Electrical Conduction in Graphene and Nanotubes E-Book

Shigeji Fujita

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 435

Veröffentlichungsjahr: 2013

Beschreibung

Written in a self-contained manner, this textbook allows both advanced students and practicing applied physicists and engineers to learn the relevant aspects from the bottom up. All logical steps are laid out without omitting steps. The book covers electrical transport properties in carbon based materials by dealing with statistical mechanics of carbon nanotubes and graphene - presenting many fresh and sometimes provoking views. Both second quantization and superconductivity are covered and discussed thoroughly. An extensive list of references is given in the end of each chapter, while derivations and proofs of specific equations are discussed in the appendix. The experienced authors have studied the electrical transport in carbon nanotubes and graphene for several years, and have contributed relevantly to the understanding and further development of the field. The content is based on the material taught by one of the authors, Prof Fujita, for courses in quantum theory of solids and quantum statistical mechanics at the University at Buffalo, and some topics have also been taught by Prof. Suzuki in a course on advanced condensed matter physics at the Tokyo University of Science. For graduate students in physics, chemistry, electrical engineering and material sciences, with a knowledge of dynamics, quantum mechanics, electromagnetism and solid-state physics at the senior undergraduate level. Includes a large numbers of exercise-type problems.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Preface

Physical Constants, Units, Mathematical Signs and Symbols

Chapter 1: Introduction

1.1 Carbon Nanotubes

1.2 Theoretical Background

1.3 Book Layout

1.4 Suggestions for Readers

References

Chapter 2: Kinetic Theory and the Boltzmann Equation

2.1 Diffusion and Thermal Conduction

2.2 Collision Rate: Mean Free Path

2.3 Electrical Conductivity and Matthiessen’s Rule

2.4 The Hall Effect: “Electrons” and “Holes”

2.5 The Boltzmann Equation

2.6 The Current Relaxation Rate

References

Chapter 3: Bloch Electron Dynamics

3.1 Bloch Theorem in One Dimension

3.2 The Kronig–Penney Model

3.3 Bloch Theorem in Three Dimensions

3.4 Fermi Liquid Model

3.5 The Fermi Surface

3.6 Heat Capacity and Density of States

3.7 The Density of State in the Momentum Space

3.8 Equations of Motion for a Bloch Electron

References

Chapter 4: Phonons and Electron–Phonon Interaction

4.1 Phonons and Lattice Dynamics

4.2 Van Hove Singularities

4.3 Electron–Phonon Interaction

4.4 Phonon-Exchange Attraction

References

Chapter 5: Electrical Conductivity of Multiwalled Nanotubes

5.1 Introduction

5.2 Graphene

5.3 Lattice Stability and Reflection Symmetry

5.4 Single-Wall Nanotubes

5.5 Multiwalled Nanotubes

5.6 Summary and Discussion

References

Chapter 6: Semiconducting SWNTs

6.1 Introduction

6.2 Single-Wall Nanotubes

6.3 Summary and Discussion

References

Chapter 7: Superconductivity

7.1 Basic Properties of a Superconductor

7.2 Occurrence of a Superconductor

7.3 Theoretical Survey

7.4 Quantum Statistical Theory of Superconductivity

7.5 The Cooper Pair Problem

7.6 Moving Pairons

7.7 The BCS Ground State

7.8 Remarks

7.9 Bose–Einstein Condensation in 2D

7.10 Discussion

References

Chapter 8: Metallic (or Superconducting) SWNTs

8.1 Introduction

8.2 Graphene

8.3 The Full Hamiltonian

8.4 Moving Pairons

8.5 The Bose–Einstein Condensation of Pairons

8.6 Superconductivity in Metallic SWNTs

8.7 High-Field Transport in Metallic SWNTs

8.8 Zero-Bias Anomaly

8.9 Temperature Behavior and Current Saturation

8.10 Summary

References

Chapter 9: Magnetic Susceptibility

9.1 Magnetogyric Ratio

9.2 Pauli Paramagnetism

9.3 The Landau States and Levels

9.4 Landau Diamagnetism

References

Chapter 10: Magnetic Oscillations

10.1 Onsager’s Formula

10.2 Statistical Mechanical Calculations: 3D

10.3 Statistical Mechanical Calculations: 2D

10.4 Anisotropic Magnetoresistance in Copper

10.5 Shubnikov–de Haas Oscillations

References

Chapter 11: Quantum Hall Effect

11.1 Experimental Facts

11.2 Theoretical Developments

11.3 Theory of the Quantum Hall Effect

11.4 Discussion

References

Chapter 12: Quantum Hall Effect in Graphene

12.1 Introduction

References

Chapter 13: Seebeck Coefficient in Multiwalled Carbon Nanotubes

13.1 Introduction

13.2 Classical Theory of the Seebeck Coefficient in a Metal

13.3 Quantum Theory of the Seebeck Coefficient in a Metal

13.4 Simple Applications

13.5 Graphene and Carbon Nanotubes

13.6 Conduction in Multiwalled Carbon Nanotubes

13.7 Seebeck Coefficient in Multiwalled Carbon Nanotubes

References

Chapter 14: Miscellaneous

14.1 Metal-Insulator Transition in Vanadium Dioxide

14.2 Conduction Electrons in Graphite

14.3 Coronet Fermi Surface in Beryllium

14.4 Magnetic Oscillations in Bismuth

References

Appendix

A.1 Second Quantization

A.2 Eigenvalue Problem and Equation-of-Motion Method

A.3 Derivation of the Cooper Equation (7.34)

A.4 Proof of (7.94)

A.5 Statistical Weight for the Landau States

A.6 Derivation of Formulas (11.16)–(11.18)

References

Index

Shigeji Fujita and Akira Suzuki

Electrical Conduction in Graphene and Nanotubes

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The Authors

Prof. Dr. Shigeji FujitaUniversity of Buffalo SUNY, Dept. of Physics 329 Fronczak Hall Buffalo, NY 14260 USA

Prof. Dr. Akira SuzukiTokyo University of Science Dept. of Physics Shinjuku-ku 162-8601 Tokyo Japan

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for

British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library.

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de.

© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

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Preface

Brilliant diamond and carbon black (graphite) are both made of carbon (C). Diamond is an insulator while graphite is a good conductor. This difference arises from the lattice structure. Graphite is a layered material made up of sheets, each forming a two-dimensional (2D) honeycomb lattice, called graphene. The electrical conduction mainly occurs through graphene sheets. Carbon nanotubes were discovered by Iijima1) in 1991. The nanotubes ranged from 4 to 30 nm in diameter and were microns (μm) in length, had scroll-type structures, and were called Multiwalled Nanotubes (MWNTs) in the literature. Single-Wall Nanotubes (SWNTs) have a size of about 1 nm in diameter and microns in length. This is a simple two-dimensional material. It is theorists’ favorite system. The electrical transport properties along the tube present, however, many puzzles, as is explained below. Carbon nanotubes are very strong and light. In fact, carbon fibers are used to make tennis rackets. Today’s semiconductor technology is based on silicon (Si) devices. It is said that carbon chips, which are stronger and lighter, may take the place of silicon chips in the future. It is, then, very important to understand the electrical transport properties of carbon nanotubes. The present book has as its principal topics electrical transport in graphene and carbon nanotubes.

The conductivity σ in individual carbon nanotubes varies, depending on the tube radius and the pitch of the sample. In many cases the resistance decreases with increasing temperature while the resistance increases in the normal metal. Electrical conduction in SWNTs is either semiconducting or metallic, depending on whether each pitch of the helical line connecting the nearest-neighbor C-hexagon centers contains an integral number of hexagons or not. The second alternative occurs more often since the pitch is not controlled in the fabrication process. The room-temperature conductivity in metallic SWNTs is higher by two or more orders of magnitude than in semiconducting SWNTs. Currents in metallic SWNTs do not obey Ohm’s law linearity between current and voltage. Scanned probe microscopy shows that the voltage does not drop along the tube length, implying a superconducting state. The prevailing theory states that electrons run through the one-dimensional (1D) tube ballistically. But this interpretation is not the complete story. The reason why the ballistic electrons are not scattered by impurities and phonons is unexplained. We present a new interpretation in terms of the model in which superconducting Bose-condensed Cooper pairs (bosons) run as a supercurrent. In our text we start with the honeycomb lattice, construct the Fermi surface, and develop Bloch electron dynamics based on the rectangular unit cell model. We then use kinetic theory to treat the normal electrical transport with the assumption of “electrons,” “holes,” and Cooper pairs as carriers.

To treat the superconducting state, we assume that the phonon-exchange attraction generates Cooper pairs (pairons). We start with a Bardeen–Cooper–Schrieffer (BCS)-like Hamiltonian, derive a linear dispersion relation for the moving pairons, and obtain a formula for the Bose–Einstein Condensation (BEC) temperature

MWNTs have open-ended circumferences and the outermost walls with greatest radii, contribute most to the conduction. The conduction is metallic (with no activation energy factor) and shows no pitch dependence.

This book has been written for first-year graduate students in physics, chemistry, electrical engineering, and material sciences. Dynamics, quantum mechanics, electromagnetism, and solid state physics at the senior undergraduate level are prerequisites. Second quantization may or may not be covered in the first-year quantum course. But second quantization is indispensable in dealing with phonon-exchange, superconductivity, and QHE. It is fully reviewed in Appendix A.1. The book is written in a self-contained manner. Thus, nonphysics majors who want to learn the microscopic theory step-by-step with no particular hurry may find it useful as a self-study reference.

Many fresh, and some provocative, views are presented. Experimental and theoretical researchers in the field are also invited to examine the text. The book is based on the materials taught by Fujita for several courses in quantum theory of solids and quantum statistical mechanics at the University at Buffalo. Some of the book’s topics have also been taught by Suzuki in the advanced course in condensed matter physics at the Tokyo University of Science. The book covers only electrical transport properties. For other physical properties the reader is referred to the excellent book Physical Properties of Carbon Nanotubes, by R. Saito, G. Dresselhaus and M.S. Dresselhaus (Imperial College Press, London 1998).

The authors thank the following individuals for valuable criticisms, discussions and readings: Professor M. de Llano, Universidad Nacional Autonoma de Mexico; Professor Sambandamurthy Ganapathy, University at Buffalo, Mr. Masashi Tanabe, Tokyo University of Science and Mr. Yoichi Takato, University at Buffalo. We thank Sachiko, Keiko, Michio, Isao, Yoshiko, Eriko, George Redden and Kurt Borchardt for their encouragement, reading and editing of the text.

Shigeji FujitaAkira Suzuki

Buffalo, New York, USATokyo, JapanDecember, 2012

1) Iijima, S. (1991) Nature (London), 354, 56.

2) Novoselov, K.S. et al. (2007) Science, 315, 1379.

Physical Constants, Units, Mathematical Signs and Symbols

Useful Physical Constants

Subsidiary Units

Prefixes Denoting Multiples and Submultiples

10

3

kilo (k)

10

6

mega (M)

10

9

giga (G)

10

12

tera (T)

10

15

peta (P)

10

−3

milli (m)

10

−6

micro (μ)

10

−9

nano (n)

10

−12

pico (p)

10

−15

femto (f)

Mathematical Signs

set of natural numbers

set of integers

set of rational numbers

set of real numbers

set of complex numbers

∀

x

for all

x

∃

x

existence of

x

maps to

∴

therefore

because

equals

approximately equals

≠

not equal to

≡

identical to, defined as

>

greater than

much greater than

<

smaller (or less) than

much smaller than

≥

greater than or equal to

≤

smaller (or less) than or equal to

∝

proportional to

~

represented by, of the order

(

x

)

order of

x

List of Symbols

The following list is not intended to be exhaustive. It includes symbols of frequent occurrence or special importance in this book.