Graduate Texts in Physics

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Kurt H. Becker, Polytechnic School of Engineering, Brooklyn, USAJean-Marc Di Meglio, Université Paris Diderot, Paris, FranceSadri Hassani, Illinois State University, Normal, USABill Munro, NTT Basic Research Laboratories, Atsugi, JapanRichard Needs, University of Cambridge, Cambridge, UKWilliam T. Rhodes, Florida Atlantic University, Boca Raton, USASusan Scott, Australian National University, Acton, AustraliaH. Eugene Stanley, Boston University, Boston, USAMartin Stutzmann, TU München, Garching, GermanyAndreas Wipf, Friedrich-Schiller-Universität Jena, Jena, Germany

Graduate Texts in Physics

Graduate Texts in Physics publishes core learning/teachingmaterial for graduate- andadvanced-level undergraduate courses on topics of current and emerging fields withinphysics, both pure and applied. These textbooks serve students at the MS- orPhD-level and their instructors as comprehensive sources of principles, definitions,derivations, experiments and applications (as relevant) for their mastery and teaching,respectively. International in scope and relevance, the textbooks correspond to coursesyllabi sufficiently to serve as required reading. Their didactic style, comprehensive-ness and coverage of fundamental material also make them suitable as introductionsor references for scientists entering, or requiring timely knowledge of, a research field.

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Mildred Dresselhaus • Gene DresselhausStephen B. Cronin • Antonio Gomes Souza Filho

Solid State PropertiesFrom Bulk to Nano

123

Mildred DresselhausDepartment of Electrical Engineeringand Computer Science and Departmentof Physics

Massachusetts Institute of TechnologyCambridge, MAUSA

Gene DresselhausFrancis Bitter Magnet LaboratoryMassachusetts Institute of TechnologyCambridge, CAUSA

Stephen B. CroninUniversity Park CampusUniversity of Southern CaliforniaLos Angeles, CAUSA

Antonio Gomes Souza FilhoDepartamento de FísicaUniversidade Federal do CearáFortaleza, CearáBrazil

ISSN 1868-4513 ISSN 1868-4521 (electronic)Graduate Texts in PhysicsISBN 978-3-662-55920-8 ISBN 978-3-662-55922-2 (eBook)https://doi.org/10.1007/978-3-662-55922-2

Library of Congress Control Number: 2017954490

© Springer-Verlag GmbH Germany 2018This 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, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer-Verlag GmbH, DEThe registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany

Foreword

What is solid state physics about? Take all the fundamentals of physics, includingclassical and quantum mechanics, electromagnetism, thermodynamics and statisti-cal physics, and put them all together to study a piece of matter. Most physicistsworldwide work in this field, largely interdisciplinary, overlapping with chemistry,engineering, biology and medicine. Sensors, solar panels, batteries, light emittingdiodes, flat displays, touch screens, computer devices, optical fibers, field emitters,high performance coolers; these are all examples of technologies resulting fromsolid state physics applications. Cyber-physical systems in the 4.0 industry,internet-of-things (IoT), lab-on-a-chip; all rely on the applications of solid statesystems.

Merging together all the fundamentals of physics into a piece of matter soundsbeautiful, but it also sounds very complicated! The concepts and mathematicalmachinery that are needed to understand the main electrical, magnetic, thermal andoptical properties of materials have to be in place. It is necessary to draw con-nections from the quantum mechanics to the physical properties of matter; fromcontinuity relations to transport phenomena; from the Maxwell equations to opticalobservables. To achieve such an endeavor, a researcher needs a textbook that wastailored towards the best learning experience. And this is exactly what has beendelivered by this book.

Solid State Properties—from bulk to nano was built based on the class notesof the MIT Professors Mildred Spiewak Dresselhaus and Gene Dresselhaus, withthe contribution of many of their students and post-doctoral fellows, over decades.Professors Mildred and Gene Dresselhaus had the unique view of those who pio-neered many discoveries on materials science, and followed the processes from thediscovery of the basic concepts up to applications. One example was the study anddevelopment of graphite intercalated compounds in the 1970s, a material that istoday the basis of chargeable cell phone and electric car batteries, and it keepsimproving grateful to nanotechnology. Professors Stephen B. Cronin and AntonioGomes Souza Filho were former students of Profs. Dresselhaus, and today theyhold worldwide recognition in the fields of transport and optics of nanostructures.Because I had the experience of working with Profs. Dresselhaus on a similar

v

project, the publication of their group theory class notes, I understand the importantrole played by Profs. Cronin and Souza Filho on transforming class notes into aself-contained textbook, working under the high standards of Professors Mildredand Gene Dresselhaus.

When travelling from bulk to nano, the book provides a sharp cut on the modernview of solid state science. It shows how the basic concepts that were introduced inthe physics of bulk materials over the last century are developing while nanoscienceand nanotechnology are taking place. This overview makes the reader capable notonly of understanding the present materials science, but most importantly, thereader is able to evaluate the limitations of the modern concepts, to go beyond thewell-established knowledge. Solid State Properties—from bulk to nano is thetextbook for those who want to understand modern scientific papers related to novelmaterials properties, and for those who want to work and have an impact in thefield.

Belo HorizonteApril 2017

Ado Jorio

vi Foreword

Preface

We had the great pleasure and honor of having Mildred (Millie) Dresselhaus serveas our Ph.D. advisor. She taught us more than we could ever express in words, andshe was patient as we worked to learn the material now contained in this book.Several years ago, when we approached Millie about publishing her class notes, wethought this would be an easy task since her husband Gene Dresselhaus had alreadyformatted those notes beautifully in LaTex. But Millie, as in all her work, had veryhigh standards and insisted that we update it to include new low-dimensionalmaterials that would be of particular interest to present-day students. As a result,revisions went on for two and a half years. We were just completing the finalchecking of the textbook when Millie passed away on February 20, 2017. Despiteher age, this news came as a shock because she had been working actively up untiltwo weeks before her death. In fact, we were actually having trouble keeping upwith her. We were deeply saddened to hear the news of her sudden passing. We hadlost a great scientist, mentor, advisor, and friend.

This book is based on an introductory solid state physics class (MIT coursenumber 6.732) that Millie started teaching in the mid-1970s and continued teachingand revising until 2005. Continuing in Millie’s footsteps, we have taught a versionof the course at the University of Southern California since 2007 as have several ofher other former students and postdocs, including Ado Jorio at the UniversidadeFederal de Minas Gerais (Brazil) and Antonio Gomes Souza Filho at theUniversidade Federal do Ceará (Brazil). For us, as for many of Millie’s formerstudents, this course was the single most important class that shaped our academiccareer.

Solid State Properties: From Bulk to Nano fills a gap between many of the basicsolid state physics and materials science books that are currently available. It iswritten for a mixed audience of electrical engineering and applied physics studentswho have some knowledge of elementary undergraduate quantum mechanics andstatistical mechanics. This book is organized into three parts: (I) Electronic

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Structure, (II) Transport Properties, and (III) Optical Properties. Each topic isexplained in the context of bulk materials and then extended to low-dimensionalmaterials where applicable. The chapters end with homework problems to providestudents an opportunity to engage with the material more intimately.

As we were finishing the book, Millie expressed her appreciation for the manyformer students who contributed to its development. Millie wrote: “For me, it wasformer students who took an old version of my introductory solid state coursestarting in the 1970s and continuing through decades of different students comingfrom different countries around the world with different interests and needs. Overmy (45 years) active classroom teaching career (from the 1960s to 2005), I enjoyedbeing in the classroom, learning with the many students I had the opportunity towork with.” This was a genuine trait of Millie while at the top of her field to thinkof herself as learning along with her students.

We are extremely grateful to all graduate students in Prof. Cronin’s group atUSC who from 2011 to 2017 worked diligently creating figures, implementingchanges in LaTex, and checking sources in the literature. Without their massiveeffort, the publication of this book would not have been possible.

It has been a tremendous honor to have played a role in publishing this book. Wehope that, because of Millie’s diligence and expertise, it will continue to teach andinspire another generation of scientists and engineers.

Los Angeles, CA, USA Stephen B. CroninMarch 2017 Professor of Electrical Engineering

Physics, and ChemistryUniversity of Southern California

Fortaleza, Brazil Antonio Gomes Souza FilhoMarch 2017 Professor of Physics, Universidade

Federal do Ceará

viii Preface

In Memory Of

Mildred Dresselhaus (1930–2017), also known as Queen of Carbon, had an illus-trious career that spanned six decades. She was the first female Professor to receivefull tenure at the Massachusetts Institute of Technology in 1968. She publishedmore than 1700 scientific papers, co-wrote eight books and received various awardsand accolades for her contributions to science and technology during the course ofher life. She was awarded the National Medal of Science in 1990, the 11th AnnualHeinz Award in 2005, the Oersted Medal in 2008, and the Kavli Prize in 2012. Shewas also co-recipient of the Enrico Fermi Award in 2012. She received thePresidential Medal of Freedom from President Obama in 2012. MildredDresselhaus served as the director of the Office of Science at the US Department of

Photograph courtesy of Micheline Pelletier taken for Millie’s L’Oreal/UNESCO Women inScience Prize 2007

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Energy from 2000 to 2001 and as a chair of the governing board of the AmericanInstitute of Physics from 2003 to 2008. She was also the president of AmericanPhysical Society, the first female president of the American Association for theAdvancement of Science, and the treasurer of the National Academy of Sciences.

Every morning, Millie would leave her house by 5:30 AM, and her car wasalways the first car in the parking lot at MIT. As a Ph.D. student, she studied underEnrico Fermi at the University of Chicago. He too was an early riser, and, since helived nearby, they had plenty of time to discuss science as they walked to schooltogether. Millie often spoke of those conversations and their influence on herstudies in that challenging academic program. It was at Chicago that she met herfuture husband Gene Dresselhaus, and in 1960, they were both hired by MITLincoln Laboratory. One of the main reasons Millie decided to study carbon wasthat it was relatively unpopular. She wanted to work on a project that most peoplethought was hard and not that interesting, so that it would be okay if she had to stayhome with a sick child. As an icon for women in science and a strong advocate forwomen in STEM, she worked tirelessly to expand opportunities for women inscience.

x In Memory Of

Contents

Part I Electronic Structure

1 Crystal Lattices in Real and Reciprocal Space . . . . . . . . . . . . . . . . 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Crystalline Lattices: Real Space . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Unit Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Lattices in Reciprocal Space . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Crystal Planes and Miller Indices . . . . . . . . . . . . . . . . 61.3.2 Reciprocal Lattice Vectors . . . . . . . . . . . . . . . . . . . . . 7

1.4 The Brillouin Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Graphene and Boron Nitride . . . . . . . . . . . . . . . . . . . . 81.4.2 Diamond and Zinc Blende Lattices . . . . . . . . . . . . . . . 9

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Electronic Properties of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Hamiltonian of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 The Electronic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 The Hartree Method . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Hartree–Fock (HF) Method . . . . . . . . . . . . . . . . . . . . . 152.3.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . 16

2.4 Plane Wave and Localized Basis Sets . . . . . . . . . . . . . . . . . . . 192.5 Hamiltonian Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . 212.6 Bloch Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7 The Slater–Koster Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 25References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Weak and Tight Binding Approximations for SimpleSolid State Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 One Electron EðKÞ in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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3.2.1 Weak Binding or the Nearly Free ElectronApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.2 Tight Binding Approximation . . . . . . . . . . . . . . . . . . . 373.2.3 Comparison of Weak and Tight Binding

Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.4 Tight Binding Approximation with 2

Atoms/Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Examples of Energy Bands in Solids . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Alkali Metals–e.g., Sodium . . . . . . . . . . . . . . . . . . . . . 574.2.2 Noble Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.3 Polyvalent Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3.1 PbTe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3.2 Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.3 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.4 III–V Compound Semiconductors . . . . . . . . . . . . . . . . 704.3.5 Zero Gap Semiconductors – Gray Tin . . . . . . . . . . . . . 724.3.6 Transition Metal Dichalcogenides, Such as MoS2

and WS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3.7 Molecular Semiconductors – Fullerenes . . . . . . . . . . . . 73

4.4 Semimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4.1 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4.2 Bismuth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5 Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.5.1 Rare Gas and Ionic Crystals . . . . . . . . . . . . . . . . . . . . 774.5.2 Boron Nitride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.5.3 Wide Bandgap Semiconductors . . . . . . . . . . . . . . . . . . 79

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 Effective Mass Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Wavepackets in Crystals and the Group Velocity

of Electrons in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3 The Effective Mass Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 925.4 Application of the Effective Mass Theorem to Donor

Impurity Levels in a Semiconductor . . . . . . . . . . . . . . . . . . . . . 955.5 Quasi-classical Electron Dynamics . . . . . . . . . . . . . . . . . . . . . . 98

xii Contents

5.6 Quasi-classical Theory of ElectricalConductivity – Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6 Lattice Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.2 Quantum Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . 1056.3 Phonons in 1D Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.3.1 A Monoatomic Chain . . . . . . . . . . . . . . . . . . . . . . . . . 1086.3.2 Diatomic Linear Chain . . . . . . . . . . . . . . . . . . . . . . . . 111

6.4 Phonons in 3D Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.5 Electron-Phonon Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Part II Transport Properties

7 Basic Transport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.2 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.3 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.4 Electrical Conductivity of Metals . . . . . . . . . . . . . . . . . . . . . . . 1307.5 Electrical Conductivity of Semiconductors . . . . . . . . . . . . . . . . 131

7.5.1 Ellipsoidal Carrier Pockets . . . . . . . . . . . . . . . . . . . . . 1347.6 Electrons and Holes in Intrinsic Semiconductors . . . . . . . . . . . . 1367.7 Donor and Acceptor Doping of Semiconductors . . . . . . . . . . . . 1407.8 Characterization of Semiconductors . . . . . . . . . . . . . . . . . . . . . 144Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8 Thermal Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . 1558.2.2 Thermal Conductivity for Metals . . . . . . . . . . . . . . . . . 1598.2.3 Thermal Conductivity for Semiconductors . . . . . . . . . . 1618.2.4 Thermal Conductivity for Insulators . . . . . . . . . . . . . . 163

8.3 Thermoelectric Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . 1648.3.1 Thermoelectric Phenomena in Metals . . . . . . . . . . . . . 1688.3.2 Thermopower for Intrinsic Semiconductors . . . . . . . . . 1708.3.3 Effect of Thermoelectricity on the Thermal

Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Contents xiii

8.4 Thermoelectric Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 1748.4.1 Seebeck Effect (Thermopower) . . . . . . . . . . . . . . . . . . 1748.4.2 Peltier Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1758.4.3 Thomson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1768.4.4 The Kelvin Relations . . . . . . . . . . . . . . . . . . . . . . . . . 1778.4.5 The Thermoelectric Figure of Merit . . . . . . . . . . . . . . . 178

8.5 The Phonon Drag Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

9 Electron and Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.1 Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.2 Scattering Processes in Semiconductors . . . . . . . . . . . . . . . . . . 188

9.2.1 Electron-Phonon Scattering in Semiconductors . . . . . . . 1889.2.2 Ionized Impurity Scattering . . . . . . . . . . . . . . . . . . . . . 1929.2.3 Other Scattering Mechanisms . . . . . . . . . . . . . . . . . . . 1939.2.4 Screening Effects in Semiconductors . . . . . . . . . . . . . . 194

9.3 Electron Scattering in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . 1979.3.1 Electron-Phonon Scattering in Metals . . . . . . . . . . . . . 1979.3.2 Other Scattering Mechanisms in Metals . . . . . . . . . . . . 201

9.4 Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2029.4.1 Phonon-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . 2029.4.2 Phonon-Boundary Scattering . . . . . . . . . . . . . . . . . . . . 2049.4.3 Defect-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . 2059.4.4 Electron-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . 205

9.5 Temperature Dependence of the Electrical and ThermalConductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

10 Magneto-Transport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . 21110.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21110.2 Magneto-Transport in the Classical Regime (xcs \1) . . . . . . . 211

10.2.1 Classical Magneto-Transport Equations . . . . . . . . . . . . 21210.2.2 Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

10.3 The Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21410.4 Derivation of the Magneto-Transport Equations from the

Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21610.5 Two Carrier Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21810.6 Cyclotron Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22010.7 Effective Masses for Ellipsoidal Fermi Surfaces . . . . . . . . . . . . 22210.8 Dynamics of Electrons in a Magnetic Field . . . . . . . . . . . . . . . 222Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

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11 Transport in Low Dimensional Systems . . . . . . . . . . . . . . . . . . . . . 23111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23111.2 Observation of Quantum Effects in Reduced Dimensions . . . . . 23111.3 Density of States in Low Dimensional Systems . . . . . . . . . . . . 233

11.3.1 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23411.4 Ballistic Transport and the Landauer Formula . . . . . . . . . . . . . . 235

11.4.1 Relationship Between the Mean Free Path and theTransmission Coefficient . . . . . . . . . . . . . . . . . . . . . . . 237

11.4.2 Relationship to the Boltzmann Transport . . . . . . . . . . . 23911.4.3 Relationship to Mobility Calculations . . . . . . . . . . . . . 23911.4.4 Dependence of the Fermi Energy on Gate Voltage . . . . 24111.4.5 Ballistic Phonon Transport . . . . . . . . . . . . . . . . . . . . . 241

11.5 Quantum Point Contacts (QPC) Effects . . . . . . . . . . . . . . . . . . 24211.6 Coulomb Blockade and Single Electron Transistors (SETs) . . . . 243Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

12 Two Dimensional Electron Gas, Quantum Wells andSemiconductor Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24712.1 Two-Dimensional Electronic Systems . . . . . . . . . . . . . . . . . . . . 24712.2 MOSFETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24712.3 Two-Dimensional Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

12.3.1 Quantum Wells and Superlattices . . . . . . . . . . . . . . . . 25312.4 Bound Electronic States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25612.5 Review of Tunneling Through a Potential Barrier . . . . . . . . . . . 25712.6 Quantum Wells of Different Shape and the WKB

Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25812.7 The Kronig–Penney Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 26112.8 3D Motion Within a 1–D Rectangular Well . . . . . . . . . . . . . . . 26312.9 Resonant Tunneling in Quantum Wells . . . . . . . . . . . . . . . . . . 265Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

13 Magneto-Oscillatory and Other Effects Associated with LandauLevels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27513.1 Introduction to Landau Levels . . . . . . . . . . . . . . . . . . . . . . . . . 27513.2 Quantized Magnetic Energy Levels in 3D . . . . . . . . . . . . . . . . 275

13.2.1 Degeneracy of the Magnetic Energy Levels in kx . . . . . 27713.2.2 Dispersion of the Magnetic Energy Levels Along

the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 27813.2.3 Band Parameters Describing the Magnetic Energy

Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28113.3 Overview of Landau Level Effects . . . . . . . . . . . . . . . . . . . . . . 28213.4 Quantum Oscillatory Magnetic Phenomena . . . . . . . . . . . . . . . 285

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13.5 Selection Rules for Landau Level Transitions . . . . . . . . . . . . . . 28913.6 Landau Level Quantization for Large Quantum Numbers . . . . . 290Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

14 The Quantum Hall Effect (QHE) . . . . . . . . . . . . . . . . . . . . . . . . . . 29514.1 Introduction to the Quantum Hall Effect . . . . . . . . . . . . . . . . . . 29514.2 Basic Relations for 2D Hall Resistance . . . . . . . . . . . . . . . . . . 29714.3 The 2D Electron Gas and the Quantum Hall Effect . . . . . . . . . . 29914.4 Effect of Edge Channels and the Quantum Field Effect . . . . . . . 30414.5 Precision of the Quantized Hall Effect and Applications . . . . . . 30714.6 Fractional Quantum Hall Effect (FQHE) . . . . . . . . . . . . . . . . . . 308Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Part III Optical Properties

15 Review of Fundamental Relations for Optical Phenomena . . . . . . . 31715.1 Introductory Remarks on Optical Probes . . . . . . . . . . . . . . . . . 31715.2 The Complex Dielectric Function and the Complex Optical

Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31715.2.1 Propagating Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

15.3 Relation of the Complex Dielectric Function to Observables . . . 32115.4 Units for Frequency Measurements . . . . . . . . . . . . . . . . . . . . . 325Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

16 Drude Theory–Free Carrier Contribution to the OpticalProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32916.1 The Free Carrier Contribution . . . . . . . . . . . . . . . . . . . . . . . . . 32916.2 Low Frequency Response: xs � 1 . . . . . . . . . . . . . . . . . . . . . 33216.3 High Frequency Response: xs � 1 . . . . . . . . . . . . . . . . . . . . . 33316.4 The Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33316.5 Plasmon Resonant Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . 33716.6 Surface Plasmon Polaritons in Graphene . . . . . . . . . . . . . . . . . 338Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

17 Interband Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34517.1 The Interband Transition Process . . . . . . . . . . . . . . . . . . . . . . . 34517.2 Hamiltonian for a Charge in an Electromagnetic Field . . . . . . . 34817.3 Relation Between Momentum Matrix Elements and the

Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35017.4 The Joint Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

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17.5 Connecting Optical Properties and the Joint Densityof States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

17.6 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35617.7 Critical Points in Low Dimensional Materials . . . . . . . . . . . . . . 360Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

18 Absorption of Light in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36518.1 The Absorption Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 36518.2 Free Carrier Absorption in Semiconductors . . . . . . . . . . . . . . . 36618.3 Free Carrier Absorption in Metals . . . . . . . . . . . . . . . . . . . . . . 36918.4 Direct Interband Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 370

18.4.1 Temperature Dependence of Eg . . . . . . . . . . . . . . . . . . 37418.4.2 Dependence of the Absorption Edge on Fermi

Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37418.4.3 Dependence of the Absorption Edge on Applied

Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37618.4.4 Dependence of the Absorption Edge on Applied

Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37718.5 Conservation of Crystal Momentum in Direct Optical

Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37918.6 Indirect Interband Transitions . . . . . . . . . . . . . . . . . . . . . . . . . 380Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

19 Optical Properties of Solids over a Wide Frequency Range . . . . . . 39119.1 Kramers–Kronig Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39119.2 Optical Properties and Band Structure . . . . . . . . . . . . . . . . . . . 39719.3 Modulated Reflectivity Experiments . . . . . . . . . . . . . . . . . . . . . 39819.4 Ellipsometry and Measurement of the Optical Constants . . . . . . 40319.5 Kramers-Kronig Relations in 2D Materials . . . . . . . . . . . . . . . . 40619.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

20 Impurities and Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41120.1 Impurity Level Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 41120.2 Shallow Impurity Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41220.3 Departures from the Hydrogenic Model . . . . . . . . . . . . . . . . . . 41620.4 Vacancies, Color Centers and Interstitials . . . . . . . . . . . . . . . . . 416

20.4.1 Schottky Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

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20.5 The Concept and Spectroscopy of Excitons . . . . . . . . . . . . . . . 42020.5.1 Exciton Effects in Bulk Materials . . . . . . . . . . . . . . . . 42420.5.2 Classification of Excitons . . . . . . . . . . . . . . . . . . . . . . 42620.5.3 Optical Transitions in 2D Systems: Quantum Well

Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43020.5.4 Excitons in 0D and 1D Systems: Fullerene C60

and Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . 43520.5.5 Excitons and Trions in Transition Metal

Dichalcogenides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43720.5.6 Excitons in Transition Metal Dichalcogenide

Heterojuctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

21 Luminescence and Photoconductivity . . . . . . . . . . . . . . . . . . . . . . . 44321.1 Classification of Luminescence Processes . . . . . . . . . . . . . . . . . 44321.2 Emission and Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44521.3 Photoconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45121.4 Photoluminescence in 2D Materials . . . . . . . . . . . . . . . . . . . . . 454Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

22 Optical Study of Lattice Vibrations . . . . . . . . . . . . . . . . . . . . . . . . 45722.1 Lattice Vibrations in Semiconductors . . . . . . . . . . . . . . . . . . . . 457

22.1.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . 45722.2 Dielectric Constant and Polarizability . . . . . . . . . . . . . . . . . . . . 46022.3 Polariton Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . 46122.4 Light Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47122.5 Feynman Diagrams for Light Scattering . . . . . . . . . . . . . . . . . . 47522.6 Raman Spectra in Quantum Wells and Superlattices . . . . . . . . . 47822.7 Raman Spectroscopy of Nanoscale Materials . . . . . . . . . . . . . . 480Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

Appendix A: Time–Independent Perturbation Theory . . . . . . . . . . . . . . . 489

Appendix B: Time–Dependent Perturbation Theory . . . . . . . . . . . . . . . . 499

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

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