and ever changing; therefore, we limit our discussions here on the basic and most widely used technologies. The most common hardware architectures adapted in parallel computations include shared memory (a single computer with multiple processors), distributed memory (an arbitrary number of computers connected by a network), or, hybrid distributed-shared memory (a combination of both). The hybrid architecture combines the advantages of both shared and distributed and is mostly used architecture [97]. There are many methods of parallel programming and implementation [97]. Generally, the parallel tasks require interprocessor communication, which can be very tedious and tricky even for solving a simple problem. The interprocessor communication is achieved by using parallel programming software like MPI. Basically MPI is a library of subprograms in C, C++ and FORTRAN language [146] that makes transfer of data from one processor to other possible. There are two types of MPI communication: point-to-point and collective communication. Point-to-point communication is between one processor to another. On the other hand, collective communication can send or receive data from/to one processor to/from many processors.

8.2 Parallelization of the TTM for Micro/Nanomachining This section presents parallel solution to the sparse linear system arising from the finite difference discretization of the partial differential equation governing transient heat transport at the nano-scale level. The problem requires solving for temperature of each node that depends on the temperature of neighboring nodes at the same time level as well as from previous time level. Therefore, a preferred way of implementing parallelization is through task decomposition (also known as domain decomposition) and solving for each task on different processors using MPI libraries simultaneously. The numerical method chosen for parallelization is a relaxation iterative scheme, such as Gauss–Seidel (GS) and successive-over-relaxation (SOR), for solving banded linear systems on distributed memory or multiprocessor platforms. The SOR method is chosen because it is an important solver for a class of large linear systems [67, 95, 145, 221]. It can also instructive to develop a quick approximation of the solution in intermediate steps for more powerful methods (e.g., Conjugate Gradient method and other multi-grid methods) or even replace direct methods in parallel applications [91]. With a rapid development of parallel computers, various parallel versions of the SOR method have been developed for solving large scale linear systems on a large number of processors. Defined by multicolor ordering technique, the multicolor SOR method is a widely used parallel version of SOR and has been widely studied by many authors [1–3]. In multicolor SOR, the unknowns are colored in such a way that no two unknowns of the same color are coupled by

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an equation. In the simplest case of 5-point stencil arising from the centered difference discretization of the Laplacian in two or three dimensional spaces, only two colors are needed and they are commonly referred as “red” and “black”; but for some complicated problems more than two colors are required to define a multicolor ordering. Xie and Adams [214] proposed and analyzed a new parallel SOR (PSOR) method, formulated by using domain partitioning and interprocessor data communication. They compared the performance of PSOR with Red–Black SOR (R/B SOR) and R/G/B/O SOR. Their numerical results indicated that PSOR was more efficient in both computation and interprocessor data communication. As shown recently by McDonough et al. [131], the Douglas–Gunn time splitting method is most efficient numerical procedure for solving 3-D microand nano-scale heat transport equation. Hence, parallelization of Douglas– Gunn time splitting method would the best way to reduce the wall clock time required for solving the governing transient equation. Douglas–Gunn time splitting can be employed to split the entire 3-D grid into series of planes in each of the x, y and z grid directions. This requires decomposing the 3-D coefficient matrix in three separate sets of tri-diagonal systems which must be solved on different processors in parallel in each time-split step (McDonough et al. [130]). However, the current work is limited to implementing parallelization using Red/Black SOR method. As discussed above, there are numerous ways of introducing parallelism for solving systems of linear equations Ax = B. The research area is extensive and requires further work to be done in order to simulate the nano-scale heat transport phenomena using parallel computing paradigm efficiently. Here we will mainly be investigating parallelism through domain decomposition and interprocessor communication for solving nano-scale heat transport equation using MPI on four different computer networking architecture.

8.3 Parallel Computing Resources In this section we discuss the computing recourses used for parallelization of nano-scale thermal transport phenomena during micro/nanomachining just to give an idea about the basic requirements. In our studies, the parallel computing experiment is conducted on four different computer clusters available at the University of Kentucky: Kentucky Fluid Cluster I (KFC1), Kentucky Fluid Cluster II (KFC2), UK HP Superdome Cluster (SDX) and UK HP Linux cluster (XC). KFC1 is a cluster of 20 dual-processor nodes each of them powered by 1,400 65 MHz AMD Athlon processors with cache sizes of 64 and 256 KB for the L1 and L2 cache, respectively. Each node contains 1 GB of main memory and 40 GB hard disks, and four 100 Mb s Fast Ethernet network interface cards (NIC). If a node has several NICs, the node can share data with several neighborhood nodes. For two nodes to communicate directly, they simply use NIC of a third node that is common to both other two nodes

8.4 Implementation of Parallelization

171

thus resulting in increased bandwidth making interprocessor communication faster. Three 22-port switches are used to form a channel bonded network and one 22-port switch forms a second network for NFS traffic. KFC2 is a cluster of 48 interprocessor nodes powered by AMD Athlon XP 2000+ processors with cache sizes, like KFC1, of 64 and 256 KB for L1 and L2, respectively. Each node contains 256 MB of main memory and four Fast Ethernet NICs. KFC2 has three channel bonded network and a single network for NFS traffic. The network configuration of KFC2 is similar to KFC1 except that KFC2 uses 48-port switch. KFC2 falls under distributed memory architecture category in which all the nodes have their own memory and can perform computations independent of other nodes. On the other hand, KFC1 falls under hybrid memory architecture in which at least one node has multiple processors and a single memory to share. The other two clusters SDX and XC also can be categorized as of hybrid memory architecture. SDX comprises of four HP Superdomes with 256 processors (64 processors per node) and powered by Itanium-2 (Madison) processors. Each processor has 2 GB of memory with 7 TB of total disk space. All processors are networked through high speed, low latency infiniband internal interconnect. The XC cluster has 248 3.4 GHz Intel Xexon em64 t processors @ 2 GB per processor with a total of 8 TB of high-speed disk storage and offer Myrinet high speed message passing interconnect for internode communication. These machines are more powerful and efficient as compared to KFCs and uses latest hardware architecture. However, the cost of SDX and XC are significantly higher compared to KFCs.

8.4 Implementation of Parallelization Parallel implementation for solving nano-scale heat transport equation is based on overlapping domain decomposition, i.e., splitting the computational grid into sub-blocks, which are then distributed to each processor or node. The work piece under consideration is assumed to be thin gold film, and the heating source bombards the target either with photons (for micro-scale calculations) or electrons (for nano-scale calculations). The origin is considered to be at a corner of the film with x − y plane as the front surface exposed to heat source and increasing thickness along the z-axis. Hence the entire computational grid can be viewed as series of x − y planes made of 3-D cubical cell each of size dx × dy × dz, where dx, dy, and dz are spatial step sizes, and the computational grid point is at the center of the each cubical cell. Since the workpiece used for experiments has simple 3-D Cartesian geometry, it is a not a tough exercise to distribute task, usually called balance the load, to each processor evenly. Load balancing is an important factor of parallel computing and might affect computational efficiency of the overall code significantly if it is not done properly in case of complex geometries. In this work, Single Program Multiple Data (SPMD) model is used which

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means all the nodes will use the same program but may produce different data depending on the given input. In parallel programming, since all the processors perform computation and communicate with each other, it is also important to clarify the scope of global and local variables used by each processor. Since it is SPMD model, all the nodes have same set of local and global variables. The main advantage of defining variables in parallel code as compared to serial code is that each node requires variable size corresponding to the size of sub-block assigned to it for computation. In this way, the memory bottleneck that was a major hurdle in solving for large computational grid using serial code is overcome. One of the nodes is treated as master node, also called node 0, and others as slave node. The master node reads the input data and distributes them to all the nodes as specified in the code. Once the data are distributed to all the nodes, parallel computation occurs until specified criteria for number of iterations meet. Communication between nodes occurs when the sub-blocks exchange data about the temperature variables at the boundaries. Temperature on the edge of one grid blocks is communicated to the dummy points of the neighboring grid blocks, and vice versa. Nanoscale heat conduction code typically requires frequent communication steps, normally after each sub-iteration. The whole idea of parallelization for nanoscale heat transport equation can be summarized in the procedures below: • MPI function calls start with specified number of processors to be used. • If the node is master processor (node = 0) – Master processor reads the input data (such as material properties, geometry configuration, time steps size, heat source etc.). – Master processor broadcasts/scatters input data to rest of the processors. • Each processor, including master, performs the calculation based on initial temperature, material properties and heat source assigned to it. • All the processors exchange boundary values with neighboring processors after each update or sub-iteration. • Each processor sends local maximum error norm to master processor. Master processor compares all the gathered local maximum error norms in order to find and broadcasts global maximum error norm to rest of the processors. This makes a complete iteration at a particular time step. • Each processor repeats the above steps until global maximum error norm reaches tolerance limit at a time step. • The master processor gathers/collects output data from all the processors once they finish calculation for desired number of time steps. • Master processor prints the data collected from all other processors to an output file. • MPI calls shutdown to terminate execution. Communication block size and the frequency of exchanging blocks with neighboring processors affect the parallel performance a lot. Knowing the

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optimum block size and the frequency with which the blocks can be exchanged can help increase the performance gain significantly. Another important issue in parallel performance is minimizing the time spent in interprocessor communication. The communication cost is significant and the problem is dependent on hardware (particularly the hardware architecture), software, geometry complexities and solution methodology. The solution methodology under consideration involves interprocessor communication at each sub-iteration and makes computation costly. If parallel performance tuning is not done to reduce the communication time, each node might spend less time on computation between the communication steps resulting in poor scalability. Though true linear scalability is often not achievable, near-linear scalability is the ultimate objective of most of the parallel computing problems. In order to achieve this type of parallel performance, communication is overlapped with the computation showed in the flow chart given in Fig. 8.1. The dashed portion in the above flowchart shows overlapped computationcommunication part in the entire parallel computing flow being performed at each discrete time step. Arrows between two dashed rectangles indicate communication between two neighboring nodes whereas portion inside dashed rectangles show computations being performed. More details about the overlapped computation-communication procedure can be summarized as follows: • If node is even numbered – Update last Red boundary values – Start sending (using MPI Isend) updated last Red boundary values to ghost plane of the next neighboring node – If node is zero update rest of the Red planes else update last half Red planes while MPI Isend is in progress – Wait until MPI Isend is complete • Else-If node is odd numbered – Start receiving (using MPI Irecv) values being sent from neighboring previous node into ghost plane – If node is last node update first half Black planes else update all the Black planes while MPI Irecv is in progress – Wait until MPI Irecv is complete • If node is odd numbered – Update first Red boundary values – Start sending (using MPI Isend) updated first Red boundary values to ghost plane of the previous neighboring node – If node is last node update rest of the Red planes else update first half Red planes while MPI Isend is in progress – Wait until MPI Isend is complete • Else-If node is even numbered – Start receiving (using MPI Irecv) values being sent from neighboring next node

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– If node is zero update all the Black planes else update last half Black planes while MPI Irecv is in progress – Wait until MPI Irecv is complete • If node is even numbered – Update first Red boundary values – Start sending (using MPI Isend) updated first Red boundary values to ghost plane of the previous neighboring node – If node is last node update rest of the Red planes else update first half Red planes while MPI Isend is in progress – Wait until MPI Isend is complete • Else-If node is odd numbered – Start receiving (using MPI Irecv) values being sent from neighboring next node into ghost plane

Fig. 8.1. The flowchart showing the parallel computing procedures and communication between nodes

8.4 Implementation of Parallelization

(a) The flowchart for process ‘A’

(b) The flowchart for process ‘B’ Fig. 8.1. (continued)

175

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8 Parallel Computations for Two-Temperature Model

– If node is zero update rest of the half Black planes else update the last half Black planes while MPI Irecv is in progress – Wait until MPI Irecv is complete • If node is odd numbered – Update last Red boundary values – Start sending (using MPI Isend) updated last Red boundary values to ghost plane of the next neighboring node – If node is last node update rest of the Red planes else update first half Red planes while MPI Isend is in progress – Wait until MPI Isend is complete • Else-If node is even numbered – Start receiving (using MPI Irecv) updated Red boundary values being sent from neighboring previous node – If node is zero update all the Black planes else update last half Black planes while MPI Irecv is in progress – Wait until MPI Irecv is complete The above four steps encompasses all the possibilities an odd and an even numbered node can have in order to exchange outer updated planes (communicate) with their next and previous neighboring nodes while updating (computing) inner planes. In other words, there are two types of node viz. odd numbered and even numbered. Each node has two ends and each end can have two possibilities; send data or receive data. That makes total of eight possibilities. Each node except the last node is assigned odd number of planes (x − y plane containing grid points). Since each node has odd number of planes, the first and last plane is always Red with alternate Black planes in between. The last node is assigned remaining number of planes that may not be an odd number.

8.5 Parallel Computing Experiment This section demonstrates the performance of parallel code for solving 3-D micro- and nano-scale heat transport equation using relaxation iterative methods, such as Gauss–Siedel and Red/Black SOR on four distinct cluster architecture. In order to verify the parallel code results, the example problem chosen for parallel computing experiment is dual-phase lag (DPL) equation. Since the results for DPL equation using pulsed laser source is already available in literature, results of the parallel code can be verified. Thickness of the gold film used for experiments is 0.1 µm whereas the length and width are 0.5 µm each. Properties of the gold used are conductivity, k = 315 W m−1 K−1 , phase lag temperature gradient, τ = 90 ps, phase lag of heat flux, τ = 8.5 ps, −1 and diffusivity, α = 1.25 × 10−4 m2 s . The heat source is pulsed laser with −1 luminous intensity, J = 13.4 J m , reflectivity, R = 0.93, penetration depth,

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δ = 15.3 nm, and t = 96 fs as the only heating source. The grid resolution considered is: 513 and 1013 corresponding to dx = dy = 10 nm; dz = 2 nm and dx = dy = 5 nm; dz = 1 nm spatial step sizes, respectively. The following results demonstrated are for 2.5 ps of simulation time with temporal step size dt = 0.01 ps. Table 8.1 lists the wall clock time (all in seconds) lapsed in parallel computing on three different computer clusters KFC1, KFC2 and SDX. For each cluster, there are four columns that represent communication time (Comm), Table 8.1. Execution time in seconds on three distinct computer cluster platforms for solving DPL heat transport equation for 513 grid size No. of PEs

Comm

Comp

Total

Sp

(KFC1 Cluster) 1 2 3 4 5 6 7 10 17

0.0 9.5 17.4 32.0 60.0 − − − −

235.0 110.1 72.9 56.1 47.9 − − − −

235.0 121.4 90.0 84.3 89.9 − − − −

1.0 1.9 2.6 2.8 2.6 − − − −

182.0 97.7 77.4 79.2 81.7 78.6 78.1 77.4 77.3

1.0 1.9 2.4 2.3 2.2 2.3 2.3 2.4 2.4

222.5 178.7 45.7 35.0 30.5 24.9 25.9 19.7 13.5

1.0 1.3 4.9 6.4 7.3 8.9 8.6 11.3 16.5

(KFC2 Cluster) 1 2 3 4 5 6 7 10 17

0.0 7.7 19.2 36.0 56.5 55.9 52.6 59.0 66.6

1 2 3 4 5 6 7 10 17

0.0 38.7 5.7 5.8 13.3 10.7 8.2 7.0 6.3

182.0 88.7 61.4 47.3 40.0 34.2 32.2 21.7 11.5 (SDX Cluster) 222.5 170.8 40.2 30.7 26.0 21.3 22.8 15.2 7.2

First column represents number of processors used. Comm, Comp and Sp represent inter-processor communication time, computation time and speed up, respectively

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computation time (Comp), total runtime time (Total) and speed up (Sp: ratio of serial code to parallel code execution time). The parallel code is first run on a single processor of all the cluster platforms. The total run time of the code reported in Table 8.1 is time elapsed in completing 250th time step. As seen from this table, the computation time and total runtime is dropping steadily with increasing number of processors. From a pure performance standpoint, the goal is to achieve as close to ideal speedup (desired linear speedup) as possible with increasing number of nodes. On the other hand, from practical standpoint, the desired performance is the least “wall clock” time required to perform the simulation so that one can choose the cluster. The speedup for KFC1 and KFC2 deviates far from near-linear scalability. This is because of the poor network performance of KFC1 and KFC2 as compared to SDX for parallel computing of nanoscale machining simulations. On the other hand, the performance of SDX is far superior to cluster to perform the job quickly, regardless of the efficiency of computation. The parallel algorithm, as shown in previous section, requires interprocessor communication at each sub-iteration hence makes the problem highly communication intensive. If the cluster network is capable of handling high communication traffic, the overall performance will be affected adversely. Figure 8.2 based on comparison Table 8.1 for 513 grid resolution case shows

Fig. 8.2. Speedup on KFC1, KFC2 and SDX cluster is compared against ideal speedup (desired linear speedup) for solving parallel micro-scale heat transport

8.5 Parallel Computing Experiment

179

poor scalability of KFC1 and KFC2 cluster compared to SDX KFC1 and KFC2. In parallel computing, super linear speedup is a common phenomenon that might be possible because of several reasons like networking hardware or decrease in number of iterations with increasing number of processors or due to cache effect. Venkatesh et al. [199] achieved super linear speedup of 11 on an eight-processor system using the FloSwitch for communication on a multigrid laminar Navier–Stokes code. Stiller et al. [186] reported a speed-up of nearly 140 within a 120-processor system on a finite element Navier–Stokes code. So far SDX proves to be better platform for performing further parallel computing experiments. As far as computational efficiency is concerned, all the three clusters shows decrease in computational time with number of processors, as expected. Another similar experiment performed for solving parallel micro-scale heat transport equation (DPL) includes XC cluster also. The grid resolution for this experiment is 1013 . In this case, although the speedup achieved in case of XC cluster is poor compared to SDX cluster, XC cluster shows very high overall performance gain over SDX. Because of superior interprocessor communication and faster computation, the total wall clock time on XC reduces significantly compared to time consumed by SDX and hence makes XC the best available platform for performing micro/nanomachining simulations. Figure 8.3 displays the parallel performance of Red/Black iterations as

Fig. 8.3. Wall clock time for completing 2.5 ps simulation on XC cluster

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a function of number of processors on XC cluster. The total wall clock time consumed by parallel Red/Black SOR code reduces with number of processors employed to perform the simulations using pulsed laser source. The XC cluster demonstrates very good scalability with increasing number of processors.

8.6 Parallel Computing Using Parabolic Two Step (PTS) Model The previous section demonstrated the XC cluster as the best available platform for performing simulations for transient micro-scale heat transport equation problems. This section concentrates on parallel computing experiments on a single cluster for developing a strategy to further reduce the wall clock time so that one can adopt these strategies to perform this kind of simulation faster. The governing equation chosen for performing the experiment is the parabolic two-step model since the model will be used in micro/nanomachining simulations. The PTS governing equations are solved using the Newton–Kantorovich method and the numerical methods chosen are the Douglas–Gunn time splitting (serial code) and iterative methods Red/Black SOR & G-S (parallel code). The serial code is used just for verifying the parallel code results. First, the electron energy equation, which is a nonlinear PDE, is solved iteratively until specified tolerance limit is reached and assuming phonon temperatures to be known. Then electron temperatures are plugged into phonon-energy equations to solve for phonon temperatures until specified tolerance limit is reached. These non-linear iterations are repeated until tolerance limits, (tolerance limit for Te and tolerance limit for Tl ) are reached. Before conducting the fine-tuning experiment for parallel PTS model on XC cluster, it is important to verify the results of parallel PTS code. In order to verify the results of parallel PTS model, a test is conducted that uses a pulsed laser as the heat source. The experimental and numerical results for PTS with pulsed laser as heat source are already reported in several noted publications. Qui and Tien [161] and Dai and Nassar [42] studied heat transfer mechanism during ultrafast laser heating of metals. They predicted temperature profiles in a 0.1 µm gold film during 0.1 ps laser pulse heating from parabolic two-step and hyperbolic two-step (HTS) model. The results obtained by the parallel PTS code show good agreement with those obtained by Dai and Nassar [42] and Qui and Tien [161]. Table 8.2 displays the electron temperature profile obtained for PTS model using Douglas–Gunn time splitting method (serial code) and Red/Black SOR method (parallel code). The results indicate that the parallel code computation agrees well with serial code computation and temperature profiles are correct and hence establish the credibility of the parallel code results. Now, the parallel code can be used for conducting further parallel computing experiments using electron beam source instead of pulsed laser source.

8.7 Parallel Computing Including an Electron Beam

181

Table 8.2. Electron temperature (in K) in 0.1 µm gold film during a 0.1 ps laser heating at three distinct time intervals of 0.2, 0.3 and 1.2 ps Thickness of gold film (nm)

0.2 psa

0.3 psa

1.2 psa

0.2 psb

0.3 psb

1.2 psb

0 10 20 30 40 50 60 70 80 90 100

585.4 566.8 528.0 483.8 441.7 405.4 376.3 354.5 339.6 331.0 328.2

596.8 592.1 579.4 561.0 539.0 515.5 492.7 472.4 456.5 446.3 442.9

407.6 407.6 407.5 407.5 407.5 407.5 407.5 407.5 407.5 407.5 407.5

585.4 566.8 528.0 483.8 441.7 405.4 376.3 354.5 339.6 331.0 328.2

596.8 592.1 579.4 561.0 539.0 515.5 492.7 472.4 456.5 446.4 442.9

407.6 407.6 407.6 407.5 407.5 407.5 407.5 407.5 407.5 407.5 407.5

a b

Douglas–Gunn serial code Parallel Red/Black SOR code

8.7 Parallel Computing Including an Electron Beam As discussed in Chap. 1, we consider a micro/nanomachining procedure using an electron beam emitted from a nano-probe, a carbon nanotube (CNT). The system is mainly comprised of an anode, the target work-piece, and a cathode, the machining tool CNT. Voltage applied between the anode and the cathode causes energized electrons to flow from the cathode to the anode. These energized electrons impinge the workpiece surface and transfer kinetic energy while penetrating through the workpiece. In this process, the kinetic energy of energized electrons is converted to thermal energy, which subsequently causes material removal from the workpiece. The corresponding heat transfer mechanism for this system was already discussed in Chaps. 5 and 6. First, the electron-beam transport equation (EBTE) was solved with a MC method. The profile then entered as external heat source term in the PTS equations. The simulation assumes the entire system to be placed inside a vacuum chamber with pressure of 10−8 torr. The parallel computer code simulates the heating process by an electron beam. The parallel PTS code is allowed to simulate the temperature field. Using the electron energy-deposition distribution generated by a MC method (see Chap. 3) for a 500-eV electron beam, heating phenomena for workpiece of (2,840 nm × 2,840 nm × 12 nm) size subjected is simulated on 16 processors of XC cluster. The results obtained from parallel code are compared against results obtained in Chap. 6. Table 8.3 shows the comparison of both results. The temperature fields obtained by parallel code follows the trend closely and are quite close with the results obtained using the simulation in Chap. 6.

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Table 8.3. Comparison of electron temperature field obtained from parallel PTS code with results obtained in Chap. 6 for 2,840 nm × 2,840 nm × 12 nm workpiece size and heat source as electron beam of radius 500 nm and kinetic energy of 500 eV Simulation time (ps) 1 2 3 4 5 6 7 8 9 10 11 12

Tea (K)

Tla (K)

Teb (K)

Tlb (K)

3,859.6 4,978.3 5,713.2 6,282.5 6,761.3 7,183.3 7,566.3 7,920.4 8,252 8,565.5 8,863.8 9,149.2

325.2 368.1 419.8 477.6 540.2 606.8 676.8 750 825.9 904.3 985.2 1,068.2

3,840 4,970 5,700 6,260 6,720 7,130 7,490 7,820 8,130 8,420 8,690 8,960

325 368 419 477 539 605 675 747 822 899 978 1,060

Te and Tl represent electron temperature and phonon temperature, respectively Parallel PTS Code Result b Result obtained using serial computer code described in Chap. 6 a

8.8 Comments Parallel computation strategies promise computationally efficient calculation of challenging scientific problems using variety of algorithms and hardware environment. In this chapter, we discussed a parallel computation approach based on MPI for the solution of transport equations corresponding to a typical micro/nanomachining problem. The problem treated in this work consisted of heating a very small localized area of the target work-piece. Red/Black SOR chosen for parallelization showed good performance on the computer clusters. The numerical results obtained by parallel code provide fairly better performance compared to results obtained by serial code. Scaling up the grid scheme starting from this nano-scale area to larger micro- or milli-meter sized areas proves to be very complicated. The grid spacing needs to be fine near the heating zone only and the rest of the domain can have coarse grid. In this way, the parallel code can be run for larger work-piece in shorter amount of time. The numerical experiment results for pulsed laser source match closely with the experimental results available in the literature. This suggests that the electron-beam simulations should also be modeled fairly accurately with this approach, provided that correct properties used. Parallelization of non-stationary methods, which remains unexplored so far, is expected to perform better than parallel version of stationary methods. Still the best way to solve the problem faster seems to be to parallelize

8.8 Comments

183

Douglas–Gunn time splitting method. The SDX cluster was high scalability compared to XC cluster however the computational efficiency of XC cluster was found to be much better than the SDX cluster. Other factors that need to be addressed for performing simulations faster is non-uniform grid implementation rather than uniform grid stretching. Another standard parallel programming paradigm called Parallel Virtual Machine (PVM) is equally promising like MPI and needs to be explored for transient heat transfer problems that require massive data transfer in performing computations.

9 Molecular Dynamics Simulations∗

The intricate mechanisms of electron–electron, electron–phonon and phononphonon interactions described in the previous chapters are used to simulate the evolution of the temperature profile within the material. One of the fundamental assumptions made so far is that the material is a continuum, e.g., the overall behavior of the material is the average of the atomic description. However, a more realistic analysis of the problem requires the study of the atomic picture to elucidate the mechanisms that lead to melting, phase change, ablation and material removal using an electron beam. Surface processing of materials from conventional to femtosecond pulse lasers has been studied thoroughly in the past. Photons from the laser interact with the free electrons in the material in a femtosecond time-scale. This causes a temperature rise in the electronic temperature which is in the order of several thousand Kelvin. After some time, these excited electrons interact with the phonons in the lattice, at which point, the temperature of the material begins to rise. The electron temperature and the phonon temperature evolve separately until thermal equilibrium is reached as described in the two-temperature model (TTM). The equilibration takes place in a picosecond timescale. Once both temperatures have equilibrated, the heat transfer process can be simply described using the Fourier law. As discussed in Chap. 2, the modeling of transport phenomena by electrons and phonons can be performed using the Boltzmann transport equation (BTE). To simplify the complex in-scattering term in the BTE, a timerelaxation approach is usually adapted for both electrons and phonons (see Sect. 2.4.1). This assumption can be relaxed by completely replacing the phonon equation with molecular dynamics (MD) to determine the evolution of the lattice temperature. In this chapter we provide a brief introduction to MD simulations. We also outline a methodology to couple them with the electron–phonon interactions discussed in the previous chapters.

∗

This chapter is co-authored with Jaime A. S´ anchez.

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9.1 Overview of Molecular Dynamics MD simulations, in essence, mimic real experiments. When we perform a real experiment, we prepare a sample of the material that we wish to study. We connect this sample to a measuring instrument (e.g., a thermometer, manometer, or viscometer), and we measure the property of interest during a certain time interval. If our measurements are subject to statistical noise (as most real measurements are), then the longer we average, the more accurate our measurement becomes. In MD simulations, we follow exactly the same approach. First, we prepare a sample: we select a model system consisting of N particles and we solve Newton’s equations of motion for this system until the properties of the system no longer change with time (we equilibrate the system). MD is a powerful tool that can be applicable to simulate any type of phenomena in which detailed atomic resolution is desired. One can study mechanical properties of nanowires [150], thermal conductivity through nanostructures (Lukes et al. [117]) or study in detail capillary waves in water [86]. One could also study the intricate mechanisms of melting and phase change of surfaces [19], or phase equilibria of different materials that illustrate the interplay of mechanisms that become important in the nanoscale [159]. In this chapter, our interest is on the application of MD simulations as a tool to study phase change and material removal processes during electron-beam based micro/nanomachining. We will outline a general procedure to accomplish this feat. Equilibrium MD is typically applied to an isolated system containing a fixed number of molecules N in a fixed volume V . Because the system is assumed isolated, the total energy E, that is the sum of the molecular kinetic and potential energies, remains constant. Thus the variables N, V , and E determine the thermodynamic state. In NVE-molecular dynamics for the ith particle with mass mi , its trajectory ri (t) is governed by: N

mi

 d2 ri Fij , = 2 dt j=1

(i = 1, 2, . . . , N ),

(9.1)

j=i

where N is the number of particles in the system and Fij denotes the force exerted on particle i by particle j and corresponds to the derivative of the interatomic potential between such particles. Fij = −

∂U (rij ) . ∂rij

(9.2)

The summation is carried out over all other particles in the system. Equation (9.1) represents a total of 3N equations due to the three Cartesian components of ri (only if translational motion is considered). The major tasks of a MD simulation are to solve these 3N equations, also known as

9.1 Overview of Molecular Dynamics

187

the many-body problem, and then analyze the simulation results to obtain the information of interest. The Newtonian equations of motion, though always correct in classical mechanics, have limitations, when describing different kinds of motion. For example, they are based on Cartesian coordinates and thus angular rotation and their associated coordinates cannot be easily incorporated in the formulation. To describe the rotation of an object by Newton’s equation of motion, one needs to use angular momentum equations rather than (9.1) directly. Such drawbacks can be overcome by using other expressions, particularly Lagrange’s or Hamilton’s equations of motion. In both these approaches, we use generalized coordinates r1 , r2 , . . . , rn that include position, angle, and so on, and define the generalized particle coordinate vector r = (r1 , r2 , . . . , rn ) and the generalized particle velocity r˙ = (r˙1 , r˙2 , . . . , r˙n ). Integrating (9.1) once yields the atomic momenta; integrating a second time produces the atomic positions. Repeatedly integrating for several thousand times results in the individual atomic trajectories from which the time averages A can be computed for macroscopic properties: 1 t→∞ t

A = lim



t0 +t

A(τ ).

(9.3)

t0

According to the ergodic hypothesis, the time average (9.3) provided by MD should be the same as the ensemble average [62]: 1 · · · exp[−βu(rN )]A(rN )dr1 · · · drN ,

A = (9.4) Q where β = 1/kB T, kB is the Boltzmann constant, and Q is the partition function integral: (9.5) Q = · · · exp[−βu(rN )]dr1 · · · drN . 9.1.1 Interatomic Potential A crucial step in a MD simulation is the choice of the potential energy function between atoms or the “interatomic potential” in that it determines how realistic the simulations would be. The interatomic potential is a function that depends on the distance between atomic nuclei and the distribution of the electrons in their orbits. Even if all numerical simulations and the subsequent analysis are performed perfectly, an inaccurate potential can lead to inaccurate conclusions about the phenomenon modeled. There is however no universal and rigorous way to compute the interatomic potentials. One can start either from the first-principles, which account for the full complexity of electronic effects, or empirical approaches can be adopted, where the atoms are treated as points interacting through some potential energy function U (r1 , . . . , rN ).

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One of the simplest, most widely used interatomic potentials, is the Lennard–Jones potential (LJ), given as [112]: 

6  12 σ σ , (9.6) − U (rij ) = 4ε rij rij where rij is the distance between particles, ε is the energy parameter, and σ is the length parameter. The first term in the Lennard–Jones potential models the strong repulsive force resulting from the overlapping of the inner-shell electrons or ions. The second term is the attractive electrostatic force between the instantaneous dipole of one atom and the induced instantaneous dipole of the other. The Lennard–Jones potential is a realistic representation of molecular crystals such as argon and has been widely used for liquids and various solids as a qualitative model. Real material systems are more complicated and typically cannot be adequately described the Lennard–Jones potential or other forms of the pair potential. An interatomic potential suitable for gold was developed by Ercolessi et al. [52]. It is based on the assumption that the number of neighbors of a given noble-metal represents the local electronic density, such that the coordination can be taken as characterizing the environment in which the ion is situated. If the coordination of an ion remains nearly constant during the motion, the ion interacts with the others essentially through an effective two-body potential. On the other hand, motions which tend to change the coordination appreciably are greatly discouraged by their high energetic cost. Such motions easily occur in the proximity of surfaces or defects, and the extreme case consists of removing an atom out of the system, similar to what is expected to be achieved via micro/nanomachining. In this formulation the potential is expressed as: U=

N 

N

U (ni ) +

i=1

N

1  φ(rij ). 2 i=1 j=1

(9.7)

j=1

In this formulation, a standard two-body part is still present, together with the new many-body term. Here, ni is the coordination of atom i, and the function U (n) associates an energy value to this coordination, thereby including the “gluing” effects of the conduction electrons. For this reason, U (n) has been nicknamed and (9.7) the “glue Hamiltonian.” In order to use (9.7) in computer simulations, we need a functional form for ni . The simplest choice consists in building ni as a superposition of contributions from the neighboring atoms: ni =

N 

j=1 (j=i)

ρ(rij ),

(9.8)

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189

where ρ(r) is a short-ranged monotonically decreasing function of distance. (9.8) essentially counts the number of neighbors of atom i. This is done in a continuous way, so that nearby atoms contribute more to ni than remote atoms. The final result for ni is a real number that generalizes the usual idea of coordination. In their work, Ercolessi et al. [52] fitted the functions φ(r), ρ(r) and U (n) to experimental data. Another interatomic potential suitable for face-centered-cubic (FCC) materials was proposed by Daw and Baskes [44]. Their formulation is based on the assumption that the total electron density in a metal is reasonably approximated by the linear superposition of contributions from the individual atoms. The electron density in the vicinity of each atom can then be expressed as a sum of the electron density contributed by the atom in question plus that contributed by all the surrounding atoms. This latter contribution to the electron density is a slowly varying function of position and can be approximated as constant. The potential energy of this atom is then the energy associated with the electrons of the atom plus a contribution due to the constant background electron density. This sum defines an embedding energy of an atom as a function of the background electron density and the atomic species. In addition, there is a repulsive electrostatic energy component due to core–core overlap of atoms. Under this picture, the embedded-atom method gives the total energy in the form: ⎤ ⎡ N N N   1  ⎦ ⎣ U= φij (rij ), (9.9) ρa (rij ) + ui 2 i=1 j=1 i=1 j=i

j=i

where ui is the energy needed to embed atom i in the background electron density ρ with origins in the density functional theory [103], ρa is the angleaveraged radial electronic charge density function obtained from the HartreeFock calculations for free atoms [38], and uij is the core–core pair repulsion between atoms i and j separated by the distance rij . The first term on the right-hand side of (9.9) represents a many-body type of interaction if ui is a nonlinear function, but reduces to a pair interaction otherwise [44]. The forms of the embedding function of several FCC metals are chosen to be cubic splines fitted to bulk experimental data such as the sublimation energy, equilibrium lattice constant, elastic constants, and vacancy-formation energies of the pure metals and the heats of solution of the binary alloys [56]. When applied to solid surfaces, the parameters for the potential are not changed, nor are additional adjustable parameters introduced. Therefore, the model is surface parameter free. This is not the case, for instance, when an effective (classical) lattice Hamiltonian is used and the interaction between two surface atoms is different from that of two bulk atoms (Ercolessi et al. [52]). Yifang et al. [219] extended the embedded atom method to body centered cubic (BCC) transition materials such as V, Nb, Ta, Cr, Mo, W and Fe. They found good agreement between the calculated dilute-solution enthalpies

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and formation enthalpies of random alloys with the experimental data available, the results from the first-principles calculations, and the results of thermodynamic calculations. For applications to micro/nanomachining of metals, the embedded atom method of Daw and Baskes [44] is the most versatile and accurate. Therefore, it is used in the simulations given below. 9.1.2 Time Integration of the Equations of Motion After we have computed all forces between the particles, we can integrate Newton’s equations of motion. The simplest integration scheme consists in the Taylor series expansion of the coordinate of a particle, around time t: r(t + ∆t) = r(t) + v(t)∆t +

f (t) 2 ∆t3 ··· r + O(∆t4 ). ∆t + 2m 3!

(9.10)

f (t) 2 ∆t3 ··· r + O(∆t4 ). ∆t − 2m 3!

(9.11)

Similarly, r(t − ∆t) = r(t) − v(t)∆t +

Summing these two equations, we obtain: r(t + ∆t) = 2r(t) − r(t − ∆t) +

f (t) 2 ∆t + O(∆t4 ). 2m

(9.12)

This is the so-called Verlet algorithm. The estimate of the new position contains an error that is of order ∆t4 , where ∆t is the time step in the MD scheme. The velocity can be derived from knowledge of the trajectory as: v(t) =

r(t + ∆t) − r(t − ∆t) + O(∆t2 ). 2∆t

(9.13)

This expression for the velocity is only accurate to order ∆t2 . To obtain more accuracy in the calculations Verlet-like algorithms, such as the velocity-Verlet algorithm [188], have been developed in which the positions are calculated as: r(t + ∆t) = r(t) + v(t)∆t +

f (t) 2 ∆t . 2m

(9.14)

and the velocities by: f (t + ∆t) + f (t) ∆t. (9.15) 2m Note that in this algorithm, we can compute the new velocities only after we have computed the new positions and, from these, the new forces. (9.15) is accurate to order ∆t4 and it can be proven rigorously that (9.14) and (9.15) are equivalent to the original Verlet algorithm [59]. A detailed discussion of other integration schemes and the importance of numerical accuracy are given by Frenkel and Smit [59]. In this chapter, we use the velocity-Verlet algorithm for the characteristics discussed above. v(t + ∆t) = v(t) +

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191

9.1.3 Molecular Dynamics in Different Ensembles In practical applications, we are interested in determining how system properties change with temperature. The microcanonical ensemble is not convenient for this purpose since one of its natural thermodynamic variables is the internal energy rather than the temperature. Instead of considering an isolated system of fixed energy, let us now consider a system of fixed number of particles N , fixed volume V and given temperature T . To maintain the system at a fixed temperature, we assume that it is in contact with a thermal reservoir at the same temperature. This situation corresponds to the canonical ensemble as discussed by Chen [30]. Several approaches have been developed to sample the canonical ensemble using MD. Andersen [6] developed a thermostat (temperature control formulation) in which the system is coupled to a heat bath that imposes the desired temperature via stochastic impulsive forces that act occasionally on randomly selected particles. However, the magnitude of the collision frequency has negative effects in the calculation of dynamic properties such as the diffusion coefficient because the stochastic collisions disturb the dynamics in a way that is not realistic – it leads to sudden random decorrelation of particle velocities. This effect will result in an enhanced decay of the velocity autocorrelation function, and hence the diffusion constant is changed [59]. Nos´e [143] has shown that one can also perform deterministic MD at constant temperature. By introducing an additional coordinate s and its corresponding mass Q in the Lagrangian of a classical N -body system, he was able to show that the ensemble average of any quantity A taken for the system under his new Lagrangian was equal to the ensemble average of A in the canonical ensemble. Later, Hoover [78] simplified the equations derived by Nos´e and therefore their result is referred to as the Nose-Hoover thermostat. The use of this thermostat directly alters the equations of motion such that the canonical ensemble is sampled during the simulation. Similarly, one can perform constant pressure MD following the IsobaricIsothermal ensemble. Following the same approach as Nos´e, Martyna et al. [127] proposed a new Lagrangian formulation that introduces the variables ξ, pξ and Q. A barostat is then introduced via these variables. Since the volume of the box can change during the simulation, they introduced an equation of motion for the volume. Details of this work are discussed by Martyna et al. [127].

9.2 Analysis of Atomic Trajectories Once an appropriate interatomic potential has been defined, one needs to integrate Newton’s equations of motion according to the statistical ensemble that wants to be sampled, e.g., constant energy, isothermal, isobaric-isothermal.

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However, the general outputs of the simulation are the positions and velocities of the atoms. For this, statistical methods can be used to find a correlation between positions and trajectories. 9.2.1 Equipartition Theorem and the Virial The equipartition theorem is a principle of classical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles at thermal equilibrium will distribute itself evenly among each of the quadratic degrees of freedom allowed to the particles of the system [68]. In MD simulations, the equipartition theorem is used to set the velocities of a group of atoms to a given temperature: N

1   3 N kB T = mi (vi,x − vCM,x )2 + (vi,y − vCM,y )2 + (vi,z − vCM,z )2 , 2 2 i=1

(9.16)

where N is the total number of atoms, kB (1.381×10−23 J K−1 ) is Boltzmann’s constant, T is the temperature of the system of N atoms, mi is the mass, viα is the instantaneous velocity of the ith atom (α = x, y, z) and vCM,α is the velocity component of the center of mass of the group of atoms considered for the temperature calculations. This consideration distinguishes between the thermal velocity of the atoms and the velocity of their collective motion (vCM,α ). In statistical mechanics and expression for the pressure is usually derived starting from the virial theorem of classical dynamics. For spheres interacting with pairwise intermolecular forces the result is: 2 3 N kB T 1  P = + (9.17) Fij · rij , V 3V i