Mechanical Engineering Series

Series Editor

Francis A. Kulacki, Department of Mechanical Engineering,University of Minnesota, Minneapolis, MN, USA

The Mechanical Engineering Series presents advanced level treatment of topics onthe cutting edge of mechanical engineering. Designed for use by students,researchers and practicing engineers, the series presents modern developments inmechanical engineering and its innovative applications in applied mechanics,bioengineering, dynamic systems and control, energy, energy conversion andenergy systems, fluid mechanics and fluid machinery, heat and mass transfer,manufacturing science and technology, mechanical design, mechanics of materials,micro- and nano-science technology, thermal physics, tribology, and vibration andacoustics. The series features graduate-level texts, professional books, and researchmonographs in key engineering science concentrations.

More information about this series at http://www.springer.com/series/1161

Zhuomin M. Zhang

Nano/Microscale HeatTransferSecond Edition

123

Zhuomin M. ZhangMarietta, GA, USA

ISSN 0941-5122 ISSN 2192-063X (electronic)Mechanical Engineering SeriesISBN 978-3-030-45038-0 ISBN 978-3-030-45039-7 (eBook)https://doi.org/10.1007/978-3-030-45039-7

1st edition: © The McGraw-Hill Companies, Inc. 20072nd edition: © Springer Nature Switzerland AG 2020This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, expressed or implied, with respect to the material containedherein or for any errors or omissions that may have been made. The publisher remains neutral with regardto jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my wife Lingyun

Preface

Right after the first edition was published in 2007, I was invited to give seminars ata number of universities in several countries. At that time, many people considerednanotechnology research to be new and emerging; however, quite a few alsowondered what it was about and questioned its real-world applications. Nowadays,the benefits of nanotechnology have entered our daily lives; it is hard for a teenageror a young adult to imagine living in an environment without modern mobiledevices, internet, big data, etc. Many universities have since used my book inteaching graduate and undergraduate courses related to the micro/nanoscale thermaltransport. Remarkable progress has been made in the last decade on the simulations,measurements, and applications of nanoscale thermal science and engineering. Thesecond edition is an update of the first edition, covering the recent advances in thisfield. The structure and chapters are not changed, and the revision follows the samephilosophy: to put the readers first and to make it easy to understand. Moreadvanced topics are covered as overviews with pertinent references so that readerscan seek further details from the literature and other resources.

Over the past thirty years, there have been tremendous developments in micro-electronics, microfabrication technology, MEMS and NEMS, quantum structures(e.g., superlattices, nanowires, nanotubes, graphene and other two-dimensionalmaterials, and nanoparticles), optoelectronics and lasers including ultrafast lasers,and molecular- to atomic-level imaging techniques (such as high-resolution electronmicroscopy, scanning tunneling microscopy, atomic force microscopy, near-fieldoptical microscopy, and scanning thermal microscopy). The field is fast moving intoscaling up and systems engineering to explore the unlimited potential thatnanoscience and nanoengineering may offer to restructure the technologies in thenew millennia. When the characteristic length becomes comparable to the mecha-nistic length scale, continuum assumptions that are often made in conventionalthermal analysis may break down. Similarly, when the characteristic time becomescomparable to the mechanistic time scale, traditional equilibrium approaches maynot be appropriate. Understanding the energy transport mechanisms in smalldimensions and short timescale is crucial for the future advancement of nanotech-nology. In recent years, a growing number of research publications have been in

vii

nano/microscale thermophysical engineering. Timely dissemination of the knowl-edge gained from contemporary research to educate future scientists and engineers isof emerging significance. For this reason, more and more universities are offeringcourses in microscale/nanoscale thermal transport. A self-contained textbook suit-able for engineering students is much needed. Many practicing engineers who havegraduated earlier wish to learn what is going on in this fascinating area, but are oftenfrustrated due to the lack of a solid background to comprehend the contemporaryliterature. A book that does not require prior knowledge in statistical mechanics,quantum mechanics, solid-state physics, and electrodynamics is extremely helpful.Nevertheless, such a book should cover all these subjects in some depth withoutsignificant prerequisites.

This book is written for engineering senior undergraduates and graduate students,practicing engineers, and academic researchers who have not been extensivelyexposed to nanoscale sciences but wish to gain a solid background in the thermalphenomena occurring at small length scales and short timescales. The basic phi-losophy behind this book is to logically integrate the traditional knowledge inthermal engineering and physics with newly developed theories in an easy-to-understand approach, with ample examples and homework problems. The materialshave been used in the graduate courses and undergraduate electives that I havetaught for a number of times at two universities since 1999. While this book can beused as a text for a senior elective or an entry-level graduate course, it is not expectedthat all the materials will be covered in a one-semester course. The instructors havethe freedom to select materials from the book according to students’ backgroundsand interests. Some chapters and sections can also be used to integrate with tradi-tional thermal science courses in order to update the current undergraduate andgraduate curricula with nanotechnology contents.

The content of this book includes microscopic descriptions and approaches, aswell as their applications in thermal science and engineering, with an emphasis onenergy transport in gases and solids by conduction (diffusion) and radiation (with orwithout a medium), as well as convection in micro/nanofluidics. Following theintroduction of Chap. 1, an in-depth overview of the foundation of macroscopicthermodynamics, heat transfer, and fluid mechanics is given in Chap. 2. Chapter 3summarizes the well-established theories in statistical mechanics, including clas-sical and quantum statistics. Thermal properties of ideal gases are described in thecontent of statistical thermodynamics, followed by a concise presentation ofquantum mechanics. Chapter 4 focuses on microfluidics and introduces theBoltzmann transport equations. The heat transfer and microflow regimes fromcontinuous flow to free molecule flow are described. In Chaps. 5–7, heat transfer insolid nanostructures is discussed. Chapter 5 presents the classical and quantum sizeeffects on specific heat and thermal conductivity without involving detailedsolid-state physics, which are introduced in Chap. 6. This arrangement allows amore intuitive learning experience. Chapter 7 focuses on transient as well asnonequilibrium energy transport processes in nanostructures. The next threechapters deal with thermal radiation at nanoscales. Chapter 8 provides the funda-mental understanding of electromagnetic waves and the dielectric functions of

viii Preface

various materials, including metamaterials with exotic properties. Theories ofblackbody radiation, radiation thermometry, and radiation entropy are present.Radiative properties of bulk materials and their relationships are discussed.Chapter 9 describes interference effects of thin films and multilayers, the bandstructure of photonic crystals, diffraction from surface-relief gratings, scatteringfrom rough surfaces, as well as plasmonics and surface polaritons. Chapter 10explores evanescent waves and the coupling phenomena in the near field for energytransfer. Recent advances in nanophotonics and nanoscale radiative heat transfer arealso summarized. In the second edition, significant enhancements have been madeto heat conduction in solids and nanostructures as well as to nanoscale thermalradiation and radiative properties. The dual nature of particles and waves isemphasized throughout the book in explaining the energy carriers, such as mole-cules in ideal gases, electrons in metals, phonons in dielectric crystalline materials,and photons for radiative transfer. Examples in the text and end-of-chapterhomework problems should enhance the understanding of how to apply the for-mulations and methodology to develop problem-solving skills. Selected homeworksolutions will be posted on the author’s website (http://zhang-nano.gatech.edu/),which also contains author’s contact information.

I am deeply in debt to Dr. Markus Flik, my doctoral father, who brought me tothe micro/nano world through the three intense and fruitful years at MIT. Aftergraduation, late Professor and Chancellor of Berkeley, Chang-Lin Tien, my aca-demic grandfather, offered immense support and encouragement for me to write thisbook. I am grateful to my master thesis advisors, Profs. Xin-Shi Ge and YifangWang of the University of Science and Technology of China (Hefei), for giving meearly research training in thermal radiation. I wish to thank my postdoctoral mentorsDr. Raju Datla and Prof. Dennis Drew (University of Maryland) for providing me avaluable opportunity to do research at NIST, where I also benefited from workingwith many outstanding researchers, including Drs. Leonard Hansson, Jack Hsia, JoeRice, Ben Tsai, and late Prof. Dave DeWitt (Purdue).

I am grateful to my colleagues and collaborators at both University of Florida(UF) and Georgia Institute of Technology (GT). I have been fortunate to have verysupportive supervisors from Prof. William Tiederman (former Department Chair atUF) to Profs. Ward Winer and William Wepfer (former School Chairs at GT), toProf. Samuel Graham (current School Chair at GT). I greatly enjoyed the collab-oration with Prof. David Tanner (Physics, UF), as well as the valuable interactionswith Profs. C. K. Hsieh, Yogi Goswami, Sherif Sherif, Jacob Chung, JamesKlausner, and David Hahn while at UF. I cherish the friendship and collaborationwith my colleagues Profs. Yogendra Joshi, G. P. “Bud” Peterson, Peter Hesketh,William King (now at UIUC), Bara Cola, Peter Loutzenhiser, Devesh Ranjan, andShannon Yee at Georgia Tech.

I have also benefited greatly from the support, encouragement, and friendship ofa large number of peers and colleagues in the heat transfer and thermophysicscommunity; too many to list here. I wish to thank members of the ASME Heat

Preface ix

Transfer Division’s K-9 Committee on Nanoscale Thermal Transport for inspiringdiscussions and comments.

This book would not have been possible without my graduate students’ hardwork and dedication. Many of them have taken my classes and proofread differentversions of the manuscripts. Some materials in the last three chapters are generatedbased on their thesis research. I would like to thank my former graduate students atUF Ravi Kumar, Brian Johnson, Donghai Chen, David Pearson, Yihui Zhou,Ferdinand Rosa, Yu-Jiun Shen, Jorge Garcia, and Linxia Gu for helping meestablish my early academic career. My first Ph.D. students graduated at GT,Qunzhi Zhu (currently at Shanghai University of Electric Power) and Ceju Fu(Peking University), made my transition from UF to GT smoother. They werefollowed by many wonderful Ph.D. graduates: Hyunjin Lee (Kookmin University),Yu-Bin Chen (National Tsing Hua University), Keunhan Park (University of Utah),Bong Jae Lee (KAIST), Soumya Basu (PsiQuantum Ltd.), Xiaojia Wang(University of Minnesota), Liping Wang (Arizona State University), AndrewMcNamara (AMD), Trevor Bright (Aerospace Corp.), Richard Z. Zhang(University of North Texas), Jesse Watjen (Knolls Atomic Power Lab), XiangleiLiu (Nanjing University of Aeronautics and Astronautics), Bo Zhao (post-docStanford University), Peiyan Yang (Apple Inc.), and Eric Tervo (NREL director’spost-doc fellow). I am glad to see that most of them have developed their ownindependent research and academic careers and become excellent teachers. Mycurrent Ph.D. students Dudong Feng, Chuyang Chen, Shin Young Jeong, andChiyu Yang have also provided great help during the revision. I am also thankful tomany visiting scholars, post-doc researchers, visiting students, master’s students,and undergraduate students who have worked with me. Many graduate andundergraduate students who have taken my classes also provided constructivesuggestions. I enjoyed working with all of them.

I wish to thank the Thermal Transport Program of NSF for the continuoussupport of my research and educational endeavor since 1998 and the ProgramDirectors Drs. Ashley Emery, Richard Smith, Alfonso Ortega, Patrick Phelan,Theodore Bergman, Sumanta Acharya, José Lage, and Ying Sun. The ongoinggrant number is CBET-1603761. I also gratefully appreciate the Physical Behaviorof Materials Program of DOE (Basic Energy Science) and the Program ManagerDr. Refik Kortan for the confidence and support in the past decade. The ongoinggrant number is DE-SC0018369. I do take full responsibility for any inadvertenterrors or mistakes.

I must thank the Chief Editor of Springer Mechanical Engineering Series, Prof.Francis Kulacki of the University of Minnesota for his encouragement, patience,and valuable comments during the past two years. The Editor at Springer, MichaelLuby, and the editorial team are acknowledged for their hard work putting this bookto print.

x Preface

Finally, I thank my family for their understanding and support throughout thewriting journey. I can’t thank enough my parents and parents-in-law for theirunselfish love and support to me and my family. My children Emmy, Angie, andBryan, now grown-ups, have given me great happiness and made my life mean-ingful. This book is dedicated to my wife Lingyun for the unconditional love andmeticulous care she has provided to me and to our children.

Marietta, GA, USAJanuary 2020

Zhuomin M. Zhang

Preface xi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Limitations of the Macroscopic Formulation . . . . . . . . . . . . . . . 21.2 The Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 From Ancient Philosophy to Contemporary Technologies . . . . . 6

1.3.1 Microelectronics and Information Technology . . . . . . . 71.3.2 Lasers, Optoelectronics, and Nanophotonics . . . . . . . . . 101.3.3 Microfabrication and Nanofabrication . . . . . . . . . . . . . 131.3.4 Probe and Manipulation of Small Structures . . . . . . . . 161.3.5 Energy Conversion and Storage . . . . . . . . . . . . . . . . . 181.3.6 Biomolecule Imaging and Molecular Electronics . . . . . 21

1.4 Objectives and Organization of This Book . . . . . . . . . . . . . . . . 25References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Overview of Macroscopic Thermal Sciences . . . . . . . . . . . . . . . . . . 352.1 Fundamentals of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . 35

2.1.1 The First Law of Thermodynamics . . . . . . . . . . . . . . . 362.1.2 Thermodynamic Equilibrium and the Second Law . . . . 372.1.3 The Third Law of Thermodynamics . . . . . . . . . . . . . . 42

2.2 Thermodynamic Functions and Properties . . . . . . . . . . . . . . . . 442.2.1 Thermodynamic Relations . . . . . . . . . . . . . . . . . . . . . . 442.2.2 The Gibbs Phase Rule . . . . . . . . . . . . . . . . . . . . . . . . 472.2.3 Specific Heats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3 Ideal Gas and Ideal Incompressible Models . . . . . . . . . . . . . . . 512.3.1 The Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3.2 Incompressible Solids and Liquids . . . . . . . . . . . . . . . 54

2.4 Heat Transfer Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4.1 Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4.2 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.4.3 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

xiii

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3 Elements of Statistical Thermodynamics and Quantum Theory . . . 753.1 Statistical Mechanics of Independent Particles . . . . . . . . . . . . . 76

3.1.1 Macrostates Versus Microstates . . . . . . . . . . . . . . . . . . 773.1.2 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.1.3 Quantum Mechanics Considerations . . . . . . . . . . . . . . 793.1.4 Equilibrium Distributions for Different Statistics . . . . . . 81

3.2 Thermodynamic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.2.1 Heat and Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.2.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.2.3 The Lagrangian Multipliers . . . . . . . . . . . . . . . . . . . . . 883.2.4 Entropy at Absolute Zero Temperature . . . . . . . . . . . . 893.2.5 Macroscopic Properties in Terms of the Partition

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.3 Ideal Molecular Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.3.1 Monatomic Ideal Gases . . . . . . . . . . . . . . . . . . . . . . . 933.3.2 Maxwell’s Velocity Distribution . . . . . . . . . . . . . . . . . 953.3.3 Diatomic and Polyatomic Ideal Gases . . . . . . . . . . . . . 97

3.4 Statistical Ensembles and Fluctuations . . . . . . . . . . . . . . . . . . . 1043.5 Basic Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.5.1 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . 1053.5.2 A Particle in a Potential Well or a Box . . . . . . . . . . . . 1073.5.3 A Rigid Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103.5.4 Atomic Emission and the Bohr Radius . . . . . . . . . . . . 1133.5.5 A Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 115

3.6 Emission and Absorption of Photons by Moleculesor Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.7 Energy, Mass, and Momentum in Terms of Relativity . . . . . . . . 1203.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4 Kinetic Theory and Micro/Nanofluidics . . . . . . . . . . . . . . . . . . . . . 1294.1 Kinetic Description of Dilute Gases . . . . . . . . . . . . . . . . . . . . . 130

4.1.1 Local Average and Flux . . . . . . . . . . . . . . . . . . . . . . . 1314.1.2 The Mean Free Path . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.2 Transport Equations and Properties of Ideal Gases . . . . . . . . . . 1384.2.1 Shear Force and Viscosity . . . . . . . . . . . . . . . . . . . . . 1384.2.2 Heat Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1404.2.3 Mass Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

xiv Contents

4.3 Intermolecular Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.3.1 Intermolecular Attractive Forces . . . . . . . . . . . . . . . . . 1464.3.2 Total Intermolecular Pair Potentials . . . . . . . . . . . . . . . 148

4.4 The Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . 1494.4.1 Hydrodynamic Equations . . . . . . . . . . . . . . . . . . . . . . 1514.4.2 Fourier’s Law and Thermal Conductivity . . . . . . . . . . . 154

4.5 Micro/Nanofluidics and Heat Transfer . . . . . . . . . . . . . . . . . . . 1554.5.1 The Knudsen Number and Flow Regimes . . . . . . . . . . 1574.5.2 Velocity Slip and Temperature Jump . . . . . . . . . . . . . . 1604.5.3 Gas Conduction—From the Continuum to the Free

Molecule Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

5 Thermal Properties of Solids and the Size Effect . . . . . . . . . . . . . . 1755.1 Specific Heat of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.1.1 Lattice Vibration in Solids: The Phonon Gas . . . . . . . . 1765.1.2 The Debye Specific Heat Model . . . . . . . . . . . . . . . . . 1795.1.3 Free-Electron Gas in Metals . . . . . . . . . . . . . . . . . . . . 183

5.2 Quantum Size Effect on Specific Heat . . . . . . . . . . . . . . . . . . . 1895.2.1 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . 1895.2.2 General Expressions of Lattice Specific Heat . . . . . . . . 1905.2.3 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.2.4 Thin Films and Nanowires . . . . . . . . . . . . . . . . . . . . . 1935.2.5 Nanoparticles or Nanocrystals . . . . . . . . . . . . . . . . . . . 1965.2.6 Graphite, Graphene, and Carbon Nanotubes . . . . . . . . . 197

5.3 Electrical and Thermal Conductivities of Solids . . . . . . . . . . . . 1985.3.1 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . 1995.3.2 Thermal Conductivity of Metals . . . . . . . . . . . . . . . . . 2035.3.3 Derivation of Conductivities from the BTE . . . . . . . . . 2055.3.4 Thermal Conductivity of Insulators . . . . . . . . . . . . . . . 208

5.4 Thermoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2175.4.1 The Seebeck Effect and Thermoelectric Power . . . . . . . 2185.4.2 The Peltier Effect and the Thomson Effect . . . . . . . . . . 2205.4.3 Thermoelectric Generation and Refrigeration . . . . . . . . 2215.4.4 Onsager’s Theorem and Irreversible

Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255.5 Classical Size Effect on Conductivities . . . . . . . . . . . . . . . . . . . 227

5.5.1 Simple Geometric Considerations . . . . . . . . . . . . . . . . 2275.5.2 Conductivity Along a Thin Film Based on the BTE . . . 2315.5.3 Conductivity Along a Thin Wire Based on the BTE . . . 2365.5.4 Size Effects on Crystalline Insulators . . . . . . . . . . . . . . 2375.5.5 Mean-Free-Path Distribution . . . . . . . . . . . . . . . . . . . . 238

Contents xv

5.6 Quantum Conductance and the Landauer Formalism . . . . . . . . . 2395.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

6 Electron and Phonon Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 2556.1 The Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2566.2 General Classifications of Solids . . . . . . . . . . . . . . . . . . . . . . . 258

6.2.1 Electrons in Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . 2586.2.2 Insulators, Conductors, and Semiconductors . . . . . . . . . 2606.2.3 Atomic Binding in Solids . . . . . . . . . . . . . . . . . . . . . . 263

6.3 Crystal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2656.3.1 The Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 2666.3.2 Primitive Vectors and the Primitive Unit Cell . . . . . . . 2696.3.3 Basis Made of Two or More Atoms . . . . . . . . . . . . . . 271

6.4 Electronic Band Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 2756.4.1 Reciprocal Lattices and the First Brillouin Zone . . . . . . 2756.4.2 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2766.4.3 Band Structures of Metals and Semiconductors . . . . . . 2816.4.4 Electronic Properties of Graphene . . . . . . . . . . . . . . . . 284

6.5 Phonon Dispersion and Scattering . . . . . . . . . . . . . . . . . . . . . . 2896.5.1 The 1D Diatomic Chain . . . . . . . . . . . . . . . . . . . . . . . 2896.5.2 Dispersion Relations for Real Crystals . . . . . . . . . . . . . 2916.5.3 Scattering Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 2946.5.4 Phononics and Coherent Phonons . . . . . . . . . . . . . . . . 300

6.6 Atomistic Simulation of Lattice Thermal Properties . . . . . . . . . 3036.6.1 Interatomic Force Constants (IFCs) . . . . . . . . . . . . . . . 3046.6.2 Lattice Dynamics and Fermi’s Golden Rule . . . . . . . . . 3076.6.3 Evaluation of Thermal Conductivity . . . . . . . . . . . . . . 310

6.7 Electron Emission and Tunneling . . . . . . . . . . . . . . . . . . . . . . . 3136.7.1 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 3136.7.2 Thermionic Emission . . . . . . . . . . . . . . . . . . . . . . . . . 3156.7.3 Field Emission and Electron Tunneling . . . . . . . . . . . . 318

6.8 Electrical Transport in Semiconductor Devices . . . . . . . . . . . . . 3216.8.1 Number Density, Mobility, and the Hall Effect . . . . . . 3216.8.2 Generation and Recombination . . . . . . . . . . . . . . . . . . 3266.8.3 The p-n Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3286.8.4 Optoelectronic Applications . . . . . . . . . . . . . . . . . . . . 332

6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

xvi Contents

7 Nonequilibrium Energy Transfer in Nanostructures . . . . . . . . . . . . 3457.1 Phenomenological Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

7.1.1 Hyperbolic Heat Equation . . . . . . . . . . . . . . . . . . . . . . 3497.1.2 Dual-Phase-Lag Model . . . . . . . . . . . . . . . . . . . . . . . . 3547.1.3 Two-Temperature Model . . . . . . . . . . . . . . . . . . . . . . 360

7.2 Heat Conduction Across Layered Structures . . . . . . . . . . . . . . . 3657.2.1 Equation of Phonon Radiative Transfer (EPRT) . . . . . . 3657.2.2 Solution of the EPRT . . . . . . . . . . . . . . . . . . . . . . . . . 3717.2.3 Thermal Boundary Resistance (TBR) . . . . . . . . . . . . . . 3787.2.4 Atomistic Green’s Function (AGF) . . . . . . . . . . . . . . . 382

7.3 Heat Conduction Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3847.4 Thermal Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

7.4.1 Microbridge and Suspended Microdevices . . . . . . . . . . 3877.4.2 Scanning Probe Microscopic Techniques . . . . . . . . . . . 3907.4.3 Noncontact Optical Techniques . . . . . . . . . . . . . . . . . . 391

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

8 Fundamentals of Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . 4078.1 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

8.1.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . 4108.1.2 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 4128.1.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4148.1.4 Energy Flux and Density . . . . . . . . . . . . . . . . . . . . . . 4168.1.5 Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 4178.1.6 Propagating and Evanescent Waves . . . . . . . . . . . . . . . 420

8.2 Blackbody Radiation: The Photon Gas . . . . . . . . . . . . . . . . . . . 4218.2.1 Planck’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218.2.2 Radiation Thermometry . . . . . . . . . . . . . . . . . . . . . . . 4268.2.3 Radiation Pressure and Photon Entropy . . . . . . . . . . . . 4308.2.4 Limitations of Planck’s Law . . . . . . . . . . . . . . . . . . . . 437

8.3 Radiative Properties of Semi-infinite Media . . . . . . . . . . . . . . . 4378.3.1 Reflection and Refraction of a Plane Wave . . . . . . . . . 4388.3.2 Total Internal Reflection and the Goos–Hänchen

Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4448.3.3 Bidirectional Reflectance Distribution Function . . . . . . 4488.3.4 Emittance (Emissivity) and Kirchhoff’s Law . . . . . . . . 450

8.4 Dielectric Function Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 4538.4.1 Kramers–Kronig Dispersion Relations . . . . . . . . . . . . . 4548.4.2 The Drude Model for Free Carriers . . . . . . . . . . . . . . . 4568.4.3 The Lorentz Oscillator Model for Phonon

Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4598.4.4 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

Contents xvii

8.4.5 Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4678.4.6 Metamaterials with a Magnetic Response . . . . . . . . . . . 469

8.5 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4728.5.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4738.5.2 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4758.5.3 Dispersive Instruments . . . . . . . . . . . . . . . . . . . . . . . . 4798.5.4 Fourier-Transform Infrared Spectrometer . . . . . . . . . . . 4828.5.5 BRDF and BTDF Measurements . . . . . . . . . . . . . . . . . 4858.5.6 Ellipsometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

9 Radiative Properties of Nanomaterials . . . . . . . . . . . . . . . . . . . . . . 4979.1 Radiative Properties of a Single Layer . . . . . . . . . . . . . . . . . . . 497

9.1.1 The Ray-Tracing Method for a Thick Layer . . . . . . . . . 4989.1.2 Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5029.1.3 Partial Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5079.1.4 Effect of Surface Scattering . . . . . . . . . . . . . . . . . . . . . 512

9.2 Multilayer Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5149.2.1 Thin Films with Two or Three Layers . . . . . . . . . . . . . 5159.2.2 The Matrix Formulation . . . . . . . . . . . . . . . . . . . . . . . 5179.2.3 Thin Films on a Thick Substrate . . . . . . . . . . . . . . . . . 5209.2.4 Waveguides and Optical Fibers . . . . . . . . . . . . . . . . . . 521

9.3 Photonic Crystals and Periodic Gratings . . . . . . . . . . . . . . . . . . 5269.3.1 Photonic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5279.3.2 Periodic Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5329.3.3 Rigorous Coupled-Wave Analysis (RCWA) . . . . . . . . . 5339.3.4 Effective Medium Formulations . . . . . . . . . . . . . . . . . . 537

9.4 Bidirectional Reflectance Distribution Function (BRDF) . . . . . . 5389.4.1 The Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . . 5409.4.2 The Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . 5419.4.3 Surface Characterization . . . . . . . . . . . . . . . . . . . . . . . 5449.4.4 Comparison of Modeling with Measurements . . . . . . . . 545

9.5 Plasmon, Polariton, and Electromagnetic Surface Wave . . . . . . . 5499.5.1 Surface Plasmon (or Phonon) Polariton . . . . . . . . . . . . 5519.5.2 Localized Surface Plasmon Resonance . . . . . . . . . . . . . 5579.5.3 Polaritons in Thin Films and Layered Structures . . . . . 5609.5.4 Magnetic Polariton . . . . . . . . . . . . . . . . . . . . . . . . . . . 5649.5.5 Graphene: Optical Properties and Graphene

Plasmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5719.5.6 Hyperbolic (Plasmon or Phonon) Polariton . . . . . . . . . 5779.5.7 General Effective Medium Theory . . . . . . . . . . . . . . . . 583

xviii Contents

9.6 Spectral and Directional Control of Thermal Radiation . . . . . . . 5859.6.1 Polariton-Enhanced Transmission . . . . . . . . . . . . . . . . 5869.6.2 Perfect Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919.6.3 Tailoring Thermal Emission with Nanostructures . . . . . 599

9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

10 Near-Field Energy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62310.1 From Near-Field Optics to Nanoscale Thermal Radiation . . . . . 62310.2 Photon Tunneling and Near-Field Radiative Heat Transfer . . . . 627

10.2.1 Photon Tunneling by Coupled Evanescent Waves . . . . 62710.2.2 Thermal Energy Transfer Between Closely Spaced

Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63010.2.3 Resonance Tunneling Through Periodic Dielectric

Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63410.2.4 Photon Tunneling with Negative Index Materials . . . . . 636

10.3 Energy Streamlines and Superlens . . . . . . . . . . . . . . . . . . . . . . 63910.4 Radiative Transfer Between Two Semi-Infinite Media . . . . . . . . 644

10.4.1 Fluctuational Electrodynamics . . . . . . . . . . . . . . . . . . . 64510.4.2 Near-Field Radiative Heat Transfer Between

Two Parallel Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . 64910.4.3 Effect of Surface Plasmon Polaritons (SPPs) . . . . . . . . 65310.4.4 Effect of Surface Phonon Polaritons (SPhPs) . . . . . . . . 65910.4.5 The Landauer-Like Formulism . . . . . . . . . . . . . . . . . . 662

10.5 Multilayers, Anisotropic Media, and 2D Materials . . . . . . . . . . 66310.5.1 Multilayers and Hyperbolic Modes . . . . . . . . . . . . . . . 66310.5.2 Graphene and Hexagonal Boron Nitride . . . . . . . . . . . 66710.5.3 Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . 67210.5.4 Green’s Functions for Multilayer Structures . . . . . . . . . 673

10.6 Nanostructures and Numerical Methods . . . . . . . . . . . . . . . . . . 67810.6.1 The Scattering Theory for Periodic Structures . . . . . . . 67810.6.2 Finite-Difference Time-Domain (FDTD) Method . . . . . 68410.6.3 Boundary Element Method (BEM) . . . . . . . . . . . . . . . 68810.6.4 Multiple Dipole Approaches . . . . . . . . . . . . . . . . . . . . 689

10.7 Measurements and Applications . . . . . . . . . . . . . . . . . . . . . . . . 69410.7.1 Measurements of Near-Field Thermal Radiation . . . . . . 69510.7.2 Application Prospects of Nanoscale Thermal

Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70410.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710

Contents xix

Appendix A: Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

Appendix B: Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 725

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745

xx Contents

List of Symbols

A Area, m2; Helmholtz free energy, JAc Cross-sectional area, m2

A0k Spectral, directional absorptance of a semitransparent material

a Acceleration, m/s2

a Lattice constant, m; magnitude of acceleration, m/s2

a0 Bohr radius, 0.0529 nmak Absorption coefficient, m−1

B Magnetic induction or magnetic flux density, T (tesla) or Wb/m2

C Volumetric heat capacity (qcp), J/K m3

c Phase speed of electromagnetic wave, m/sc0 Speed of light in vacuum, 2:998� 108 m/scp or cv Specific heat for constant pressure or constant volume, J/kg KD Dynamic matrix; electric displacement, C/m2

D Density of states, m−3; diameter, mDAB Binary diffusion coefficient, m2/sd Diameter or film thickness, mE Electric field vector, N/C or V/mE Energy, J; magnitude of electric field, V/mEF Fermi energy, JEg Bandgap energy, Je Electron charge (absolute value), 1:602� 10�19 Ceb Blackbody emissive power, W/m2

F, F Force, NF Normalized distribution functionf Distribution functionG Reciprocal lattice vector, m−1; dyadic Green’s functionG Gibbs free energy, J; electron-phonon coupling constant, W/m3 Kg Degeneracy�g Molar specific Gibbs free energy, J/kmolH Magnetic field vector, A/m

xxi

H Enthalpy, J; magnetic field strength, A/mh Mass specific enthalpy, J/kg; convective heat transfer coefficient,

W/m2 K; Planck’s constant, 6:626� 10�34 J shm Mass transfer coefficient, m/s�h Planck’s constant divided by 2p, h/2p�h Molar specific enthalpy, J/kmolI Unit matrix; unit dyadicI Moment of inertia, kg m2; intensity or radiance, W/m2 lm sr; current, Ai

ffiffiffiffiffiffiffi�1p

i; j; k Indices used in seriesJ or J Flux vector or magnitude (quantity transferred per unit area per unit

time)J or Je current density (or electric charge flux), A/m2

K Block wavevector, m−1

K Spring constant, N/m; Thomson’s coefficient, V/Kk Wavevector, m−1

k Magnitude of the wavevector, m−1

kB Boltzmann’s constant, 1:381� 10�23 J/KL, l Length or characteristic length, mL0 Average distance between molecules or atoms, mLk Radiation entropy intensity, W/K m2 lm srl, m, n Index numberM Molecular weight, kg/kmolm Mass of a system or a single particle, kgmr Reduced mass, kgm* Effective mass, kgN Number of particles; number of phonon oscillatorsNA Avogadro’s constant, 6:022� 1026 kmol�1; acceptor concentration,

m�3

ND Donor concentration, m�3

_N Particle flow rate, s�1

n Number density, m−3; quantum number; index number; refractive index;or real part of the complex refractive index

�n Amount of substance, kmol~n Complex refractive indexP Propagation matrix; polarization vector or dipole moment per unit

volume, C/m2

P Pressure, Pa (N/m2 or J/m3)Pij Momentum flux component, Pap Momentum vector (mv or �hk), kg m/sp Momentum (mv or �hk), kg m/s; probability; specularityQ Heat, J; quality factor of a resonance_Q Heat transfer rate, Wq Number of coexisting phases; number of atoms per molecule

xxii List of Symbols

_q Thermal energy generation rate, W/m3

q00 Heat flux vector, W/m2

q00 Heat flux, W/m2

R Gas constant, J/kg K; electric resistance, X or V/AR0 Directional–hemispherical reflectanceR00b Thermal boundary resistance, m2 K/W

R00t Thermal resistance, m2 K/W

�R Universal gas constant, 8314.4 J/kmol Kr Distance or radius, m; reflection coefficientre Electrical resistivity, Xm~r Complex Fresnel’s reflection coefficientS Poynting vector, W/m2

S Entropy, J/KSj Strength of the jth phonon oscillator_S Entropy transfer rate, W/K_Sgen Entropy generation rate, W/Ks Specific entropy J/kg K; entropy density, J/m3 K�s Molar specific entropy, J/kmol K_sgen Volumetric entropy generation rate, W/m3 Ks00 Entropy flux, J/m2 KT Temperature, KT 0 Directional–hemispherical transmittancet Time, s; transmission coefficient~t Complex Fresnel’s transmission coefficientU Internal energy, J; potential, Jud Drift velocity, m/su Specific internal energy, J/kg; energy density, J/m3

�u Molar specific internal energy, J/kmolV Volume, m3; voltage, Vv Velocity, m/svB Bulk or mean velocity, m/svR Random or thermal velocity, m/sv Speed, m/s; specific volume, m3=kgva Speed of sound or average speed of phonons, m/svF Fermi velocity, m/svg Magnitude of group velocity (dx=dk), m/svl; vt Longitudinal, transverse phonon speed, m/svp Phase speed (x=k), m/svx; vy; vz Velocity components, m/s�v Average speed, m/s; molar specific volume, m3=kmolW Work, J; width, mx; y; z Coordinates, mZ Partition function

List of Symbols xxiii

Dimensionless Parameters

Kn Knudsen number, K=LLe Lewis number, DAB=a ¼ Pr=ScLz Lorentz number, j=rTMa Mach number, v=vaNu Nusselt number, hL=jPe Peclet number, RePr ¼ v1L=aPr Prandtl number, m=aRe Reynolds number, qv1L=lSc Schmidt number, m=DAB

ZT Dimensionless figure of merit for thermoelectricity

Greek Symbols

a Thermal diffusivity, m2/s; some constant; polarizabilitya and b Lagrangian multipliers; indicesaT Thermal accommodation coefficientav (Tangential) momentum accommodation coefficientav0 Normal momentum accommodation coefficienta0k Spectral, directional absorptivity or absorptanceb Parallel wavevector component, m−1; various coefficientsbP Isobaric thermal expansion coefficient, K−1

bT 2cð2� aTÞKn=½aTðcþ 1ÞPr�bv ð2� avÞKn=avCij Hemispherical transmissivity for phonons from media i to jCS Seebeck’s coefficient, V/Kc Specific heat ratio; scattering rate (1/s), rad/scs Sommerfeld constant, J/kg K2

d Differential small quantity; delta function; boundary layer thickness, mdk Radiation penetration depth, me Energy of a particle or quasiparticle, J; electric permittivity, F/m; relative

permittivity; emittance; or emissivity~e Complex dielectric function or relative permittivitye0k spectral, directional emissivity (or emittance)f Dummy variable; perpendicular wavevector component, m−1

g Various efficiencies; imaginary part of perpendicular wavevector, m−1

gH Hall coefficient Ey�JxB, m3

�C

H Characteristic temperature, K; mean energy of Planck’s oscillatorHD Debye temperature, Kh Zenith angle, radhB Brewster’s angle, rad

xxiv List of Symbols

hc Critical angle, radj Thermal conductivity, W/m K; extinction coefficient or imaginary part

of the refractive indexjT Isothermal compressibility, Pa−1

K Mean free path, m; period of a grating or photonic crystal, mKa Average collision distance, mk Wavelength in vacuum, m (often expressed in lm)l (Dynamic) viscosity, N s/m2; chemical potential, J; electron or hole

mobility, m2=V s; magnetic permeability, N/A2; relative magneticpermeability

lF Fermi energy, Jm Kinematic viscosity, m2=s; frequency, Hz�m Wavenumber, m−1 (often expressed in cm−1)n Energy transmission coefficient or photon tunneling probability; certain

coordinate or variableP Peltier’s coefficient, Vq Density, kg/m3

qe Charge density, C/m3

q0 Directional–hemispherical reflectanceq0k Spectral, directional reflectivityr Electrical conductivity, ðXm)�1; standard deviationrrms Root-mean-square surface roughness, mrSB Stefan–Boltzmann constant, 5:67� 10�8 W/m2 K4

r0SB Phonon Stefan–Boltzmann constant, W/m2 K4

s Relaxation time, s; transmission coefficients0 Directional–hemispherical transmittanceU Scattering phase function; viscous dissipation function; potential function/ Number of degrees of freedom; azimuthal angle, rad; intermolecular

potentialv susceptibilityW Schrödinger’s wave function; various functionsw Molecular quantity; wave function; work function, J; phase shift, radX Solid angle, sr; thermodynamic probabilityx Angular frequency, rad/sxp Plasma frequency, rad/s- Velocity space, d- ¼ dvxdvydvz

List of Symbols xxv

Subscripts

0 Vacuum or free space1,2,3 Medium 1,2,3b Blackbody; boundaryE Energye Electron, electrich Holei Incidentm Bulk or mean; maximum; mediummp Most probablen normal directionn or p n-type or p-type semiconductorp TM wave, p (parallel) polarizationr reflected; rotationals Lattice; scattered; solid; surfaces TE wave, s (perpendicular) polarizationt Transmitted; translationalth Thermalv Vibrationalw Wall1 Free steamk; m; or x Spectral quantity in terms of wavelength, frequency, or angular

frequency

xxvi List of Symbols