Graphene Photonics 9781108656870

Table of contents :
Contents......Page 8
Preface......Page 10
1.1 Graphene Band Structure......Page 12
1.2 Density of States and Carrier Concentration......Page 21
1.3 Fermi Energy, Chemical Potential, and Fermi Level......Page 22
1.4 Temperature Dependence of Carrier Concentration......Page 24
1.5 Carrier Velocity and Effective Mass......Page 25
1.6 Band Structure of Multilayer Graphene......Page 26
References......Page 36
2.1 Current and Conductivity of a Two-Dimensional Material......Page 38
2.2 Boltzmann Transport Equation......Page 41
2.3 Scattering of Carriers......Page 44
2.4 Surface Resistivity and Mobility......Page 62
2.5 Transport Experiments......Page 65
2.6 Multilayer Graphene......Page 72
References......Page 75
3.1 Optical Fields......Page 77
3.2 Susceptibility and Permittivity of a Two-Dimensional Material......Page 81
3.3 Optical Transitions in Graphene......Page 87
3.4 Hamiltonian of Graphene in an Electromagnetic Field......Page 90
3.5 Optical Conductivity of Monolayer Graphene......Page 91
3.6 Optical Conductivity of Multilayer Graphene......Page 102
3.7 Permittivity of Monolayer and Multilayer Graphene......Page 109
3.8 Absorbance of Monolayer and Multilayer Graphene......Page 112
References......Page 117
4.1 Dispersion in Frequency and Momentum......Page 118
4.2 Drude Model......Page 119
4.3 Polarizability Function......Page 122
4.4 Random-Phase Approximation......Page 131
4.5 Polarizability Function of Bilayer Graphene......Page 152
References......Page 155
5.1 Nonlinear Susceptibility and Nonlinear Conductivity......Page 156
5.2 Semiclassical Approach for Intraband Transitions......Page 163
5.3 Bistability......Page 168
5.4 Quantum Mechanical Approach for Interband Transitions......Page 169
5.5 Second-Order Optical Nonlinearity......Page 173
5.6 Third-Order Optical Nonlinearity......Page 175
5.7 Experiments on Nonlinear Optical Properties......Page 185
References......Page 187
6.1 Plasmons, Surface Plasmons, and Surface Plasmon Polaritons......Page 189
6.2 Graphene Surface Excitations......Page 196
6.3 Surface Plasmons of Two Graphene Sheets......Page 215
6.4 Excitation and Detection of Graphene SPPs and SPs......Page 220
References......Page 225
7.1 Plasmonic Waveguides......Page 227
7.2 Photodetectors......Page 231
7.3 Optical Modulators......Page 241
7.4 Terahertz Modulators......Page 246
7.5 Saturable Absorber......Page 255
References......Page 259
Index......Page 262

Citation preview

Graphene Photonics Understand the fundamental concepts, theoretical background, major experimental observations, and device applications of graphene photonics with this self-contained text. Systematically and rigorously developing each concept and theoretical model from the ground up, it guides readers through the major topics, from basic properties and band structure to electronic, optical, optoelectronic, and nonlinear optical properties, and plasmonics and photonic devices. The connections between theory, modeling, experiment, and device concepts are demonstrated throughout, and every optical process is analyzed through formal electromagnetic analysis. Suitable for both self-study and a one-semester or one-quarter course, this is the ideal text for graduate students and researchers in photonics, optoelectronics, nanoscience and nanotechnology, and optical and solid-state physics, working in this rapidly developing field. Jia-Ming Liu is Distinguished Professor of Electrical and Computer Engineering, and Associate Dean of the Henry Samueli School of Engineering and Applied Science, at the University of California, Los Angeles. He is the author of Principles of Photonics (Cambridge University Press, 2016), and Photonic Devices (Cambridge University Press, 2005), and a fellow of the Optical Society of America, the American Physical Society, the IEEE, and the Guggenheim Foundation. I-Tan Lin is a hardware engineer at Intel, and his current research interests include graphene-based optoelectronics, terahertz frequency devices, and plasmonics.

Graphene Photonics JIA-MING LIU University of California, Los Angeles

I-TAN LIN Intel

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108476683 DOI: 10.1017/9781108656870 © Cambridge University Press 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed in United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Liu, Jia-Ming, 1953– author. | Lin, I-Tan, 1983– author. Title: Graphene photonics / Jia-Ming Liu (University of California, Los Angeles), I-Tan Lin (Intel, California). Description: Cambridge : Cambridge University Press, 2018. | Includes bibliographical references and index. Identifiers: LCCN 2018023253 | ISBN 9781108476683 (alk. paper) Subjects: LCSH: Photonics. | Graphene. | Nanostructured materials – Optical properties. Classification: LCC TA1530 .L58 2018 | DDC 621.36/5–dc23 LC record available at https://lccn.loc.gov/2018023253 ISBN 978-1-108-47668-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To our families

Contents

Preface 1

2

3

Basic Properties and Band Structure

page ix 1

1.1 Graphene Band Structure 1.2 Density of States and Carrier Concentration 1.3 Fermi Energy, Chemical Potential, and Fermi Level 1.4 Temperature Dependence of Carrier Concentration 1.5 Carrier Velocity and Effective Mass 1.6 Band Structure of Multilayer Graphene References

1 10 11 13 14 15 25

Electronic Properties

27

2.1 Current and Conductivity of a Two-Dimensional Material 2.2 Boltzmann Transport Equation 2.3 Scattering of Carriers 2.4 Surface Resistivity and Mobility 2.5 Transport Experiments 2.6 Multilayer Graphene References

27 30 33 51 54 61 64

Optical Properties

66

3.1 Optical Fields 3.2 Susceptibility and Permittivity of a Two-Dimensional Material 3.3 Optical Transitions in Graphene 3.4 Hamiltonian of Graphene in an Electromagnetic Field 3.5 Optical Conductivity of Monolayer Graphene 3.6 Optical Conductivity of Multilayer Graphene 3.7 Permittivity of Monolayer and Multilayer Graphene 3.8 Absorbance of Monolayer and Multilayer Graphene References

66 70 76 79 80 91 98 101 106

viii

Contents

4

Optoelectronic Properties

107

4.1 Dispersion in Frequency and Momentum 4.2 Drude Model 4.3 Polarizability Function 4.4 Random-Phase Approximation 4.5 Polarizability Function of Bilayer Graphene References

107 108 111 120 141 144

Nonlinear Optical Properties

145

5.1 Nonlinear Susceptibility and Nonlinear Conductivity 5.2 Semiclassical Approach for Intraband Transitions 5.3 Bistability 5.4 Quantum Mechanical Approach for Interband Transitions 5.5 Second-Order Optical Nonlinearity 5.6 Third-Order Optical Nonlinearity 5.7 Experiments on Nonlinear Optical Properties References

145 152 157 158 162 164 174 176

Plasmonics

178

6.1 Plasmons, Surface Plasmons, and Surface Plasmon Polaritons 6.2 Graphene Surface Excitations 6.3 Surface Plasmons of Two Graphene Sheets 6.4 Excitation and Detection of Graphene SPPs and SPs References

178 185 204 209 214

Photonic Devices

216

7.1 Plasmonic Waveguides 7.2 Photodetectors 7.3 Optical Modulators 7.4 Terahertz Modulators 7.5 Saturable Absorber References

216 220 230 235 244 248

Index

251

5

6

7

Preface

Graphene is a single-layer crystal of carbon. It has unique electronic and photonic properties, as well as unusual mechanical and thermal properties. As the thinnest twodimensional material, graphene has a nanostructure that can be integrated with various devices and systems, offering the opportunity to transform devices and systems in many areas. Intensive research on graphene started only after it was first isolated in 2004. Since then, graphene has become one of the most studied materials. It has opened up many new fields of scientific research and technological development, which continue to expand rapidly today. Research and development in these young fields promise to revolutionize electronics and photonics technologies in addition to creating new graphene-based technologies. Graphene has a unique hexagonal crystal structure of two carbon atoms per unit cell. Intrinsic graphene is a semiconductor of zero bandgap, with its charged carriers behaving like Dirac fermions of zero mass, resulting in many extraordinary properties that are very different from those of other materials. Such properties can be controllably modified by proper impurity doping or by electrical or optical modulation, making graphene extremely attractive for novel device applications. The salient electronic and photonic properties of graphene, together with its unique nanostructure, offer innovative opportunities for many potentially revolutionary applications in high-speed/highfrequency electronic and photonic devices, terahertz oscillators and sensors, and ultrafast nonlinear optical elements. This book aims to systematically cover the subject of graphene photonics in an organized and comprehensive manner by guiding the reader through fundamental concepts, the theoretical framework, existing experimental observations, and many demonstrated device applications. As such, it starts by describing the basic properties, band structure, and electronic properties of graphene in the first two chapters to lay the necessary foundation for discussions on the optical properties, optoelectronic properties, and nonlinear optical properties of graphene in the following three chapters. Because graphene is a truly two-dimensional system that can serve as a conductor at frequencies up to the terahertz region, it naturally has unique plasmonic properties. Discussions on plasmonics based on graphene are given in a following chapter. The final chapter gives an overview of the graphene-based photonic devices that have been developed or proposed to date. The development of each concept and theoretical model is thorough and rigorous. Nonetheless, the approach is systematic and often pedagogical for the reader to easily

x

Preface

follow. The theoretical models of the various properties are described and developed through rigorous quantum mechanical formulations and calculations, usually through both semiclassical and full quantum mechanical approaches. Every optical process is analyzed through formal electromagnetic analysis. Detailed figures are used to illustrate the concepts, theoretical results, and experimental measurements. As such, this book is useful to both researchers and students. The required background for a good understanding of this book includes basic concepts of solid-state physics, a solid foundation of quantum mechanics, a firm grounding in electromagnetic wave and fields, and a basic knowledge of physical optics. This book is primarily aimed at researchers, technologists, and graduate students in the fields of photonics, electronics, nanoscience, nanotechnology, optical physics, solidstate physics, and materials science. It can be used as a reference or a guide to facilitate self-study for these readers. This book is also useful as a textbook for a one-semester or one-quarter graduate course on the specific topic of graphene photonics because it is written such that the concepts are systematically developed from the ground up and it is self-contained. A course utilizing this text would ideally be targeted toward beginning graduate students who have the required background knowledge as described above, though advanced undergraduate students with the required background could also attend. This book was developed through our research on graphene photonics over the past 12 years. We would especially like to express our gratitude to Kuang-Hsiung Wu and ChihWei Luo of National Chiao-Tung University for their friendship and efforts in our collaborative research on graphene photonics. We would like to thank our editor, Julie Lancashire, for her help in the publication of this book. We would also like to thank Vida Liu for creating an original oil painting for the cover art.

1

Basic Properties and Band Structure

1.1

Graphene Band Structure Graphene is a two-dimensional structure of carbon atoms. Single-layer graphene consists of only one layer of carbon atoms. A carbon atom has six electrons occupying the 1s2, 2s2, 2px, and 2py atomic orbitals. When carbon atoms are brought together, one electron from the 2s orbital is promoted to the 2pz orbital for the formation of hybrid orbitals. In diamonds, the 2s, 2px, 2py, and 2pz orbitals are mixed to form four sp3 hybrid orbitals for each carbon atom; therefore, each carbon atom is joined with four neighbors by overlapping their sp3 hybrid orbitals. In graphite, instead of four sp3 hybrid orbitals, three sp2 hybrid orbitals are formed through mixing of the 2s, 2px, and 2py orbitals, while the fourth orbital remains as 2pz. Overlapping sp2 hybrid orbitals from two adjacent atoms creates a strong σ covalent bond (C‒C bond); these in-plane σ bonds connect each carbon atom to three neighbors. The remaining 2pz orbitals of these carbon atoms form π bonds, which are responsible for binding carbon layers together in graphite. Because π bonds are much weaker than σ bonds, graphite has a low shear strength so that its carbon layers can be easily detached. For monolayer graphene, these nearly free π electrons are responsible for most of its experimentally observed electronic and optical properties. Because the Pauli exclusion principle requires that π electrons from different carbon atoms do not occupy the same state, the large number of closely packed carbon atoms in graphene causes degenerate energy levels to split into continuously distributed nondegenerate levels of allowed energy states, forming energy bands. The real-space two-dimensional honeycomb lattice of graphene is shown in Figure 1.1(a). The distance between two neighboring carbon atoms in graphene is a ≈ 0:142 nm:

ð1:1Þ

The primitive lattice vectors are pffiffiffi a1 ¼ 3 a

pffiffiffi  pffiffiffi  pffiffiffi 3 3 1 1 ^x  ^y and a2 ¼ 3a ^x þ ^y : 2 2 2 2

Therefore, the lattice constant is

ð1:2Þ

2

Basic Properties and Band Structure

(a)

(b)

ky b2

A

B K

3

y

2

kx a2

1

K

x a1 b1

Figure 1.1 (a) Real-space honeycomb graphene lattice. The lattice consists of two overlapping Bravais sublattices, A (gray dots) and B (black dots). The primitive unit cell is drawn as a shaded area. a1 and a2 are the primitive lattice vectors. δ1 , δ2 , and δ3 are vectors pointing from a B atom to its nearest A atoms. (b) Brillouin zone of graphene drawn as a shaded area. b1 and b2 are the primitive vectors of the reciprocal lattice. Dirac points Κ (black dots) and Κ0 (gray dots) are marked.

a0 ¼ ja1 j ¼ ja2 j ¼

pffiffiffi 3a ≈ 0:246 nm:

The area Ap of a primitive unit cell can be obtained as pffiffiffi pffiffiffi 3 2 3 3 2 a; a ¼ Ap ¼ ja1  a2 a3 j ¼ 2 0 2

ð1:3Þ

ð1:4Þ

where a3 ¼ a1  a2 =Ap ¼ ^z is the unit vector that points in the direction perpendicular to the plane of the two-dimensional graphene lattice. We also find that the vectors connecting adjacent carbon atoms are δ1 ¼ a

pffiffiffi  pffiffiffi    3 3 1 1 ^y ; δ2 ¼ a ^x þ ^y ; and δ3 ¼ a^x : ^x  2 2 2 2

ð1:5Þ

as shown in Figure 1.1(a). The corresponding Brillouin zone of the lattice in Figure 1.1(a) is shown in Figure 1.1(b). The primitive vectors of the reciprocal lattice are pffiffiffi   3 a2  a3 4π 1 ^y ; ^x  b1 ¼ 2π ¼ 2 a1  a2 a3 3a 2 pffiffiffi   3 a3  a1 4π 1 ^y : ^x þ ¼ b2 ¼ 2π 2 a1  a2 a3 3a 2

ð1:6Þ

We can also find the vector b3 ¼ 2πa1  a2 =ða1  a2 a3 Þ ¼ 2π^z that is perpendicular to the plane of the two-dimensional reciprocal lattice. Therefore, ai bj ¼ 2πδij ;

ð1:7Þ

1.1 Graphene Band structure

3

where δij is the Kronecker delta function: δij ¼ 1 if i ¼ j, and δij ¼ 0 if i ≠ j. The honeycomb lattice of graphene is not a Bravais lattice because the locations of all carbon atoms cannot be generated by the translation R ¼ ma1 þ na2 ;

ð1:8Þ

where m and n are integers. The lattice structure of graphene consists of two overlapping Bravais sublattices A and B, and thus a primitive unit cell contains two carbon atoms, as shown in Figure 1.1(a). All of the carbon atoms on the same sublattice, but not those on the two different sublattices, are connected by the vector R. The electronic band structure of graphene can be obtained by using the tight-binding ^ for the Schrödinger equation: model. We start from a tight-binding Hamiltonian H ^ ψðrÞ ¼ EψðrÞ: H

ð1:9Þ

The wave function ψðrÞ is given by the linear superposition of orbital functions ϕl ðrÞ: 1 X ik  Rl 1 X ik  Rl ψðrÞ ¼ pffiffiffiffi cl e ϕl ðrÞ ¼ pffiffiffiffi cl e ϕðr  Rl Þ; ð1:10Þ N l N l where N is the number of unit cells, ϕl ðrÞ ¼ ϕðr  Rl Þ is the ground state of the 2pz electron of an isolated carbon atom that is located at Rl , and the index l runs over all carbon atom points on the graphene lattice. Equation (1.10) is of the Bloch form because ψðr þ ai Þ ¼ expðikai ÞψðrÞ, where i ¼ 1 or 2. From the symmetry point of view, all atoms on sublattice A are geometrically identical; in other words, the surrounding is the same when viewing the graphene lattice from any atom on sublattice A. The same can be said for all atoms on sublattice B. No difference can be seen when viewing the graphene lattice from different atoms on the same sublattice. However, the atoms on sublattice A are not geometrically identical to those on sublattice B. By comparing the scenery in a certain direction, it is possible to tell the difference between viewing the graphene lattice from one atom on sublattice A and viewing it from one on sublattice B. Therefore, the wave function in (1.10) has only two independent coefficients for cl : cA and cB , which respectively represent the amplitudes of the wave functions at carbon sites on sublattices A and B. Thus, (1.10) can be rewritten as a linear superposition of two Bloch functions ψA and ψB that are respectively wave functions for sublattices A and B: cA X ikRl cB X ik  Rl B ϕ ðrÞ; ð1:11Þ e A ϕlA ðrÞ þ pffiffiffiffi e ψðrÞ ¼ cA ψA ðrÞ þ cB ψB ðrÞ ¼ pffiffiffiffi lB N lA N lB where lA and lB run over all of the carbon atoms on sublattices A and B, respectively. The wave functions ψA and ψB are normalized, and the overlap of ψA and ψB is negligible [1], so that 〈ψi jψj 〉 ¼ δij . Therefore, jψA 〉 and jψB 〉 form the basis for the graphene wave functions. By multiplying both sides of (1.9) by ψA ðrÞ, we obtain     ^ ψA 〉 þ cB 〈ψA H ^ ψB 〉 ¼ cA E: cA 〈ψA H

ð1:12Þ

4

Basic Properties and Band Structure

In (1.12), all of the interactions among A atoms and those between A atoms and B atoms are considered. However, it is very difficult to solve for the eigenenergy from (1.12) where all interactions are accounted for. To avoid such difficulty, (1.12) is simplified by taking the approximation of considering only the self-interaction of each carbon atom at a lattice point and the interactions of the atom with its three nearest neighbors; all longrange interactions are assumed to be much weaker than these interactions and are thus ignored. Within this approximation, (1.12) takes the simple form: ^ jϕl 〉 þ cB 〈ϕl jH ^ jϕl 〉ðeikδ1 þ eikδ2 þ eikδ3 Þ ¼ cA E: cA 〈ϕlA jH A A B

ð1:13Þ

By multiplying both sides of (1.11) by ψB ðrÞ and taking the same approximation, we have ^ jϕl 〉 þ cA 〈ϕl jH ^ jϕl 〉ðeikδ1 þ eikδ2 þ eikδ3 Þ ¼ cB E: cB 〈ϕlB jH B B A

ð1:14Þ

Equations (1.13) and (1.14) can be expressed in a matrix form for the eigenenergy equation of the system: "

ε

γ0 ðeikδ1 þ eikδ2 þ eikδ3 Þ

#   γ0 ðeikδ1 þ eikδ2 þ eikδ3 Þ c  c A ¼E A ; c cB B ε ð1:15Þ

where ^ jϕl 〉 ¼ 〈ϕl jH ^ jϕl 〉 ε ¼ 〈ϕlA jH A B B

ð1:16Þ

is the self-interaction energy, and ^ jϕl 〉 ¼ 〈ϕl jH ^ jϕl 〉 γ0 ¼ 〈ϕlA jH B B A

ð1:17Þ

is the nearest-neighbor hopping energy in graphene. The nearest-neighbor hopping energy γ0 is experimentally measured to have a value of γ0 ¼ 3:16 eV [2–4], which is the same as that found for graphite. The nontrivial eigenvalues of (1.15) yield two eigenenergies: EðkÞ ¼ ε  γ0 jeikδ1 þ eikδ2 þ eikδ3 j;

ð1:18Þ

where the plus sign is for the conduction band, which has higher energy, and the minus sign is for the valence band, which has lower energy. The conduction band and the valence band described by (1.18) are also called the π band and the π band, respectively. Equation (1.18) can be written in another form that is frequently seen in the literature [5,6]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi    pffiffiffi 3 3 EðkÞ ¼ ε  γ0 3 þ 2 cosð 3ky aÞ þ 4 cos ky a cos kx a : 2 2

ð1:19Þ

The energy band structure described by (1.19) is plotted in Figure 1.2(a). At the Г point (k ¼ 0), we find the eigenenergies E ¼ ε  3γ0 from (1.19), which are associated

1.1 Graphene Band structure

5

(b)

(a) 3

K K

2 1

3

E(

0

)

0 1

0

K

0

2 2 3

0 2

0 k x (1 a )

2

2

k y (1 a )

3 1 2

(c) 5 *

*

E (eV )

0

5

10

K M

15 M

K

Figure 1.2 (a) Band structure of graphene plotted using (1.19) while setting the self-interaction energy to be ε ¼ 0. The top surface of a higher energy is the conduction band (π band), and the bottom surface of a lower energy is the valence band (π band). The conduction band is projected onto a twodimensional plane on the top. The circled region near the Dirac point Κ0 is enlarged in (b). (c) Full band structure of graphene. The thick curves are the π and π bands, and the thin curves are the σ and σ bands [7].

6

Basic Properties and Band Structure

1 with the eigenstates ½cA ; cB  ¼ pffiffiffi ½1; 1, respectively. The eigenenergy E ¼ ε  3γ0 2 for the valence band is associated with symmetric bonding orbitals such that the amplitudes cA and cB of the wave functions for sublattices A and B have the same sign. By contrast, the eigenenergy E ¼ ε þ 3γ0 for the conduction band is associated with antibonding orbitals such that the amplitudes cA and cB of the wave functions for sublattices A and B have opposite signs. An eigenstate of graphene is not necessarily bonding or antibonding, however; most eigenstates are a mixture of both bonding and antibonding orbitals. As we shall see in the following, only along a specific set of directions can the eigenstates be described by purely bonding or antibonding orbitals. The conduction band (π band) and the valence band (π band) are symmetric with respect to the E ¼ 0 plane, as shown in Figure 1.2(a). Note that the long-range interactions of carbon atoms beyond the nearest neighbors are ignored in the above analysis. When these long-range interactions are accounted for, we find that the π and π bands are actually not symmetric to each other, as seen in Figure 1.2(c), where we also show the σ and σ bands contributed by the σ bonds. As can be seen in Figure 1.2(a), the conduction band touches the valence band at E ¼ ε. The self-interaction energy ε can be taken as the reference level by setting it to be 0 for simplicity so that an electron in a state on the conduction band has a positive energy and that in a state on the valence band has a negative energy. Then, (1.19) can be simplified as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi   ffi pffiffiffi 3 3 EðkÞ ¼ γ0 3 þ 2 cos ð 3ky aÞ þ 4 cos ky a cos kx a ; 2 2

ð1:20Þ

where the plus sign represents the conduction band and the minus sign represents the valence band. The points at EðkÞ ¼ 0 are called the Dirac points; the vicinities of these points are referred to as the valleys, where the graphene band structure can be described by the relativistic Dirac equation, which is further discussed in the following section. Corresponding to the two distinct sublattices A and B in the real space, there are two distinct groups of Dirac points K and Κ0 in the k space. Inside the Brillouin zone drawn in Figure 1.1(b), two Dirac points can be found at the k-space locations represented by the reciprocal space vectors: 4π K ¼ pffiffiffi 3 3a

pffiffiffi  3 1 ^x þ ^y ; 2 2

4π K ¼ pffiffiffi 3 3a 0

pffiffiffi  3 1 ^x  ^y : 2 2

ð1:21Þ

Another popular choice of the k-space locations of K and Κ0 in the literature is to select the K and Κ0 points on the ky axis so that they are mirror symmetric with respect to the kx axis; the physics and the results obtained in the following are unchanged by this alternative selection. It can be easily shown using (1.20) that EðKÞ ¼ EðK0 Þ ¼ 0 at these Dirac points. For perfectly intrinsic graphene, the chemical potential, i.e., the Fermi level, is located at the energy level of the Dirac points, E ¼ 0, so that the valence band is completely filled and the conduction band is completely empty. As most physics and carrier

1.1 Graphene Band structure

7

transitions happen at energy levels around the chemical potential, we shall rewrite (1.20) by shifting the origin of the k space to one of the Dirac points and assume a small k to simplify (1.20). Substituting k in (1.20) with K þ k or K0 þ k, we obtain, for jkja ≪ 1 in the vicinity of a Dirac point, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 2 2 9 2 2 3 E ¼  γ0 ky a þ kx a ¼  aγ0 jkj ¼ ℏvF k; 4 4 2

ð1:22Þ

where ℏ is the reduced Planck’s constant and vF ¼

3aγ0 ≈ 1:0  106 m s1 2ℏ

ð1:23Þ

is the Fermi velocity in graphene, the physical meaning of which will become clear in Section 1.5. Equation (1.22) indicates that there is a region in the k space around each Dirac point where the carrier energy is linearly proportional to the wave number measured with respect to the given Dirac point. This region is called the Dirac cone, as shown in the insert in Figure 1.2(b). Because (1.22) can also be derived from the massless Dirac equation, electrons on the Dirac cone are also called Dirac electrons. The Hamiltonian around the K point is obtained in the same way as (1.22) is obtained. By substituting k with K þ k in (1.15) to set the origin of the k space at the K point, the off-diagonal matrix elements for jkja ≪ 1 become  ^ K jψB 〉 ¼ γ0 eiðKþkÞδ1 þ eiðKþkÞδ2 þ eiðKþkÞδ3 〈ψA jH ¼

pffiffiffi i ℏv F hpffiffiffi 3kx þ ky þ iðkx  3ky Þ 2

ð1:24Þ

and ^ K jψA 〉 ¼ 〈ψA jH ^ K jψB 〉: 〈ψB jH

ð1:25Þ

The coefficients of kx and ky can be simplified by rotating both the kx and ky coordinate axes counterclockwise by 30 degrees through the unitary transformation 2 π π 3   0 pffiffiffi  sin cos kx 1 3kx þpkffiffiyffi 6 7 kx 6 6 ¼4 ð1:26Þ π π 5 ky ¼ 2 kx þ 3ky : k 0y cos  sin 6 6 ^ K jψB 〉 ¼ ℏv F ðk 0  ik 0 Þ ¼ 〈ψB jH ^ K jψA 〉 Using (1.26), we obtain the relation 〈ψA jH x y 0 0 from (1.24) and (1.25). By dropping the prime symbols, k x → kx and k y → ky , to simplify the notation, the Hamiltonian near the K point for jkja≪1 can be expressed as ^K ¼ H



0 ℏv F ðkx þ iky Þ

 ℏvF ðkx  iky Þ : 0

ð1:27Þ

8

Basic Properties and Band Structure

(a)

ky

K

ky

(b) K

k

kx

k

K

30°

K

K

kx ky

K

ky 30°

K K

K

k k

kx

K kx

K

K 0

Figure 1.3 Wave vector k near (a) the K point, and (b) the Κ point. The Hamiltonian near the K or Κ0 point is obtained by rotating the coordinate from the gray axes to the black axes. The angle θk is measured with respect to the rotated kx axis.

The corresponding eigenfunctions are

ψK k; ðrÞ

" #   1 eikr eikr 1 kx þ iky ¼ pffiffiffi ¼ pffiffiffi ; iθ 2  2 e k k

ð1:28Þ

where the  signs correspond to the signs of the eigenenergies given in (1.22), and θk is the polar angle measured between k and the rotated kx axis as shown in Figure 1.3(a) for the K point. Similarly, the Hamiltonian near the Κ0 point is obtained by substituting k with K0 þ k in (1.15) and taking the unitary transformation by rotating the axes 30 degrees clockwise; the result for jkja ≪ 1 is   0 ℏvF ðkx þ iky Þ ^ ^ T: 0 HK ¼ ð1:29Þ ¼H K ℏvF ðkx  iky Þ 0 The eigenfunctions are   0 eikr eiθk p ffiffi ffi ψK ðrÞ ¼ ; k; 1 2

ð1:30Þ

where the  signs correspond to the signs of the eigenenergies given in (1.22), and θk is the polar angle measured between k and the rotated kx axis at the K0 point, as shown in Figure 1.3(b). From (1.28) and (1.30), it is clear that for the conduction band, the eigenstate is antibonding at θk ¼ π and bonding at θk ¼ 0, as shown in Figure 1.4(a). By contrast, the eigenstate is bonding at θk ¼ π and antibonding at θk ¼ 0 for the valence band, as shown in Figure 1.4(b).

1.1 Graphene Band structure

(a)

9

(b)

K

K

k

K

k

K

k

K

K

k

k

k K

K

k

K

K

k

K

K

Figure 1.4 Antibonding (solid thin lines) and bonding (dotted lines) states in (a) the conduction band and (b) the valence band.

Equations (1.27) and (1.29) are often written as ^ K ¼ vF σ  p H

ð1:31Þ

^ K0 ¼ v F σ   p H

ð1:32Þ

for an electron near a K point and as

for an electron near a K0 point, where p ¼ ℏk and σ is the Pauli vector:       0 1 0 i 1 0 ^x þ ^y þ σ¼ ^z : i 0 0 1 1 0

ð1:33Þ

Note that p ¼ ℏk ¼ ℏðkx^x þ ky^y Þ with kz ¼ 0 for the π electrons in monolayer graphene because they only move through the graphene lattice on the xy plane of a monolayer graphene sheet. In fact, the Hamiltonian of graphene near a K or Κ0 point has the form identical to that of a spin one-half particle in a magnetic field; its eigenfunctions are two-component spinors, as seen in (1.28) and (1.30). In the case of graphene, σ does not really represent an electronic spin, but a pseudospin that indicates the sublattice on which the electron is located. We can define a helicity operator ^h ¼ σp=p to find the projection of the

K momentum along the pseudospin direction; then it can be shown that ^hψK k; ¼ ψk; . The positive sign gives positive helicity, meaning that the momentum of an electron on the conduction band is parallel to the pseudospin, whereas the negative sign gives negative helicity, meaning that the momentum of an electron on the valence band is

10

Basic Properties and Band Structure

(a)

(b) p

p

p

p

h =1

h= 1

K

K k

k

h =1

h= 1 p

p

p

p

Figure 1.5 Helicity h, pseudospin σ, and momentum p of an electron near (a) the K point, and (b) the Κ0

point. The directions of p and σ are shown. In the case of a hole, the direction of momentum is reversed. Due to the conservation of pseudospin, an electron or a hole cannot be scattered into an electron or a hole of a different pseudospin.

antiparallel to the pseudospin. The relation among the helicity, the carrier momentum, and the pseudospin are shown in Figure 1.5. Because of the conservation of pseudospin, a right-moving electron on the conduction band cannot be scattered into a left-moving electron on the conduction band or into a right-moving electron on the valence band. Such scattering requires the pseudospin to be flipped from þ1 to 1 for an electron near the K point, or flipped from 1 to þ1 for an electron near the Κ0 point, neither of which is allowed. Similarly, scattering of a left-moving electron on the conduction band into a right-moving electron on the conduction band or into a left-moving electron on the valence band is also forbidden. Nevertheless, such scattering events can happen when there exists a short-range potential that acts differently on sublattices A and B, thus breaking the symmetry between the two sublattices.

1.2

Density of States and Carrier Concentration With the band structure near the Dirac point described by (1.22), the density of electron states of graphene in the energy range between E and E þ dE for E > 0 near the conduction band edge is ð 1 X g 2E e ρ c ðEÞdE ¼ δðEk  EÞdE ¼ 2 δðℏv F k  EÞ2πkdkdE ¼ 2 2 dE; A k; E >0 4π πℏ vF ∞

k

0

ð1:34Þ and that for E < 0 near the valence band edge is

1.3 Fermi Energy, Chemical Potential, and Fermi Level

11

ð 1 X g 2E e δðEk EÞdE ¼ 2 δðℏv F kEÞ2πkdkdE ¼  2 2 dE; ð1:35Þ ρ v ðEÞdE ¼ A k; E μ. Therefore, for jμj≫ kB T, ð∞

ð0

e p0 ≈ e ρ c ðEÞHðμ  EÞdE þ n0 þ e 0

 e ρ v ðEÞ 1  Hðμ  EÞ dE ¼

∞

μ2 πℏ2 v 2F

: ð1:38Þ

In this book, we use a tilde for a 2D quantity to clearly distinguish it from its 3D counterpart in order to avoid confusion. Therefore, the surface electron density e n, the surface hole density e p, and the densities of states e ρ c ðEÞdE and e ρ v ðEÞdE of graphene are all numbers per unit area in the unit of m2 , or cm2 ; whereas their 3D counterparts, n, p, ρc ðEÞdE, and ρv ðEÞdE, are all numbers per unit volume in the unit of m3 , or cm3.

1.3

Fermi Energy, Chemical Potential, and Fermi Level Here we clarify the terminologies and notations that are sometimes swapped or not clear in some books and literature: the Fermi energy EF, the chemical potential μ, and the Fermi level. The Fermi energy EF is defined as the energy difference between the highest energy level and the lowest energy level that are occupied by electrons at the absolute zero temperature, T ¼ 0. Therefore, the Fermi energy is defined only at T ¼ 0; it is

12

Basic Properties and Band Structure

theoretically defined but experimentally cannot be exactly measured because the absolute temperature can never truly reach zero. For a metal, the Fermi energy is usually taken to be the kinetic energy of the electron that occupies the highest energy level at T ¼ 0; therefore, it is always positive. For graphene, we take the Fermi energy to be the total energy of the electron that occupies the highest energy level at T ¼ 0, which can be either positive or negative, or zero, if the reference point of zero energy is defined at the Dirac point, as seen below. By contrast, the chemical potential is meaningful at any temperature because it is a statistical reference for the total energy, including the potential energy and the kinetic energy, of a particle in a system in thermal equilibrium. In solid-state physics, the chemical potential μ usually refers to the change of free energy when an additional electron is added to a system. The effect of μ is best observed in the Fermi–Dirac distribution f0 ðEÞ, where one can see that at zero temperature, the minimum energy required to add an additional electron is μ because below it electronic states are all occupied. In the electronic devices context, the term Fermi level usually refers to the chemical potential μ. However, in semiconductor physics, the Fermi level is usually denoted by the symbol EF though it refers to the chemical potential μ. To avoid confusion with the Fermi energy, in this book we use the symbol EF to represent only the Fermi energy while using the symbol μ to represent the chemical potential and the Fermi level. From the above discussions, we see that μ is a function of temperature: μðTÞ. In the case when μð0Þ lies within the conduction band, EF is positive and can be found from the definition EF ≡ μð0Þ at zero temperature. In the case when μð0Þ lies within the valence band, we define EF to be negative using the definition EF ≡ μð0Þ at zero temperature. The temperature dependence of μðTÞ of graphene can be found by recognizing the fact that each additional thermally excited electron always comes with the creation of a hole to result in an electron–hole pair. Therefore, for a given piece of graphene, the difference between the electron concentration and the hole concentration, e n0  e p 0 , is a conserved quantity that is independent of temperature. In the case when μð0Þ ¼ EF > 0 at zero temperature so that the chemical potential lies within the conduction band, we have e p 0 ¼ 0 and e n0  e p0 ¼ e n0 þ e p 0 ¼ EF2 =πℏ2 v 2F from (1.38). Therefore we can relate μ and EF by combining (1.36) and (1.38) to write ð∞

EF2 πℏ2 v 2F

ð0

¼ e ρ c ðEÞ f0 ðEÞdE 

 e ρ v ðEÞ 1  f0 ðEÞ dE;

ð1:39Þ

∞

0

which can be written in a compact form: ð∞ EF2 E sinh ðμ=kB TÞ ¼ dE; cosh ðE=kB TÞ þ cosh ðμ=kB TÞ 2

ð1:40Þ

0

from which μ can be numerically obtained. In the case when μð0Þ lies within the valence band so that EF ≡ μð0Þ < 0, we have e n 0 ¼ 0 and e n0 þ e p0 ¼ e n0 þ e p 0 ¼ EF2 =πℏ2 v 2F from (1.38). Then, following a procedure similar to the above, we obtain

1.4 Temperature Dependence of Carrier Concentration

13

5

(

Carrier density 1011 cm

2

)

4.5 4 3.5 3 2.5 2 1.5

0

100

200

300

400

T (K) Figure 1.6 Carrier concentration for EF ¼ 50 meV as a function of temperature calculated under different assumptions. The dotted line and the dashed curve are obtained under the lowtemperature approximation and the constant μ approximation, respectively, while the solid curve is the exact result.

ð∞ EF2 E sinhðμ=kB TÞ ¼ dE: coshðE=kB TÞ þ coshðμ=kB TÞ 2

ð1:41Þ

0

When jμj ≫ kB T, the integrals for both (1.40) and (1.41) approach μ2 =2, and we have μ ≈ EF .

1.4

Temperature Dependence of Carrier Concentration The dependence of the free carrier concentration e n 0 of graphene on temperature is shown in Figure 1.6 for EF ¼ 50 meV. The characteristics obtained under two different approximations are compared to the exact temperature dependence of the carrier concentration obtained without approximations. The dotted line is the low-temperature approximation given by (1.38), and the dashed curve is obtained from (1.36) by assuming μ ¼ EF independent of temperature. The solid curve is the exact result calculated from (1.36), with μ varying with temperature obtained from (1.40). As can be seen, the dotted line is accurate only when the temperature is low, and the dashed curve overestimates the carrier concentration. In general, a higher temperature leads to a higher carrier concentration because of thermally excited electron–hole pairs.

14

Basic Properties and Band Structure

1.5

Carrier Velocity and Effective Mass In the preceding sections, we regard a charge carrier in graphene as a wave that has a momentum of ℏk and an energy of E. In this section, we treat the wave packet as a particle that has a velocity equal to the group velocity of its wave packet. For an isotropic two-dimensional material such as graphene, the phase velocity v p and the group velocity vg of an electron that has a momentum of ℏk and an energy of E are, respectively, vp ¼

E ∂E and v g ¼ : ℏk ℏ∂k

ð1:42Þ

For an electron in graphene near a Dirac point, the carrier energy is linearly proportional to the momentum: E ¼ ℏvF k; thus, we find that v p ¼ v g ¼ v F ≈ 1  106 m s1 , independent of the electron energy. The same conclusion applies to a hole in graphene near a Dirac point. Therefore, the parameter v F represents both the group velocity and the phase velocity of a carrier in graphene. The parameter vF also represents the Fermi velocity because a carrier that has a kinetic energy equal to the Fermi energy also moves at a group velocity of vg ¼ vF in the vicinity of a Dirac point. This result is distinctively pffiffiffiffiffiffiffiffiffiffiffiffiffiffi different from the energy-dependent group velocity v g ¼ 2E=m of a carrier in an ordinary material, where m is the effective mass of an electron in the conduction band of the material or that of a hole in the valence band. The linear energy–momentum relation of the carriers in graphene resembles the relativistic Dirac equation. Unlike the Schrödinger equation, the Dirac equation accounts for special relativity through the relativistic energy–momentum relation: E2 ¼ p2 c2 þ m20 c4 ;

ð1:43Þ

where p ¼ ℏk is the particle momentum, m0 is the rest mass of the particle, and c is the speed of light. If we compare the energy–momentum relation given in (1.22) for a carrier in graphene with the relativistic energy–momentum relation given in (1.43), we find that a carrier in graphene has an effective speed of c ¼ v F and a zero effective rest mass of m0 ¼ 0. Therefore, the carriers in graphene are often called massless Dirac fermions. In contrast to the effective rest mass, an effective mass associated with the acceleration of a carrier in graphene under the influence of an external force can be defined. For example, when a carrier in graphene is accelerated by an external magnetic field, the carrier follows a circular orbit dictated by Maxwell’s equations. If the area enclosed by the orbit is AðEÞ, the cyclotron effective mass is defined as [6,8,9] m ¼

ℏ2 ∂AðEÞ : 2π ∂E

With E ¼ ℏv F k and AðEÞ ¼ πk 2 ¼ πE2 =ðℏvF Þ2 , we obtain

ð1:44Þ

1.6 Band Structure of Multilayer Graphene

m ¼

E ℏk ¼ : v 2F v F

15

ð1:45Þ

Therefore, the cyclotron effective mass is energy dependent, and (1.45) manifests the mass–energy equivalence from special relativity, namely, E ¼ mc2 with c ¼ vF . In solid-state physics, the effective mass of a charge carrier is often calculated as m ¼ ℏ2 ð∂2 E=∂k 2 Þ1 because the energy of a carrier is quadratically dependent on its momentum near a conduction band minimum or a valence band maximum of an ordinary three-dimensional solid-state material. However, in the case of graphene, this relation is invalid because it gives a mathematically divergent result due to the fact that ∂2 E=∂k 2 ¼ 0 for a carrier in graphene. The reason is that the energy of a carrier in graphene near a Dirac point varies with its momentum not quadratically but linearly: E ¼ ℏv F k from (1.22). Thus the assumption of a constant effective mass given by m ¼ ℏ2 ð∂2 E=∂k 2 Þ1 is not valid for a carrier in graphene. Instead, m for a carrier in graphene can be calculated using the momentum relation mvg ¼ mv F ¼ ℏk so that m ¼

ℏk ℏk E ¼ ¼ 2; vg vF vF

ð1:46Þ

which gives a result that is consistent with (1.45). This relation yields m ¼ 0 at the Dirac point where k ¼ 0; this result is also conceptually consistent with the zero effective rest mass for a carrier in graphene discussed above. The comparison of parabolic 2D/3D systems and monolayer graphene for various physical quantities is presented in Table 1.1.

1.6

Band Structure of Multilayer Graphene When multiple layers of graphene are brought together to form a sheet of multilayer graphene, the interlayer interaction can fundamentally change the band structure. In this section, the band structures of AA and AB (Bernal) stacking orders are derived. Other possible stacking arrangements are discussed at the end.

1.6.1

AB Stacking Order In AB-stacked multilayer graphene, the arrangement of layers follows the αβαβ . . . order. Figure 1.7 illustrates the AB stacking by showing one set of α and β layers. Nearest-neighbor intralayer and interlayer interactions are considered, as shown in Figure 1.7(a); high-order interactions beyond nearest neighbors are neglected for simplicity. The intralayer interaction energy is γ0 , defined in (1.17), whereas the interlayer interaction energy is characterized by a different off-diagonal parameter γ1 in the Hamiltonian. The top layer (β layer) is shifted relative to the bottom layer (α layer) by one C–C distance, as shown in Figure 1.7(b).

16

Basic Properties and Band Structure

Table 1.1 Physical quantities for parabolic 2D/3D systems and monolayer graphene.a Quantity EF ¼ μð0Þ ≥ 0 kF ρ c ðEÞ ρc ðEÞ; e

Parabolic 3D  2=3 ℏ2 6π2 n g 2m  2 1=3 6π n g

2ℏ πe n mg   4πe n 1=2 g

gð2mÞ3=2

gm

4π2 ℏ3

ρ c ðEF Þ ρc ðEF Þ; e

E1=2

 2 1=3 3g n πℏ2 4π m

m

Parabolic 2D

Graphene   4πe n 1=2 ℏvF g  1=2 4πe n g

2

gE

2πℏ2 gm

2πℏ2 v2F  1=2 1 ge n ℏvF π

2πℏ2

Energy independent   2EF 1=2  m 1=2 2E m

vF vg

Equation (1.38) – (1.34) –

E v2F

(1.45)

Energy independent

–

vF

–

a: In this table, the parameters of an electron in a conduction band are listed; those of a hole in a valence band can be similarly expressed. E is the energy of the electron measured with respect to the conduction band edge, by setting Ec ¼ 0 for the conduction band minimum of a parabolic system and EDirac ¼ 0 for graphene. n is the electron density of a 3D system and e n is the electron density of a 2D system including graphene, in the limit that T → 0. g is the total degeneracy including spin degeneracy.

(a)

(b)

(c)

A

0

B 1 1

B

0

A z

y x

z

y x

x

Figure 1.7 (a) Intralayer (γ0 ) and interlayer (γ1 ) interactions of AB-stacked multilayer graphene showing one set of α and β layers. (b) Top view of (a), showing the overlap between α (bottom) and β (top) layers. (c) Side view of (a). A chain of alternating A and B carbon atoms connected through interlayer interactions of energy γ1 along the z direction is shown.

The band structure of N-layer AB-stacked multilayer graphene can be obtained in a manner similar to the derivation of the band structure of monolayer graphene. As shown in (1.27), after proper transformation the Hamiltonian at the K point for intralayer interactions can be expressed in terms of the energy parameters γ ¼ ℏvF ðkx  iky Þ ¼ 3aγ0 ðkx  iky Þ=2 with ℏv F ¼ 3aγ0 =2. Therefore, the tight-

1.6 Band Structure of Multilayer Graphene

17

binding Hamiltonian at the K point for AB-stacked multilayer graphene can be expressed in a form similar to (1.27) as [10] 3 2 0 0 0 0 γ 0 7 6γþ 0 γ1 0 0 0 7 6 7 6 0 γ1 0 γ 0 γ1 7 6 AB 7 6 0 0 γþ 0 . . . 0 0 ^ ð1:47Þ HK ¼ 6 7: 7 60 0 0 0 0 γ  7 6 7 6 0 0 γ1 0 γþ 0 5 4 .. .

B A B A B In (1.47), the basis to be jψA 1 〉; jψ 1 〉; jψ2 〉; jψ 2 〉; . . . ; jψ N 〉; jψN 〉 , with the 

Ais taken T B A B A;B A;B eigenvector jψ〉 ¼ c1 ; cB1 ; cA correspond to the 2 ; c2 ; . . . ; cN ; cN , where jψn 〉 and cn ^ AB jψ〉 ¼ Ejψ〉, we sublattice A or B of layer n. By applying the eigenvalue equation H K find from (1.47) the following coupled difference equations, ( (

B EcA n ¼ γ cn A A EcBn ¼ γ1 ðcA n1 þ cnþ1 Þ þ γþ cn

B B B EcA n ¼ γ1 ðcn1 þ cnþ1 Þ þ γ cn

EcBn ¼ γþ cA n

for odd n;

ð1:48Þ

for even n;

ð1:49Þ

B A B where n ¼ 1; 2; . . . ; N and cA 0 ¼ c0 ¼ cNþ1 ¼ cNþ1 ¼ 0.

We first check if E ¼ 0 is a nontrivial solution for the coupled equations of (1.48) and (1.49). For E ¼ 0, we have γ cBn ¼ 0 for odd n and γþ cA n ¼ 0 for even n. If we assume nonzero γ and γþ , we obtain from (1.48) and (1.49) a trivial solution with all c coefficients being zero. However, if we assume γ ¼ γþ ¼ 0, we have rows in (1.47) that are identically zero; thus, cBn for even n and cA n for odd n are decoupled and can assume any value. Other coefficients can be either zero or nonzero, determined by (1.48) and (1.49) and the number of layers. Therefore, E ¼ γþ ¼ γ ¼ 0

ð1:50Þ

represents a nontrivial solution for the coupled equations of (1.48) and (1.49). For E ≠ 0, we can simplify (1.48) and (1.49) as A ðE  γþ γ =EÞcBn ¼ γ1 ðcA n1 þ cnþ1 Þ

for odd n;

B B ðE  γþ γ =EÞcA n ¼ γ1 ðcn1 þ cnþ1 Þ for even n:

ð1:51Þ ð1:52Þ

Then, by defining cn ¼ cBn for odd n and cn ¼ cA n for even n, the coupled equations of (1.51) and (1.52) can be expressed in a single simplified form: E0 cn ¼ γ1 ðcn1 þ cnþ1 Þ;

ð1:53Þ

where E0 ¼ E  γþ γ =E. Thus we have transformed the original problem expressed in the coupled equations of (1.48) and (1.49) into a chain of N atoms with eigenenergy E0

18

Basic Properties and Band Structure

while considering only the nearest-neighbor interactions characterized by the interlayer interaction energy γ1, as shown in Figure 1.7(c). With the constraint that c0 ¼ cNþ1 ¼ 0, we find the solutions of (1.53) as   πn cn ¼ sin r ; ð1:54Þ N þ1 where r ¼ 1; 2; . . . ; N. Therefore, we find N eigenvectors for N-layer graphene. By plugging (1.54) back into (1.53), we find the eigenenergies:   π E0 ¼ 2γ1 cos r : ð1:55Þ N þ1 To gain more physical insight for (1.55), we can regard N-layer graphene as a Fabry– Pérot cavity in which an electromagnetic wave is confined in the direction perpendicular to the multilayer graphene surface, which is taken to be the z direction [11]. The boundary condition c0 ¼ cNþ1 ¼ 0 implies that the wave function vanishes at n ¼ 0 and n ¼ N þ 1 beyond the first and Nth layers. The distance between the locations of n ¼ 0 and n ¼ N þ 1 layers is ðN þ 1Þd; therefore, the effective length of the Fabry–Pérot cavity is deff ¼ ðN þ 1Þd, where d ¼ 0:335 nm is the distance between two neighboring graphene layers in AB stacking. A standing wave can form within the multilayer graphene structure with two opposite wave vectors of equal magnitude in the z direction: kz ¼ r

π π : ¼ r deff ðN þ 1Þd

ð1:56Þ

Therefore, as the thickness of the multilayer graphene increases with the number N of layers, the number of the allowed values of quantized kz also increases. By using (1.56), we can rewrite (1.55) as E0 ¼ 2γ1 coskz d;

ð1:57Þ

which gives the eigenenergy of N interacting carbon atoms with kz being the wave number along the direction normal to the graphene surface. Indeed, if the in-plane wave numbers have zero values so that kx ¼ ky ¼ 0 and γþ ¼ γ ¼ 0, the original problem represented by (1.48) is tantamount to the chain of carbon atoms along the z direction shown in Figure 1.7(c). The eigenenergy is simply E ¼ E0 because γþ ¼ γ ¼ 0 in the relation E0 ¼ E  γþ γ =E. When kx and ky are not both zero so that γþ ≠ 0 and γ ≠ 0, the eigenenergy E can be found from the relation E0 ¼ E  γþ γ =E, which gives E¼

1 E0  2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ðℏvF k∥ Þ2 þ E02 ;

ð1:58Þ

where k∥ ¼ ðkx2 þ ky2 Þ1=2 is the in-plane wave number, in contrast to the out-of-plane wave number kz. In the limit that jE0 j ≫ ℏvF k∥ , (1.58) has two small-k∥ solutions:

1.6 Band Structure of Multilayer Graphene

19

ðℏv F k∥ Þ2 E0

ð1:59Þ

ðℏv F k∥ Þ2 : E0

ð1:60Þ

E ¼ E0 þ and E¼

Note that in both (1.59) and (1.60), E varies quadratically, not linearly, with k∥ . By defining the effective mass for these quadratic bands as m ¼ jE0 j=2v2F , the first solution (1.59) has the form of E ¼ E0  ðℏk∥ Þ2 =2m, and the second solution (1.60) has the form of E ¼  ðℏk∥ Þ2 =2m, where the upper signs are taken when E0 > 0 and the lower signs are taken when E0 < 0. Accordingly, the energy bands given by (1.59) and (1.60) are called massive bands, in contrast to the massless Dirac bands of monolayer graphene. For even N, there are N different values of E0 given by (1.55). Therefore, (1.59) and (1.60) respectively give N energy bands and thus together a total of 2N massive energy bands. Because half of E0 values are positive and the other half are negative, there are N conduction bands and N valence bands. As an example, the band structure of ABstacked bilayer graphene (N ¼ 2) is shown in Figure 1.8(a), where the solid curves are obtained from (1.58) without approximation and the dashed curves are obtained from (1.59) and (1.60) under the small-k∥ approximation, using γ0 ¼ 3:16 eV and (a)

(b)

2

r=

3

1

r=

2

1

r=

1

0

E

E

1

r

r

=1

r=

1

=2

0

r=

r=

1

2 0.2

2

2

0 k a0

0.2

3 0.2

2

3

1

r=

r=

1

1

2

3

0 k a0

0.2

Figure 1.8 Band structures of AB-stacked (a) bilayer and (b) trilayer graphene. The solid curves in

(a) for both r ¼ 1 and 2 and those in (b) for r ¼ 1 and 3 are obtained from (1.58), and the dashed curves are the small-k∥ approximation obtained from (1.59). The linear bands for r ¼ 2 in (b) are the Dirac bands.

20

Basic Properties and Band Structure

γ1 ¼ 0:37 eV obtained from experiments [4]. As can be seen, there are two bands for each value of r (marked next to the curves), and there are a total of 2N ¼ 4 bands for the bilayer system, all of which are massive bands. The effective mass in the small-k∥ limit is m ¼ 0:032me , where me ¼ 9:11  1031 kg is the rest mass of a free electron. The eigenvectors can be obtained accordingly from (1.54) for cB1 ¼ c1 and cA 2 ¼ c2 ,  T γ c2 γ c1 B ; c1 ; c2 ; þ which in turn give cA , where the 1 and c2 from (1.48): jψ〉 ¼ E E eigenenergy E is given in (1.58). For odd N, we can always find a value of r such that E0 ¼ 0 given by (1.57), for which (1.58) gives the linear bands E ¼ ℏvF k∥ . Therefore, there are 2ðN  1Þ massive conduction and valence bands described by (1.59) and (1.60), and two linear bands defining one massless Dirac cone given by E ¼ ℏvF k∥ . As an example, the band structure of AB-stacked trilayer of N ¼ 3 is shown in Figure 1.8(b), where six bands can be seen; four are massive bands and two are massless linear Dirac bands. From the above discussions, N-layer AB-stacked graphene has 2N bands. For even N, all 2N bands are massive bands. For odd N, there are 2ðN  1Þ massive bands plus two massless linear Dirac bands. The band structure of natural graphite follows the AB stacking order; it can be obtained from (1.56) and (1.57) by taking the limit that N → ∞ in (1.54). In this limit, kz is no longer discrete but forms a continuous spectrum. In a reduced Brillouin zone scheme, kz spans from π=2d to π=2d, where the H point is located, as shown in Figure 1.9 [12]. At k∥ ¼ 0, there are three energy bands: one energy band for E ¼ 0 given by (1.50), and two energy bands for E ¼ E0 given by (1.57) with E0 > 0 for a conduction band and E0 < 0 for a valence band, as shown in Figure 1.9. By using

(a)

(b) 2

H

H

0

2

H

H

E

1

0

E

1

2

2

k 0

k 0 2d

0 kz

2d

2d

0 kz

2d

Figure 1.9 Band structures of AB-stacked (a) bilayer graphene and (b) trilayer graphene, obtained

by cutting the band structure of graphite at certain kz values.

1.6 Band Structure of Multilayer Graphene

21

(1.58), one can map out the band structure of graphite as a function of k∥ at different values of kz . The band structures of bilayer and trilayer graphene are shown in Figure 1.8. For bilayer graphene, two bands at kz ¼ π=3d, respectively, in the reduced zone scheme are shown in Figure 1.9(a). For trilayer graphene, the massive bands at kz ¼ π=4d and the massless Dirac bands at kz ¼ π=2d are also plotted, as shown in Figure 1.9(b). Note that the mirrored band structure does not represent double degeneracy because the positive and negative values of kz do not represent two independent eigenstates but represent two counter-propagating waves that form the standing wave in multilayer graphene. As can be seen from (1.54) and (1.56), for any solution of kz , the values of cn obtained using the positive and negative values of kz only differ by a negative sign. Therefore, both sets of the coefficients cn obtained from the positive and negative values of kz give the same eigenstate jψ〉. The truly independent states are the standing waves of different wave numbers represented by different values of jkz j.

1.6.2

AA Stacking Order In the AA stacking order, the sublattice A of the upper layer is directly above the sublattice A of the lower layer, and the sublattice B of the upper layer is also directly above the sublattice B of the lower layer, as shown in Figure 1.10. Nearest-neighbor intralayer and interlayer interactions are considered. The intralayer interaction energy γAA and the interlayer interaction energy γAA of 0 1 AA-stacked multilayer graphene are not generally the same as γ0 and γ1 of AB-stacked multilayer graphene. The tight-binding Hamiltonian at the K point for AA-stacked multilayer graphene is constructed as [10]

AA 0

AA 1

B A

z y x

Figure 1.10 Atomic structure of AA-stacked bilayer graphene.

22

Basic Properties and Band Structure

2

^ AA H K

0

6γAA 6þ 6γAA 61 6 ¼6 0 6 0 6 6 0 4

γAA  0 0 γAA 1 0 0

γAA 1 0 0 γAA þ γAA 1 0

0 γAA 1 γAA  0 0 γAA 1 .. .

0 0 γAA 1 0 0 γAA þ

0 0 0 ...

γAA 1 γAA  0

3 7 7 7 7 7 7; 7 7 7 5

ð1:61Þ

AA AA AA AA where γAA  ¼ ℏv F ðkx  iky Þ ¼ 3aγ0 ðkx  iky Þ=2 with ℏv F ¼ 3aγ0 =2. The basis for AA A B A B A B ^ H K is again taken to be fjψ1 〉; jψ  1 〉; jψ2 〉; jψ2 〉; . . . ; jψN 〉; jψN 〉g, with the eigenvector B A B A B T ^ AB jψ〉 ¼ cA ; c ; c ; c ; . . . ; c ; c 1 1 2 2 N N , both of which have the same form as those for H K ^ AA jψ〉 ¼ Ejψ〉, we find from (1.61) the of (1.47). By applying the eigenvalue equation H K following difference equations: AA A A AA B EcA n ¼ γ1 ðcnþ1 þ cn1 Þ þ γ cn ;

ð1:62Þ

B B AA A EcBn ¼ γAA 1 ðcnþ1 þ cn1 Þ þ γþ cn ;

ð1:63Þ

B A B where n ¼ 1; 2; . . . ; N and cA 0 ¼ c0 ¼ cNþ1 ¼ cNþ1 ¼ 0. Now we want to transform (1.62) and (1.63) into a form similar to that of (1.53), whose solutions are already known. To do so, we first multiply (1.63) by an unknown variable x that gives us the freedom to adjust the coefficient. We then add (1.62) to (1.63). With some arrangements followed by collecting terms of the same n index, we get " #    A   A  γAA AA A B AA B B  ðE  xγþ Þcn þ E  cn1 þ xcn1 þ cnþ1 þ xcnþ1 : ð1:64Þ xcn ¼ γ1 x

AA If xγAA þ ¼ γ =x, which can be satisfied by taking x ¼  simplified:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AA γAA  =γþ , (1.64) can be

AA ðE  xγAA þ Þcn ¼ γ1 ðcn1 þ cnþ1 Þ;

ð1:65Þ

B where cn ¼ cA n þ xcn . Equation (1.65) has the form of (1.53); therefore, we obtain the eigenenergies: AA AA E ¼ ℏvAA ; F jk∥ j þ 2γ1 cos kz d

ð1:66Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AA AA γAA where the identity γAA  =γþ ¼ ℏv F jk∥ j is used, kz is given by (1.56) with d þ AA AA replaced by d , and d ¼ 0:36 nm is the distance between two AA-stacked graphene layers [13,14]. Different from the bands of AB-stacked multilayer graphene, the energy bands of AA-stacked graphene given in (1.66) are all massless linear bands. These linear bands form Dirac cones of shifted energies. As examples, the band structures of AA-stacked bilayer graphene (N ¼ 2) and trilayer graphene (N ¼ 3) are shown in Figures 1.11(a) and (b), respectively, using

1.6 Band Structure of Multilayer Graphene

(a)

(b)

5

5

1 r=

AA 1

r=

r=

5 0.5

0 k a0

E

E

0

r=

1

r=

2

r=

3

2

AA 1

r=

23

0

1

2

0.5

5 0.5

r=

1

r=

2

r=

3

0 k a0

0.5

Figure 1.11 Band structures of AA-stacked (a) bilayer graphene and (b) trilayer graphene.

AA γAA 0 ¼ 2:59 eV and γ1 ¼ 0:217 eV [13,15]. As can be seen, there are two bands for each value of r (marked next to the curves) and a total of 2N bands for the system. The shift of the Dirac point for each band is determined by the second term in (1.66). Note that each Dirac cone of a given r value has its unique kz value. Therefore, unless an out-of-plane momentum is provided, carrier transitions between two different Dirac cones are forbidden.

1.6.3

Misorientation and Other Stacking Orders For chemically deposited multilayer graphene, misorientation might happen among graphene layers [15,16]. When the relative rotation is small, large areas of locally AA-stacked and AB-stacked regions are formed, as shown in Figure 1.12 for misoriented bilayer graphene. In such a case, each of these regions can be locally treated as AA or AB stacking if the mean free path of the carriers is much smaller than the physical size of the region. Otherwise, one must identify the unit cell of the misoriented multilayer system and numerically calculate the band structure. It is found that while the energy band is still linear around the K point, the Fermi velocity is greatly reduced and therefore localization of electrons is possible in misoriented multilayer graphene [17]. When the number of layers increases, the number of different possible stacking arrangements also increases. For example, in ABC-stacked multilayer graphene, there is a continuous, constant shift from one layer to the next, as shown in Figure 1.13.

24

Basic Properties and Band Structure

Figure 1.12 Misoriented bilayer graphene with locally AA-stacked and AB-stacked regions. Reprinted figure with permission from I. T. Lin, J. M. Liu, K. Y. Shi, P. S. Tseng, K. H. Wu, C. W. Luo, and L. J. Li, “Terahertz optical properties of multilayer graphene: Experimental observation of strong dependence on stacking arrangements and misorientation angles,” Physical Review B, Vol. 86, 235446 (2012). Copyright 2012 by the American Physical Society.

c

Figure 1.13 ABC stacking of multilayer graphene formed by three distinct relative positions of graphene layers, labeled α, β, and c, each successively displaced by one carbon‒carbon distance.

The Hamiltonian of ABC-stacked graphene is given as 2

^ ABC H K

0

6γABC 6þ 6 0 6 6 ¼6 0 6 0 6 6 0 4

γABC  0 γABC 1 0 0 0

0 γABC 1 0 γABC þ 0 0

0 0 γABC  0 γABC 1 0 .. .

0 0 0 γABC 1 0 γABC þ

0 0 0 0 γABC  0

3 7 7 7 ... 7 7 7; 7 7 7 5

ð1:67Þ

References

(a) 4

(b)

ABC 1

0

E

E

4

2

ABC 1

2

25

2

0

2

4 0.5

0 k a0

0.5

4

0.5

0 k a0

0.5

Figure 1.14 Band structures of ABC-stacked (a) trilayer graphene and (b) four-layer graphene.

where γABC ¼ ℏv ABC ðkx  iky Þ ¼ 3aγABC ðkx  iky Þ=2 with ℏvABC ¼ 3aγABC =2, and  F 0 F 0 ABC ABC γ1 is the interlayer interaction energy. Again, γ0 is not generally equal to γ0 or ABC γAA is not generally equal to γ1 or γAA 0 , and γ1 1 [18]. Simple as it might seem for the Hamiltonian shown in (1.67), there do not exist simple difference equations as those obtained for AB- and AA-stacked multilayer graphene [10]. Instead, the eigenenergies of (1.67) are numerically obtained for the ABC trilayer and four-layer graphene, as shown in Figure 1.14. As can be seen, regardless of the number of layers, there are always two bands that touch at the K point. The other energy bands are shifted in energy and momentum in different manners but overlap at γABC for k∥ ¼ 0. 1

References 1. S. Reich, J. Maultzsch, C. Thomsen, and P. Ordejón, “Tight-binding description of graphene,” Physical Review B, Vol. 66, 035412 (2002). 2. N. M. R. Peres, “Colloquium: The transport properties of graphene: an introduction,” Review of Modern Physics, Vol. 82, pp. 2673–2700 (2010). 3. D. Chung, “Review graphite,” Journal of Materials Science, Vol. 37, pp. 1475–1489 (2002). 4. D. S. L. Abergel, V. Apalkov, J. Berashevich, K. Ziegler, and T. Chakraborty, “Properties of graphene: A theoretical perspective,” Advances in Physics, Vol. 59, pp. 261–482 (2010). 5. P. R. Wallace, “The band theory of graphite,” Physical Review, Vol. 71, pp. 622–634 (1947). 6. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Review of Modern Physics, Vol. 81, pp. 109–162 (2009). 7. O. V. Sedelnikova, L. G. Bulusheva, and A. V. Okotrub, “Ab initio study of dielectric response of rippled graphene,” Journal of Chemical Physics, Vol. 134, 244707 (2011).

26

Basic Properties and Band Structure

8. D. L. Miller, K. D. Kubista, G. M. Rutter, et al., “Observing the quantization of zero mass carriers in graphene,” Science, Vol. 324, pp. 924–927 (2009). 9. K. S. Novoselov, A. K. Geim, S. V. Morozov, et al., “Two-dimensional gas of massless Dirac fermions in graphene,” Nature, Vol. 438, pp. 197–200 (2005). 10. H. Min and A. H. MacDonald, “Electronic structure of multilayer graphene,” Progress of Theoretical Physics Supplement, Vol. 176, 227 (2008). 11. K. F. Mak, M. Y. Sfeir, J. A. Misewich, and T. F. Heinz, “The evolution of electronic structure in few-layer graphene revealed by optical spectroscopy,” Proceedings of the National Academy of Sciences, Vol. 107, pp. 14999–15004 (2010). 12. J. C. Slonczewski and P. R. Weiss, “Band structure of graphite,” Physical Review, Vol. 109, pp. 272–279 (1958). 13. I. Lobato and B. Partoens, “Multiple Dirac particles in AA-stacked graphite and multilayers of graphene,” Physical Review B, Vol. 83, 165429 (2011). 14. J. K. Lee, S. C. Lee, J. P. Ahn, et al., “The growth of AA graphite on (111) diamond,” Journal of Chemical Physics, Vol. 129, 234709 (2008). 15. I. T. Lin, J. M. Liu, K. Y. Shi, et al., “Terahertz optical properties of multilayer graphene: Experimental observation of strong dependence on stacking arrangements and misorientation angles,” Physical Review B, Vol. 86, 235446 (2012). 16. E. J. Mele, “Interlayer coupling in rotationally faulted multilayer graphenes,” Journal of Physics D: Applied Physics, Vol. 45, 154004 (2012). 17. G. Trambly de Laissardière, D. Mayou, and L. Magaud, “Localization of Dirac electrons in rotated graphene bilayers,” Nano Letters, Vol. 10, pp. 804–808 (2010). 18. F. Zhang, B. Sahu, H. Min, and A. H. MacDonald, “Band structure of ABC-stacked graphene trilayers,” Physical Review B, Vol. 82, 035409 (2010).

2

Electronic Properties

2.1

Current and Conductivity of a Two-Dimensional Material The flow of free charge carriers, i.e., electrons in the conduction band or holes in the valence band, in a semiconductor that is subject to an electric field is accelerated by the electric field but is hindered by scattering events. In a semiconductor, free carriers accelerate in the presence of an electric field. The randomly distributed scattering centers, such as impurities and defects, act as a counter force that decelerates and deflects the carriers. When the steady state is eventually reached under a constant electric field, a constant flow of carriers is achieved. In graphene, by contrast, charge carriers on the Dirac cone have a constant speed and do not accelerate or decelerate in response to the electric field or the scattering centers; instead, the effect of the electric field is rather to align the motion of carriers to the direction of the electric field, and the scattering centers act as a source to disturb this alignment process. This process is well captured by the Boltzmann transport equation, which has successfully described many statistical behaviors of carriers in metals and semiconductors. The purpose of this chapter is to describe the electronic properties of graphene, starting from the Boltzmann transport equation. In Section 2.5, the theoretically derived electronic properties are then compared with experimental observations to show the agreement between the theory and the experimental data. In this chapter, only the DC electronic properties are considered. Therefore, the electric field and the electric current density considered in this chapter are both timeindependent real quantities, though both of them can vary with space to account for their spatial distributions. In this book, real field vectors are represented in italic, bold, capital Roman characters, such as Eðr; tÞ and Jðr; tÞ, whereas the complex field vectors of a harmonic field are represented in regular bold, capital Roman characters, such as Eðr; tÞ and Jðr; tÞ. The DC real field vectors considered in this chapter have no complex counterparts because they are not harmonic fields. However, to maintain consistency in notation and to avoid confusion, we represent these DC field vectors in the form of EðrÞ and JðrÞ, which does not vary with time but can vary with space. Unlike most three-dimensional (3D) materials in which the electric current can flow in any direction, the electric current of a two-dimensional (2D) material such as graphene is confined on a surface. In the following discussion, the graphene sheet is taken to be flat

28

Electronic Properties

and located at z ¼ 0 on the xy plane; then the current is confined on the flat surface defined by the xy plane. We can write the current density as Jðx; y; zÞ ¼ Jx ðx; y; zÞˆx þ Jy ðx; y; zÞˆy J y ðx; yÞδðzÞˆy ¼e J x ðx; yÞδðzÞˆx þ e e ¼ J ðx; yÞδðzÞ;

ð2:1Þ

where the tilde is added to distinguish 2D physical quantities from 3D quantities. From (2.1), the 2D surface current density is thus given by Je ðx; yÞ ¼ e J x ðx; yÞˆx þ e J y ðx; yÞˆy :

ð2:2Þ

As can be seen, for monolayer graphene, the current is confined on the z ¼ 0 plane; the z component of Je is zero, and Je is not a function of z. Because the current density is a vector, the conductivity is generally defined as a tensor: 2 3 σ xx ðx; y; zÞ σ xy ðx; y; zÞ σ xz ðx; y; zÞ σðx; y; zÞ ¼ 4σ yx ðx; y; zÞ σ yy ðx; y; zÞ σ yz ðx; y; zÞ 5: ð2:3Þ σzx ðx; y; zÞ σzy ðx; y; zÞ σ zz ðx; y; zÞ Because the tensor σ represents a physical quantity, it can always be diagonalized and represented by three diagonal elements σ x , σ y , and σ z , which are the eigenvalues of the matrix given in (2.3). The x and y principal axes on the graphene surface are determined by the hexagonal structure of the graphene lattice, and the z principal axis is normal to the graphene surface. Because the current is confined on the graphene surface as (2.1) clearly shows, the conductivity is nonzero only on the graphene surface at z ¼ 0. Furthermore, σ z ¼ 0 because an electric field perpendicular to the graphene surface does not induce a current, as can also be seen from the fact that Jz ¼ 0 in (2.1). We can thus simplify (2.3) as 2 3 e σ x ðx; yÞδðzÞ 0 0 σðx; y; zÞ ¼ 4 0 e σ y ðx; yÞδðzÞ 0 5 ¼ e σ ðx; yÞδðzÞ; ð2:4Þ 0 0 0 where the 2D surface conductivity is 2 0 e σ x ðx; yÞ e e σ y ðx; yÞ σ ðx; yÞ ¼ 4 0 0 0

3 0 0 5; 0

ð2:5Þ

which is related to the 3D conductivity by the relation σ ¼ e σ δðzÞ. Using (2.1), (2.4), and the relation J ¼ σ  E, we obtain Je ðx; yÞδðzÞ ¼ e σ ðx; yÞδðzÞ  Eðx; y; zÞ ¼ e σ ðx; yÞ  Eðx; y; 0ÞδðzÞ; or simply

ð2:6Þ

2.1 Current and Conductivity of a Two-Dimensional Material

29

e ðx; yÞ  Eðx; yÞ at z ¼ 0; Je ðx; yÞ ¼ σ

ð2:7Þ

which gives e J x ðx; yÞ ¼ e σ x ðx; yÞEx ðx; yÞ and e J y ðx; yÞ ¼ e σ y ðx; yÞEy ðx; yÞ. For intrinsic graphene that is not deformed in any manner, an electric field parallel to the graphene surface can only induce a current in the direction of the field because undeformed intrinsic graphene is electrically isotropic on the xy plane of its surface, though the principal x and y axes determined by the hexagonal graphene lattice cannot be arbitrarily chosen. In other words, though the lattice structure of graphene is not isotropic, the linear electric property of undeformed graphene is isotropic. For this reason, we have e σx ¼ e σy ¼ e σ∥ ¼ e σ for undeformed graphene, where the subscript ∥ labels the conductivity on the graphene surface; therefore, (2.5) becomes 2 3 2 3 e σ ∥ ðx; yÞ 0 0 e σ ðx; yÞ 0 0 e σ ðx; yÞ ¼ 4 0 e σ ∥ ðx; yÞ 0 5 ¼ 4 0 e σ ðx; yÞ 0 5: ð2:8Þ 0 0 0 0 0 0 Note, however, that anisotropy can be introduced by deforming the graphene lattice, for example, by stretching a sheet of graphene in a specific direction. In this situation, e σ in the form of (2.5) is still valid with x and y directions properly chosen to be aligned with the principal axes of the anisotropy, but e σx ≠ e σ y so that (2.8) is no longer valid. The comparison between the 2D and 3D physical quantities is illustrated in Figure 2.1. Because the current density J is defined as current per unit area, the unit of J is ampere per square meter (A m2 ); accordingly, the conductivity σ has the unit of siemens per meter (S m1 ). In the case of 2D graphene, the surface current density Je is defined as current per unit length and thus has the unit of ampere per meter (A m1 ). Therefore, in contrast to the usual 3D conductivity σ, the 2D surface conductivity e σ of graphene has the unit of siemens (S). Surface conductivity is also known as sheet conductivity. Similar to the relation ρ ¼ 1=σ between the resistivity and the conductivity of a 3D material, the (a)

(b)

w E Vm

1

E Vm

1

I J= E=

I wd

d

J

w

z=0 J= E=

J Am

2

Sm

1

J Am

1

I

I w S

Figure 2.1 Units of conductivity and current density for (a) a 3D material, such as gold, and (b) a 2D material, such as graphene.

30

Electronic Properties

relation between the surface resistivity, also called sheet resistivity, and the surface conductivity of a 2D material is e ρ¼

1 : e σ

ð2:9Þ

Note that, in general, surface conductivity e σ, which is defined only for a 2D material, is fundamentally different from surface conductance, Gsurface , which can be measured for a 2D or 3D material; also, sheet conductivity e σ, which is defined only for a 2D material to be the same as surface conductivity, is conceptually different from sheet conductance, Gs , which is usually defined for a thin 3D material. Similarly, surface resistivity e ρ is fundamentally different from surface resistance, Rsurface ¼ 1=Gsurface , and sheet resistivity e ρ is conceptually different from sheet resistance, Rs ¼ 1=Gs . By definition, conductivity and resistivity are respectively specific conductance and specific resistance, which are intrinsic properties of a material, but conductance and resistance depend on the geometry and the size of a material. Furthermore, surface conductance and surface resistance generally refer to the conductance and resistance measured on the surface of a 3D material, whereas sheet conductance and sheet resistance depend on the thickness d of a thin 3D material, such as a thin film, as Gs ¼ σd and Rs ¼ ρ=d respectively; therefore, surface conductance and surface resistance are practically different from sheet conductance and sheet resistance. For 2D graphene, however, these fundamental differences in definition, concept, and practice do not result in any quantitative difference if we quantitatively define a 3D conductivity as σ ¼ e σ =d and a 3D resistivity as ρ ¼ e ρ d for graphene, where d is the thickness of the graphene sheet. Then, we have Gsurface ¼ Gs ¼ e σ and Rsurface ¼ Rs ¼ e ρ for graphene.

Boltzmann Transport Equation To study the electronic transport properties of graphene, we start from the Boltzmann transport equation:

Δ

∂f þF ∂t

p

f þv

Δ

2.2

  ∂f ; rf ¼ ∂t coll

ð2:10Þ

where f ðp; r; tÞ is a distribution function that gives the probability of an electron having a momentum p at location r and time t, F is an external force, v is the velocity of the electron, and the term on the right-hand side of the equation is the rate of change of f ðp; r; tÞ due to scattering events. Here, we consider a homogeneous system subject to a uniform electric field E in the steady state so that f ðp; r; tÞ ¼ f ðpÞ and F ¼ eE, where e is the elementary charge. Without loss of generality, we assume that the uniform DC electric field is along the x direction: E ¼ Ex xˆ . In this case, (2.10) becomes

2.2 Boltzmann Transport Equation

Δ

  ∂f ∂f ¼ þF p f þv ∂t coll ∂t ¼F pf df ¼ eEx ℏdkx df ; ¼ eEx ℏkx v2F EdE

r

31

f ð2:11Þ

Δ

Δ

where the linear energy–momentum relation, E ¼ ℏv F k, of the carriers in graphene is used. The sign of the carrier energy does not affect the result in (2.11). We further adopt the relaxation-time approximation by assuming that the scattering processes cause the deviation of the distribution function f from the thermal equilibrium Fermi–Dirac distribution f0 to decay exponentially over time with a relaxation time constant τ. The collision term can thus be expressed as   ∂f f f0 : ð2:12Þ ¼ ∂t coll τ Combining (2.11) and (2.12), we obtain f ≈ f0 þ τeEx ℏkx v2F

df0 ; EdE

ð2:13Þ

where we have assumed that the electric field is weak so that df =dE ≈ df0 =dE. The electric current density in graphene caused by the applied electric field is given by ðð ðð g g e e e J ¼ J e þ J h ¼ e vx f dk þ e v x ð1  f Þdk; ð2:14Þ ð2πÞ2 ð2πÞ2 where g ¼ 4 accounts for the valley and spin degeneracies, vx ¼ vF kx =k is the velocity along the direction of the electric field, which is taken to be the x direction, and e J e and e Jh are the current densities contributed by electrons in the conduction band and by holes in the valence band, respectively. Note that in the 2D graphene structure the current densities e J e and e J h have the unit of A m1 because a 2D structure has a 1D crosssection, as shown in Figure 2.1(b). Using (2.13), we obtain the electron current density: e J e ¼ e

¼

ð∞2π ð

g ð2πÞ2

vF 0 0

v2 g e2 Ex F 2

 kx f kdθdk k

ð∞2ðπ τ cos2 θ

ð2πÞ 0 0 ð∞ e2 Ex df0 EdE ¼ 2 τ dE πℏ

¼

τeff e2 Ex πℏ2

df0 kdθdk dE

0

kB T ln ðe μ=kB T þ 1Þ:

ð2:15Þ

32

Electronic Properties

In the final step of the above derivation, τ is pulled out of the integral and is replaced by an effective relaxation time τeff to circumvent the difficulty in the evaluation of the integral due to the complicated energy dependence of τ, as is discussed in Section 2.3. When μ ≫ kB T, df0 =dE is effectively a Dirac delta function, and the effective relaxation time is a function of the chemical potential: τeff ≈ τðμÞ. At a sufficiently high temperature such that the condition μ ≫ kB T is not valid, the integral in (2.15) has to be numerically calculated to account for the energy dependence of τ. By following a similar procedure, the hole current density is obtained:  τeff e2 Ex μ=kB T e Jh ¼ k T ln e þ 1 : B πℏ2

ð2:16Þ

J h is in the sign of the We see that the difference between the expressions for Je e and e chemical potential, which is a direct consequence of the symmetry with respect to the Dirac point between the conduction band and the valence band of graphene. Because the surface conductivity is related to the surface current density as Je ¼ e σ E, we combine (2.15) and (2.16) to obtain the surface conductivity: e σ¼

2τeff e2 kB T πℏ2

  μ ln 2cosh : 2kB T

ð2:17Þ

Note that the surface conductivity e σ of a two-dimensional material such as graphene has the unit of S in contrast to the unit of S m1 for the conductivity σ of a threedimensional material. For jμj ≫ 2kB T, τeff ≈ τðμÞ ≈ τðEF Þ as discussed above, and (2.17) is reduced to e σ≈

τðEF Þe2 πℏ2

jμj:

ð2:18Þ

According to (2.18), it seems as if the surface conductivity would approach zero as μ → 0. This statement is not true because (2.17) is a semiclassical result that is valid only for a large quantity of carriers, whereas (2.18) is valid under (2.17) only for jμj ≫ 2kB T. Indeed, even (2.17) indicates that the surface conductivity would not vanish as μ → 0, though (2.17) is not truly valid in this limit as the carrier density approaches a vanishingly low value. When μ approaches the Dirac point with a vanishing carrier density, a more sophisticated model is needed to account for the enhanced quantum effects in the calculation of graphene surface conductivity. In transport experiments, the temperature T and the chemical potential μ in (2.17) are the control parameters. The only unknown parameter is the effective relaxation time τeff , which is approximately τðμÞ when jμj is sufficiently larger than 2kB T. In the following, we theoretically derive the value of τ as a function of carrier energy. The theoretical value of τðμÞ is then used to calculate the surface resistivity and the mobility of graphene, which can be compared with the experimentally observed physical quantities.

2.3 Scattering of Carriers

2.3

33

Scattering of Carriers As can be seen in (2.17) and (2.18), it is necessary to know the value of the relaxation time before the surface conductivity of graphene can be determined. The relaxation time is determined by the existing scattering mechanisms in a graphene sample. Extrinsic scattering mechanisms include scattering from impurities and defects as these scattering centers are not the inherent property of graphene. For a graphene sample that is directly deposited on a substrate, carriers can also be scattered by the surface phonons arising from the optical phonons of the substrate. Scattering from these phonons is another extrinsic scattering channel. The carrier relaxation times for different graphene samples are usually different due to the extrinsic variations from one sample to another in impurity doping and in defect density, as well as in the type of substrate on which a graphene sheet is placed. Contrary to the extrinsic scattering mechanisms, intrinsic scattering mechanisms are inherent to graphene, such as intrinsic acoustic phonon scattering and optical phonon scattering from the phonon modes of graphene. These intrinsic scattering mechanisms cannot be eliminated by any method; thus, they set the limit for the relaxation time and the mobility of carriers in graphene. In this section, the analytical expression and the theoretical value of the relaxation time or the scattering rate due to the aforementioned scattering mechanisms are discussed. Once the value of the relaxation time is determined, the surface conductivity and surface resistivity of a graphene sample, and the mobility of the carriers in the graphene sample, can be evaluated. To obtain the relaxation time τ, we first express the collision term as the sum of all the possible transitions of carriers scattered into and out of state k in band n:       X ∂f 0 0 0 0 ¼ Sðk ; kÞf ðk Þ 1  f ðkÞ  Sðk; k Þf ðkÞ 1  f ðk Þ ; ð2:19Þ ∂t coll n0 ; k0 where the k0 state is in the n0 band, Sðk0; kÞ is the scattering transition rate of carriers scattered from k0 to k, and Sðk; k0 Þ is the scattering transition rate from k to k0 . Using (2.12), (2.13), and the fact that   0 Sðk ; kÞf0 ðk Þ 1  f0 ðkÞ ¼ Sðk; k Þf0 ðkÞ 1  f0 ðk Þ 0

0





0

ð2:20Þ

  1  f0 ðEk0 Þ τðEk Þ  n0 nτðEk0 Þcos θ ; 1  f0 ðEk Þ

ð2:21Þ

in equilibrium, (2.19) can be simplified as [1] 1¼

X n0 ; k0

Sðk; k0 Þ

where θ is the angle between k and k0 , i.e., cos θ ¼ k  k0 =kk 0 , and the band indices n and n0 take the value of 1 for the conduction band and the value of 1 for the valence band. Therefore, the carrier energy Ek of state k in band n is given by Ek ¼ nℏv F jkj, and Ek0 of state k0 in band n0 is given by Ek0 ¼ n0 ℏv F jk0 j. The sign of n0 n represents the type of scattering: n0 n ¼ 1 for intraband scattering, and n0 n ¼ 1 for interband scattering.

34

Electronic Properties

Because an interband scattering event results in a change of energy for the carrier involved in the process, elastic scattering is necessarily intraband scattering, whereas inelastic scattering can be interband or intraband. Thus, for elastic scattering, n0 ¼ n and Ek0 ¼ Ek so that (2.21) is reduced to X 1 ¼ Sðk; k0 Þð1  cos θÞ; τðEÞ k0

ð2:22Þ

where the summation over band index n0 is dropped because n0 ¼ n. As can be seen from the ð1  cos θÞ factor, the scattering rate τ1 has no contribution from forward scattering, for which θ ¼ 0. Elastic scattering of charge carriers in graphene clearly favors large-angle scattering events. However, the angular contribution to the scattering rate τ1 still has to account for the angular dependence of the transition rate Sðk; k0 Þ, which depends on the scattering mechanism, as discussed in the following subsections. Therefore, we cannot simply conclude from the ð1  cos θÞ factor in (2.22) without accounting for the angular dependence of Sðk; k0 Þ that τ1 has the largest contribution from backward scattering, for which θ ¼ π. Indeed, backward scattering is suppressed for intraband scattering because Sðk; k0 Þ∝ð1 þ cos θÞ for intraband scattering, as discussed below. From the above discussions, it is clear that τ is indeed the momentum relaxation time related to the charge carrier transport, which is characterized by the electric conductivity expressed in (2.17). For elastic scattering, the transition rate Sðk; k0 Þ can be found using Fermi’s golden rule: Sðk; k0 Þ ¼

2 2π  Hk0;k  δðEk0  Ek Þ; ℏ

ð2:23Þ

where Hk0;k is the matrix element given by Hk0;k ¼

1 A

ðð

ψ†k0 Us ðrÞψk dr;

ð2:24Þ

where A is the area of the graphene sample under consideration and Us ðrÞ is the scattering potential energy. The eigenstates ψk and ψk0 are given by (1.28) and (1.30), respectively, in which the plus or minus sign is determined by the signs of n and n0 for ψk and ψk0 , respectively. Here we limit ourselves to the calculation of intravalley scattering, and therefore the superscript K or K0 of the eigenstates is dropped for simpler expressions; the wave functions ψk and ψk0 are given by (1.28) for carriers near the K point, and by (1.30) for carriers near the K0 point. The intervalley scattering between two different Dirac points, K and K0 , are not considered in (2.24). Such intervalley scattering needs a large momentum transfer through the interaction with optical phonons, which is assumed to be inefficient due to the large phonon energy of 160–200 meV compared to the electron thermal energy of around 26 meV at room temperature.

2.3 Scattering of Carriers

35

For inelastic scattering, a carrier is scattered by emitting or absorbing a quasiparticle of energy ℏω; therefore, instead of (2.23), the transition rate Sðk; k0 Þ for inelastic scattering is Sðk; k0 Þ ¼

2 2π  Hk0;k  δðEk0 Ek  ℏωÞ; ℏ

ð2:25Þ

where Hk0;k is given by (2.24), the plus sign in the delta function is taken for the emission process with Ek0 ¼ Ek  ℏω, and the minus sign is taken for the absorption process with Ek0 ¼ Ek þ ℏω. The form of Us ðrÞ in (2.24) that determines Hk0;k depends on the scattering mechanisms, as discussed in the remainder of this section.

2.3.1

Charged-Impurity Scattering As in semiconductors, impurity scattering is considered one of the most important scattering mechanisms for carrier transport in graphene. Consider a point impurity scattering center that has unit charge e and is located at z0 below a graphene sheet, as shown in Figure 2.2. The substrate is assumed to have an electric permittivity of ϵ s , and above the graphene sheet is the free space of a permittivity ϵ 0. The scattering potential energy between an electron and a charged impurity of the same charge is given by the Coulomb potential as Us ðrÞ ¼

e2 e2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Us ðrÞ; 4πϵ z02 þ jrj2 4πϵ z20 þ r2

ð2:26Þ

where ϵ is the average permittivity that includes the screening effect. If the screening effect is ignored, ϵ is approximately ϵ avg ¼ ðϵ 0 þ ϵ s Þ=2, which is the arithmetic mean of the permittivity that accounts for the image charge located in the free space above the graphene sheet. This concept is also illustrated in Figure 2.2. In general, (2.26) should be represented by a 2  2 matrix because ψk has two components due to the existence of two sublattices in graphene. However, because of

Image charge

z0 Graphene

r z0

Charged impurity Figure 2.2 Configuration for the calculation of charged-impurity scattering.

0

s

36

Electronic Properties

the long-range nature of the Coulomb potential, the scattering potential is not expected to change rapidly within the unit cell; atoms of the two graphene sublattices A and B see approximately the same potential given by (2.26). Therefore, the mixing of their wave functions can be ignored. The scattering matrix is calculated from (2.24) [2]:   e2 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eik  r dr neiθk 4πϵ z20 þ jrj2 ð ð  1 ∞ 2π e2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi eiqr cos θ 1 þ n0 neiðθk0 θk Þ rdθdr ¼ 2A 0 0 4πϵ z2 þ r2 0 ð ∞ J ðqrÞ e2  0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi rdr 1 þ n0 neiðθk0 θk Þ ¼ 4Aϵ 0 z20 þ r2 e2  1 þ n0 neiðθk0 θk Þ eqz0 ; ¼ 4Aϵq

Hk0;k ¼

1 2A

ðð

0

eik r



1 n0 eiθk0

†

ð2:27Þ

where J0 is the Bessel function of the first kind, and q is the scattering wave number defined as ℏq ¼ jℏqj ¼ jℏk0  ℏkj:

ð2:28Þ

Here, the momentum difference ℏq of the charge carrier incurred by the scattering event is transferred to the charged impurity. In the following subsections on other scattering mechanisms, this momentum difference can also be transferred to other quasiparticles such as phonons, and thus we should not associate the variable q with the wave number of any specific quasiparticle, but rather regard q simply as the scattering wave number, defined through the momentum difference ℏq given by (2.28) of the carrier between the two states ψk and ψk0 . Charged-impurity scattering is elastic because a charged-impurity scattering center does not change the energy of a carrier but only deflects the trajectory of the carrier. Therefore, charged-impurity scattering is intraband scattering with n0 ¼ n so that n0 n ¼ 1. Plugging (2.27) into (2.23), we obtain  2 2 2π e Sðk; k Þ ¼ e2qz0 FδðEk0  Ek Þ; ℏ 2Aϵq 0

ð2:29Þ

where F ¼ ð1 þ cos θÞ=2 is called the form factor, and θ ¼ θk0  θk is the angle between k and k0 . Because F ¼ 0 for θ ¼ π, we find that backward scattering is suppressed, which is peculiar to graphene due to its sublattice symmetry. Finally, we consider randomly distributed impurity scattering centers that have a density of e ni. e Multiplying (2.29) with the number of impurities, n i A, we obtain Sðk; k0 Þ ¼

2πe ni 2 U ðqÞFδðEk0  Ek Þ; ℏA s

ð2:30Þ

2.3 Scattering of Carriers

37

where Us ðqÞ ¼ ðe2 =2ϵqÞ expðqz0 Þ is the Fourier transform of the scattering potential energy Us ðrÞ given in (2.26). Using (2.30), we obtain from (2.22) the charged-impurity scattering rate: 2ðπ e 1 niE ¼ ð1  cos2 θÞUs2 ðqÞdθ; τimp ðEÞ 4πℏ3 v2F

ð2:31Þ

0

where q ¼ k0  k and q ¼ 2k sinðθ=2Þ because k 0 ¼ k for elastic scattering. Because of the ð1  cos2 θÞ term in the integral, thepscattering rate is mostly determined by 90° ffiffiffi scattering with θ ¼ π=2 or 3π=2 for q ¼ 2k. To account for the screening effect, the screening wave number qs is introduced, and ϵ can be expressed as ϵ ¼ ϵ avg ð1 þ qs =qÞ. When q ≫ qs the screening effect can be ignored, and ϵ → ϵ avg . For jμj ≫ kB T and q ≤ 2kF , qs is approximately the Thomas– Fermi screening wave number given by qs ¼

e2 e ρ ðμÞ; 2ϵ avg

ð2:32Þ

where e ρ ðμÞ is the electron or hole density of states given by (1.34) or (1.35), respectively, depending on the sign of μ. As most of the scattering happens within the region q ≤ 2kF in the momentum space, and the condition jμj ≫ kB T is generally true for the values of μ measured in experiments for graphene at room temperature, we shall use the Thomas– Fermi screening wave number in this chapter. A detailed discussion on ϵ and the Thomas–Fermi screening wave number is presented in Chapter 4. To observe the screening effect, we drop the exp ðqz0 Þ term and ignore screening by setting ϵ ¼ ϵ avg to obtain from (2.31) the analytical expression for the impurity scattering rate:  2 2 e 1 e 1 ni ; ¼ τimp ðEÞ 4ℏ 2ϵ avg E

ð2:33Þ

where e n i is the density of randomly distributed charged impurities measured in the unit 2 of m , often quoted in cm2 . The numerical solution obtained from (2.31) for the impurity scattering rate considering screening and the analytical solution obtained from (2.33) ignoring screening are plotted in Figure 2.3. The physical parameters used for these calculations are ϵ s ¼ 12ϵ 0 , ni ¼ 4  1011 cm2 , and EF ¼ 100 meV. Two different values of the distance z0 between the graphene sheet and the impurity scattering center are considered for the numerical solution: z0 ¼ 0 nm (solid curve) and z0 ¼ 0:5 nm (dashed curve). As can be seen in Figure 2.3, the analytical solution given in (2.33) (dotted curve) that is obtained by ignoring screening greatly overestimates the scattering rate by about an order of magnitude greater than the numerical solution obtained by accounting for the screening effect. In particular, the scattering rate given in (2.31) obtained by ignoring screening diverges when the carrier energy E → 0,

38

Electronic Properties

Scattering rate,

1 imp

1

( ps )

104

102

100

10

2

0

1

2

3 Carrier energy, E EF

4

5

Figure 2.3 Impurity scattering rate calculated from (2.33) by ignoring the screening effect, shown as the dotted curve, and that calculated from (2.31) by accounting for the screening effect, shown as the solid curve for z0 ¼ 0 nm and as the dashed curve for z0 ¼ 0:5 nm. Other parameters used are ϵs ¼ 12ϵ0 , e n i ¼ 4  1011 cm2 , and EF ¼ 100 meV.

which is not physical. As the energy E of the carrier increases, the scattering wave number q also increases because q ¼ 2k sinðθ=2Þ ¼ 2ðE=ℏv F Þ sinðθ=2Þ; for q ≫ qs, the analytical solution obtained by ignoring screening gradually approaches the numerical solution that accounts for screening as the screening effect diminishes. It can also be seen that a larger distance z0 leads to a smaller scattering rate due to a weaker Coulomb potential as the impurity is located further away from the graphene sheet.

2.3.2

Defect Scattering A point defect in the graphene lattice may generate a highly localized scattering potential that can be described by a delta function as Us ðrÞ ¼ V0 δðrÞ;

ð2:34Þ

where δðrÞ has the unit of m2 because graphene has a 2D lattice structure and V0 has the unit of J m2 . By following the same procedure as described above to obtain the impurity scattering rate, the transition rate for defect scattering is found as Sðk; k0 Þ ¼

2πnd 2 V FδðEk0  Ek Þ; ℏA 0

ð2:35Þ

where the form factor F ¼ ð1 þ cos θÞ=2. The defect scattering rate, i.e., the short-range scattering rate, is found from (2.35) as 1 nd V 2 ¼ 3 02 E; τs ðEÞ 4ℏ vF

ð2:36Þ

2.3 Scattering of Carriers

39

where nd is the density of defects. The linear dependence of the short-range defect scattering rate on the carrier energy seen in (2.36) is due to the fact that the density of states linearly increases with the carrier energy, as shown in (1.34) and (1.35). As the density of states increases, the number of available states for the carriers to be scattered into also increases, thus increasing the scattering rate. The relation in (2.36) for the defect scattering rate has been widely used in the literature [3–5] and in the fitting of experiment data [6]. It is directly derived from Fermi’s golden rule expressed in (2.23). The underlying Fermi’s golden rule given in (2.23) with the matrix elements given in (2.24) is essentially the first-order Born approximation, which assumes that the scattered field is a plane wave. However, given the highly localized nature of the defect potential expressed in (2.34), the scattered field within the range of this potential must be significantly distorted from a plane wave. It is argued that instead of the first-order Born approximation, it is necessary to employ a self-consistent Born approximation (SCBA) method that accounts for multiple scattering events [7]. The results from the SCBA show a very different behavior from that expressed in (2.36) from the first-order Born approximation [8]. However, the SCBA approach itself is not free from criticism either; it is argued that the SCBA approach, although suitable for weakly disordered metals and superconductors, is not justifiable for the Dirac Hamiltonian of graphene [9,10]. In short, so far there is still debate on the form of the defect scattering rate for graphene; ongoing development is expected in the context of scattering theory. For defects such as vacancies, adatoms, and cracks, the scattering potential cannot be described by the localized delta function potential of (2.34). The scattering potential of such a defect is modeled as a circular potential well that has an infinite magnitude and a radius of Rd , where R2d is of the same order of magnitude as the area Ap of the unit cell given in (1.4) in Chapter 1. The introduced boundaries of the defects result in zeroenergy bound states, also known as midgap states, at the Dirac point. As discussed above regarding the limitation of the first-order Born approximation, the nature of the strong potential and the resultant midgap states cannot be captured by the plane-wave assumption either. A more sophisticated method is needed, which is beyond the scope of this book. The result is given directly here [3]: 1 π2 ℏv2Fe nd ¼ ; τðEÞ Eðln kRd Þ2

ð2:37Þ

where e n d is the density of defects.

2.3.3

Acoustic Phonon Scattering The momentum change ℏq ¼ jℏqj ¼ jℏk0  ℏkj given in (2.28) for a scattered carrier can be absorbed or provided by an acoustic phonon of graphene through acoustic phonon scattering. In this situation, ℏq represents the momentum of the acoustic phonon mode that is involved in the scattering process. Propagating acoustic phonons create an oscillating potential that can be described as [11]

40

Electronic Properties

Us ðrÞ ¼ Kq Aq eiðq  rωq tÞ ;

ð2:38Þ

where ωq ¼ v a q is the acoustic oscillation frequency with a speed of v a ≈ 2  104 m s1 for longitudinal acoustic phonons, Kq quantifies the deformation potential associated with the displacement of carbon atoms, and Aq is the amplitude of the acoustic oscillation. Here, we only consider longitudinal acoustic phonons because scattering from transverse acoustic phonons is much weaker [12]. The magnitudes of Kq and Aq depend on q and ωq , respectively: jKq j2 ¼ D2A q2 ; jAq j2 ¼

ℏ Nðωq Þ; 2e ρ Aωq

ð2:39Þ ð2:40Þ

where DA ¼ 18 eV is taken for the magnitude of the deformation potential energy [12], e ρ ¼ 7:6  107 kg m2 is the density of graphene, A is the area of the graphene sample, and Nðωq Þ ¼

1 kB T ≈ eℏωq =kB T  1 ℏωq

ð2:41Þ

is the number of phonons given by the Bose–Einstein statistics, for which the approximation is valid when ℏωq ≪ kB T. The condition ℏωq ≪ kB T is generally true for acoustic phonons at room temperature. Using (2.38) for (2.24) to find Hk0;k for (2.25), we obtain Sðk; k0 Þ ¼

i 2π 2 2 h Kq Aq F δðEk0  Ek þ ℏωq Þ þ δðEk0  Ek  ℏωq Þ ; ℏ

ð2:42Þ

where F ¼ ð1 þ cos θÞ=2, the δðEk0  Ek þ ℏωq Þ term accounts for scattering by phonon emission, and the δðEk0  Ek  ℏωq Þ term accounts for scattering by phonon absorption. Because v F ≫ va and therefore Ek ; Ek0 ≫ ℏωq when k, k 0 , and q are of the same order of magnitude, we find the acoustic phonon scattering to be approximately elastic so that Ek0 ≈ Ek and the approximation δðEk0  Ek þ ℏωq Þ ≈ δðEk0  Ek  ℏωq Þ ≈ δðEk0  Ek Þ is valid. Using (2.42), we obtain from (2.22) the acoustic phonon scattering rate: 1 D2 kB T ¼ A3 2 E: τac ðEÞ 4e ρ ℏ vF v 2a

ð2:43Þ

We see that the rate of acoustic phonon scattering increases with both temperature and carrier energy. The linear temperature dependence is due to the linear increase of the phonon number with temperature, as seen in (2.41), and the linear energy dependence is due to the linear increase of the density of states with carrier energy, as in the case of defect scattering seen in (2.36).

2.3 Scattering of Carriers

41

Note that (2.43) is entirely constructed on the premise that kB T ≫ ℏωq . In this regime, it is valid to take the classical picture of (2.38), which defines the total energy Eac (kinetic energy plus potential energy) of acoustic oscillation to be proportional to the square of the amplitude of oscillation as given by (2.40). At a relatively low temperature when kB T ≫ ℏωq is not valid, this classical picture fails to agree with the quantum picture, which requires the acoustic oscillation energy to be quantized as Eac ¼ ðN þ 1=2Þℏωq . To account for the quantum effect, the quantum mechanical approach has to be adopted. The resulting scattering transition rate has a form similar to the classical one given in (2.42) except that it accounts for the difference in the phonon number between emission and absorption of a phonon [12]: Sðk; k0 Þ ¼

  h    i 2π 2 ℏ Kq F N þ δ Ek0  Ek þ ℏωq þ N  δ Ek0  Ek  ℏωq ; ℏ 2ρAωq ð2:44Þ

where F ¼ ð1 þ cos θÞ=2 and N ¼

1 1 1 þ  : eℏωq =kB T  1 2 2

ð2:45Þ

At a sufficiently high temperature such that kB T ≫ ℏωq , we find that N ≈ N ≈ kB T=ðℏωq Þ, as given in (2.45), so that (2.44) reduces to (2.42), as expected. At a really low temperature such that kB T ≪ ℏωq , it can be shown from (2.44) that instead of the linear temperature dependence given by (2.43), the scattering rate without screening effects has a T 4 dependence and that with screening by the carriers themselves has a T 6 dependence [12]. In the case that kB T ≈ ℏωq , the temperature dependence of the scattering rate becomes complicated, transitioning from T 4 dependence for kB T ≪ ℏωq to T dependence for kB T ≫ ℏωq. 

2.3.4

Optical Phonon Scattering So far the discussion has been limited to elastic scattering for which Ek0 ≈ Ek . For optical phonon scattering, the relation Ek0 ≈ Ek is not valid because the optical phonon energy is comparable to the carrier energy. Therefore, optical phonon scattering is usually considered as inelastic scattering. There are two types of optical phonon scattering in graphene: intrinsic optical phonon scattering and extrinsic optical phonon scattering. Intrinsic optical phonon scattering arises from the coupling between charge carriers and optical phonon modes of graphene. Extrinsic optical phonon scattering is due to the coupling between carriers in graphene and surface optical phonons of the substrate on which the graphene sheet is deposited. In the following, we first discuss the extrinsic optical phonon scattering.

42

Electronic Properties

Extrinsic Optical Phonon Scattering The permittivity of a polar substrate in the frequency range around an optical phonon frequency can be expressed as [13] ϵ s ðωÞ ¼ ϵ high þ ðϵ high  ϵ low Þ

ω2TO ; ω2  ω2TO

ð2:46Þ

where ωTO is the transverse optical (TO) phonon frequency of the substrate, and ϵ high and ϵ low are respectively the high-frequency (ω ≫ ωTO ) and low-frequency (ω ≪ ωTO ) dielectric permittivities of the polar substrate. A substrate that has a permittivity described by (2.46) is called a polar substrate, especially when ωTO is low, because it can be easily polarized to form strong dipole groups. Examples of polar substrates are SiO2, ZrO2, HfO2, and Al2O3. Note that the polar substrates as defined above are not necessarily polar crystals that are defined by the polar crystallographic point groups. It is well known that photons can couple with TO phonons, resulting in phonon polaritons in the bulk. The polariton dispersion is given by βc ω ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ϵ s ðωÞ=ϵ 0

ð2:47Þ

where ϵ s ðωÞ is given by (2.46). The relation between ω and β for the polariton dispersion found from (2.47) is plotted as thin solid curves in Figure 2.4. The light line pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω ¼ kc= ϵ high =ϵ 0 for the substrate is also plotted as the thin dashed line using (2.47), with ϵ s ðω → ∞Þ ¼ ϵ high . As can be seen, one branch of the polariton dispersion curves starts from ω ¼ 0 and asymptotically approaches ωTO (dotted line), another pand ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi grows from the longitudinal optical (LO) phonon frequency of ωLO ¼ ϵ low =ϵ high ωTO and approaches the light line as ω increases. When an interface is introduced to divide the space into two semi-infinite dielectrics of permittivities ϵ s and ϵ 0 , respectively, an additional surface phonon mode is supported, which can also couple with photons, resulting in a surface phonon polariton mode with its electric field confined on the interface. This surface polariton mode is often referred to as a surface optical (SO) phonon mode in the literature [14]. The eigenvalue equation of the SO mode can be found by solving Maxwell’s equations with boundary conditions imposed on the interface (see Section 6.1.2 for the derivation): ϵ0 ϵs þ ¼ 0; γ0 γs

ð2:48Þ

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 γ0 ¼ β2  2 ; c

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ s ω2 : γs ¼ β2  ϵ 0 c2

ð2:49Þ

By inserting (2.49) and (2.46) in (2.48), an explicit form of ω as a function of β can be found [14]:

2.3 Scattering of Carriers

=

43

kc 0 h

Energy,

c =k

hig

LO SO TO

= c

c

= kc =

TO

s

( )

0

c

Wave number, k, Figure 2.4 Dispersion curves of bulk phonon polaritons (thin solid curves) obtained from (2.47) and the surface phonon polariton (thick solid curve) obtained from (2.50) for a polar substrate. The light line for the free space is also plotted as the thick dashed line, and that for the substrate as the thin dashed line with ϵs ðω → ∞Þ ¼ ϵhigh .

1 1 ω2 ¼ ðω02 þ ω2LO Þ  2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðω02 þ ω2LO Þ2  4ω02 ω2SO ;

ð2:50Þ

where ω02 ¼

ϵ high þ ϵ 0 2 2 c β ; ϵ high

ω2SO ¼

ϵ low þ ϵ 0 2 ω : ϵ high þ ϵ 0 TO

ð2:51Þ

The dispersion relation of the SO mode of the substrate–air interface given by (2.50) is plotted in Figure 2.4 as a thick solid curve. The light line of air, ω ¼ kc, is plotted as a thick dotted line. They intersect at the cutoff point of the SO mode, where βc ¼ kc ¼ ωTO =c. At a frequency below the TO phonon frequency such that ω < ωTO , an SO mode has a wave number below this critical wave number, β < βc , and it is smaller than the wave number k of the free space, i.e., β < k ¼ ω=c, as can be seen from Figure 2.4. In this case, γ0 becomes purely imaginary so that the SO mode is not a true surface mode but radiates into the free space. By contrast, one can find that the frequency of the SO mode has to be higher than ωTO , i.e., ω > ωTO , so that ϵ s < 0 for both γs and γ0 given in (2.48) to have real values to confine the SO mode at the interface. As β passes through the critical value of βc ¼ ωTO =c, the frequency of the SO mode quickly approaches a constant value of ωSO that is slightly higher than ωTO . For a polar substrate that has N TO modes, there are N corresponding SO modes. When a graphene sheet is placed near the polar substrate, SO phonon scattering can be induced by the coupling of these SO modes with the carriers in graphene.

44

Electronic Properties

r

Graphene z0

0 s

Figure 2.5 Geometry for the calculation of surface optical phonon scattering.

The electrons of graphene can scatter with SO phonons by phonon emission or absorption. For most substrates, the energy of a TO phonon falls in the range between 10 and 200 meV, which gives a value of βc in the range between 5  104 and 1  106 m1 . For EF ¼ 100 meV as an example, the Fermi wave number of the electrons in graphene is kF ¼ 1:5  108 m1 , which is much larger than βc . Therefore, most of the times the scattering event involves an SO phonon that has a wave number larger than βc , and the frequency of the SO phonon mode is approximately a constant, as shown in Figure 2.4. This feature makes the calculation of the SO phonon scattering rate much easier because we do not have to consider the β dependence of the SO phonon mode; according to Figure 2.4, for the SO phonon energy ℏωSO, we can always find a value for β that satisfies jℏk0  ℏkj ¼ ℏβ as long as β is larger than βc . Once the condition jEk0  Ek j ¼ ℏωSO for energy conservation is satisfied, the condition jℏk0  ℏkj ¼ ℏq for momentum conservation is automatically satisfied by setting ℏq ¼ ℏβ. In the following, we calculate the SO phonon scattering rate using this assumption, namely, β > βc . Consider the geometry shown in Figure 2.5. A semi-infinite substrate of a permittivity ϵ s is located at a distance z0 below a graphene sheet. The rest of the space is the free space of a permittivity ϵ 0. The transition rate due to SO phonon scattering is [15] Sðk; k0 Þ ¼

 2qz0  2π X Fv2 Nv e2 e 1 þ n0 n cos θ δðEk0  Ek  ℏωv Þ; 2 ℏ v; A ð1 þ qs =qÞ 2 q

ð2:52Þ

where for the  sign, the plus sign has to be chosen for phonon emission in the scattering process, and the minus sign chosen for phonon absorption; v runs through all SO phonon modes with ωv ¼ ωSOv ; Nv is given by (2.45) with ωq replaced by ωv ; qs is given by (2.32) with ϵ avg ¼ ðϵ 0 þ ϵ low Þ=2 for static screening and ϵ low ¼ ϵ s ðω → 0Þ; and the factor Fν is the electron–phonon coupling parameter of the form:   ℏωv 1 1 Fv2 ¼  : ð2:53Þ 2 ϵ high þ ϵ 0 ϵ low þ ϵ 0 In the case of two dominant SO phonon modes, it is useful to introduce an intermediate permittivity ϵ int that describes the dielectric permittivity for some intermediate frequency between the two transverse optical phonon frequencies (ωTO1 < ωint < ωTO2 ). Thus, from (2.53), F12 is obtained by replacing ϵ high with ϵ int ,

2.3 Scattering of Carriers

45

Table 2.1 Physical parameters of selected materials [16–20]. Quantity (unit)

SiO2

Al2O3

ZrO2

HfO2

SiC

h-BN

ϵlow =ϵ0 ϵint =ϵ0 a ϵhigh =ϵ0 a ℏωTO1 ðmeVÞb ℏωTO2 ðmeVÞb ℏω1 ðmeVÞc ℏω2 ðmeVÞc F12 ðmeV=ϵ0 Þd F22 ðmeV=ϵ0 Þd

3.90 3.36 2.40 55.6 138.10 58.94 156.39 0.74 5.06

12.53 7.27 3.20 48.18 71.41 61.63 100.20 1.45 5.87

24.0 7.75 4.00 16.67 57.70 28.18 76.33 1.05 3.27

22.0 6.58 5.03 12.40 48.35 21.60 54.21 0.96 0.92

9.7 – 6.5 98.82 – 118.03 – 2.35 –

5.09 4.58 4.10 97.08 187.22 101.42 195.83 0.76 1.65

a

ϵlow , ϵ int , and ϵhigh : low-, intermediate-, and high-frequency dielectric permittivities, respectively. ℏωTOν : transverse optical phonon energy of mode v. c ℏων : surface optical phonon energy of mode v with ωv ¼ ωSOv . d 2 Fv : coupling parameter for the surface optical phonon mode v. a

b

and F22 is obtained by replacing ϵ low with ϵ int . These physical parameters can be found in Table 2.1. By plugging (2.52) into (2.21), we obtain the SO phonon scattering rate: 1 So ðEÞ X X ¼ ; τðEÞ 1 þ Se;v ðEÞτðE þ ℏωv Þ þ Sa;v ðEÞτðE  ℏωv Þ v

ð2:54Þ

v

where So ðEÞ ¼

e2

X

4πℏ3 v 2F

v;

Fv2 Nv

2ðπ   1f0 ðE  ℏωv Þ 1 þ s cos θ e2q z0 ðE  ℏωv Þ n0 n dθ; 1  f0 ðEÞ q ð1 þ qs =q Þ2 0

ð2:55Þ

Se;v ðEÞ ¼

e2

Fv2 Nv 4πℏ3 v2F

2ðπ   1  f0 ðE þ ℏωv Þ 1 þ s cos θ e2q z0 ðE þ ℏωv Þ cos θdθ; 1  f0 ðEÞ q ð1 þ qs =q Þ2 0

ð2:56Þ Sa;v ðEÞ ¼

e2

Fv2 Nvþ 4πℏ3 v2F

2ðπ   1  f0 ðE  ℏωv Þ 1 þ s cos θ e2qþ z0 ðE  ℏωv Þ cos θdθ; 1  f0 ðEÞ qþ ð1 þ qs =qþ Þ2 0

ð2:57Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and q ¼ ℏ2 ω2ν þ 2Eðℏωv  EÞðn0 n cos θ  1Þ ℏv F is the scattering wave number, which is determined from the relation Ek ¼ Ek0  ℏωv required by energy conservation and the relation k ¼ k0  q required by momentum conservation. For hole scattering in the valence band, the scattering rate is calculated by changing the chemical potential

=

46

Electronic Properties

(E +

v

) Se,v

So

(E)

Sa,v

(E

v

) Dirac point

k

q k

k q

Figure 2.6 Illustration of out-scattering (gray arrows) and in-scattering (black arrows) processes for

a state of carrier wave vector k and energy E. Also shown are the relations among the carrier wave vector k (dashed arrow) before scattering, the carrier wave vector k0 (solid arrows) after scattering, and the scattering phonon wave vector q (dotted arrows). Two pairs of wave vectors k0 and q are drawn for the phonon emission scattering process k ¼ k0 þ q. Sa;v , Se;v , and So link the relaxation times of different carrier energies. Reprinted from I. T. Lin and J. M. Liu, “Surface polar optical phonon scattering of carriers in graphene on various substrates,” Applied Physics Letters, Vol. 103, 081606 (2013), with the permission of AIP Publishing.

from μ to μ without the sign change for the carrier energy, i.e., τðEÞjμ ¼ τðEÞjμ . The relations among So ðEÞ, Sa;v ðEÞ, and Se;v ðEÞ are shown in Figure 2.6. The outscattering rate So ðEÞ accounts for transitions from a state of energy E to either a state of higher energy E þ ℏωv by absorbing an SO phonon, or a state of lower energy

2.3 Scattering of Carriers

47

E  ℏωv by emitting an SO phonon. By contrast, Sa;v ðEÞ and Se;v ðEÞ are the in-scattering rates from a lower-energy state and a higher-energy state, respectively, to a state of energy E. As carriers are scattered out from the states of energy E, τ1 ðEÞ gradually decreases due to Pauli blocking as the states of energies E  ℏωv are getting populated, whereas τ1 ðE  ℏωv Þ gradually increases as more and more states of energy E become empty for carriers at E  ℏωv to be scattered into. Eventually, carriers in states of different energies reach a balance to establish the equilibrium distribution of carrier population as a function of E. Then, the energy-dependent scattering rate τ1 ðEÞ can be determined. It is clear that (2.54) has no close-form solution for τðEÞ because τðEÞ cannot be determined unless τðE  ℏωv Þ is known, which is dependent on τðEÞ and also on τðE  2ℏωv Þ. Therefore, we adopt the iteration method first formulated by Rode [21]. We first set Sa;v ðEÞ and Se;v ðEÞ to zero for every v so that the in-scattering processes are ignored; then, the zeroth-order solution τð0Þ ¼ So1 can be obtained. The nth-order solution τðnÞ is obtained iteratively by replacing τ on the right-hand side of (2.54) with the ðn  1Þth-order solution τðn1Þ : 1þ τðnÞ ðEÞ ¼

X

Se;v ðEÞτðn1Þ ðE þ ℏωv Þ þ

v

X v

So ðEÞ

Sa;v ðEÞτðn1Þ ðE  ℏωv Þ :

ð2:58Þ

The iteration process continues until the difference between the results of two successive iterations is below a certain threshold value. The SO phonon scattering rate for graphene on SiO2 and on HfO2 is plotted in Figure 2.7 as a function of carrier energy and Fermi energy. The scattering rate is calculated iteratively using (2.54) for z0 ¼ 0:335 nm, which is the thickness of monolayer graphene. The chemical potential μ is a function of temperature, which is determined by the conservation of the total electron density and found by solving (1.38). At a low temperature, the value of μ is approximately equal to EF , which is set at EF ¼ 100 meV for our numerical calculation to obtain the data plotted in Figure 2.7(a). As can be seen, due to the limited available states below the chemical potential μ for electrons to be scattered into, the scattering rate is strongly suppressed for carrier energies around μ. This dip of the scattering rate at μ translates into multiple dips at the energy locations of μ  n1 ℏω1  n2 ℏω2 , where n1 and n2 are integers, through the coupling among scattering at these energy locations described by (2.54). Also note that the location of the dip for the minimum scattering rate shifts with μ, which shifts toward the Dirac point as the temperature increases. The SO scattering rate for graphene on HfO2 is an order of magnitude larger than that for graphene on SiO2 due to the low SO phonon energies of HfO2. While the large ϵ low of a high-permittivity material, commonly known as a high-κ material, can effectively screen the Coulomb potential to reduce the charged-impurity scattering rate, the ionic nature of the highly polarizable bonds of such material implies low TO phonon energies and thus low SO phonon energies. Therefore, there is a trade-off between the charged-impurity scattering rate and the SO phonon scattering rate: Graphene on a substrate of a large ϵ low usually enjoys

48

Electronic Properties

(a)

(b) 101

8

Scattering rate,

10

SiO 2

10

2

10

3

1 1

( ps )

1

7 6.5

Scattering rate,

HfO 2

100

1

1

( ps )

7.5 HfO 2

0.4 SiO 2

0.2 0

−200

−100

0

100

200

Carrier energy, E ( meV )

50

100

150

200

Fermi energy, EF ( meV )

Figure 2.7 Surface optical phonon scattering rate for graphene on SiO2 and on HfO2 (a) as a function of carrier energy and (b) as a function of Fermi energy for the carrier energy E ¼ EF. For these plots, the separation between graphene and the substrate is assumed to be z0 ¼ 0:335 nm; other physical parameters can be found in Table 2.1. In (a), the lower three curves are for the SiO2 substrate, and the upper three curves are for the HfO2 substrate. For these curves, the Fermi energy is taken to be EF ¼ 100 meV, and the temperature is set at 100 K (solid), 200 K (dashed), and 300 K (dotted), respectively. In (b), T ¼ 300 K is assumed.

a low charged-impurity scattering rate but suffers from a high SO phonon scattering rate. By contrast, a relatively low SO phonon scattering rate is possible for graphene on a substrate of a small ϵ low , which, however, usually leads to a high charged-impurity scattering rate. As is the case for charged-impurity scattering, SO phonon scattering is also a function of Fermi energy. This dependence on Fermi energy is mostly due to the screening number qs in (2.52). As the Fermi energy increases, the density of carriers also increases, resulting in a reduced SO phonon scattering rate due to increased screening. The SO phonon scattering rate for the carrier energy E ¼ EF for graphene on SiO2 and on HfO2 as a function of Fermi energy is shown in Figure 2.7(b), which clearly shows that the SO phonon scattering rate decreases with increasing Fermi energy.

Intrinsic Optical Phonon Scattering Contrary to the coherent oscillations of acoustic phonons, intrinsic optical phonons in graphene arise from the out-of-phase vibrations of carbon atoms in the lattice. For a given phonon wave vector q, there are three optical phonon modes characterized by orthogonal directions of vibration: two TO phonon modes polarized perpendicular to q and one LO phonon mode polarized along q. Because graphene has a 2D structure, one of the two TO modes is characterized by out-of-plane vibration, which is usually called the ZO mode to be distinguished from the in-plane TO mode. With further calculation, one finds that the ZO mode can hardly couple with in-plane conduction electrons [22].

2.3 Scattering of Carriers

49

By comparison, degenerate TO and LO modes at the Γ point (Γ-TO, Γ-LO) and the TO mode at the K point (K-TO) can couple with conduction electrons. Γ-TO and Γ-LO phonons scatter electrons within the same valley due to the small wave number q of these phonon modes, while a K-TO phonon can scatter an electron between different valleys (K and K0 valleys). However, at room temperature their contributions to the scattering rate are limited because these modes are energetic, in the range of 160–200 meV, so that their phonon numbers, given by (2.45), are small. For this reason, in the usual transport experiment, intrinsic optical phonon scattering is not considered. When the temperature is high or the driving electric force is strong, intrinsic optical phonon scattering might become important and cannot be ignored.

2.3.5

Total Scattering Rate To calculate the total scattering rate due to both inelastic scattering, which is mainly SO phonon scattering, and elastic scattering, which includes charged-impurity scattering, defect scattering, and acoustic phonon scattering, we again resort to the original equation (2.21), but with the scattering transition rate given by Sðk; k0 Þ ¼ Sðk; k0 Þelastic þ Sðk; k0 Þinelastic ;

ð2:59Þ

where Sðk; k0 Þinelastic is given by (2.52) and Sðk; k0 Þelastic is the sum of the elastic scattering transition rates given by (2.30), (2.35), and (2.42). By using (2.59) for (2.21), we obtain So ðEÞ þ τ1 ðEÞ 1 elastic X X ¼ ; τðEÞ 1 þ Se;v ðEÞτðE þ ℏωv Þ þ Sa;v ðEÞτðE  ℏωv Þ v

ð2:60Þ

v

which is similar to (2.54) with an extra term for the total elastic scattering rate 1 1 1 τ1 elastic ðEÞ ¼ τ imp ðEÞ þ τ s ðEÞ þ τ ac ðEÞ:

ð2:61Þ

Again, (2.60) has to be solved iteratively due to the presence of inelastic scattering. Clearly, if we ignore inelastic scattering by setting Sðk; k0 Þinelastic ¼ 0, then So ðEÞ ¼ Sa;v ðEÞ ¼ Se;v ðEÞ ¼ 0 for every v and τ1 ðEÞ ¼ τ1 elastic ðEÞ. By iteratively solving (2.60), the total scattering rate τ 1 ðEF Þ, at the Fermi energy, is found and plotted in Figure 2.8 as a function of Fermi energy and temperature for graphene on SiO2 and on HfO2 for a graphene–substrate separation of z0 ¼ 0:335 nm and a charged-impurity density of e n i ¼ 4  1011 cm2 , which is of the same order of magnitude as the carrier density. Defect scattering is not considered for the moment. As can be seen in Figures 2.8(a) and (b), at 300 K the total scattering rate τ1 ðEF Þ at the Fermi energy for graphene on an SiO2 substrate is higher than that for graphene on an HfO2 substrate when the Fermi energy is low, but the situation is reversed when the Fermi energy is high. This is due to the fact that for the case of the SiO2 substrate, the scattering rate is mostly contributed by impurity scattering, which decreases rapidly with

50

Electronic Properties

(a)

(b)

SiO 2 20 1

(E ) (E )+ (E )

HfO 2 20

(E )

1

F

F

10

Scattering rate,

Scattering rate,

1 ac

( ps )

1 imp

F

15

1

15

1

1

( ps )

F

1 imp

5

10

5

0

0 50

150

(d)

SiO 2

150

200

HfO 2 10

1

1

8

Scattering rate,

1

8

1

Scattering rate,

100

Fermi energy, EF ( meV )

( ps )

8.2

50

200

Fermi energy, EF ( meV )

(c)

( ps )

100

7.8

7.6 100

200 Temperature, T (K)

300

6

4

2 100

200

300

Temperature, T (K)

Figure 2.8 Scattering rate for graphene (a) on SiO2 and (b) on HfO2 as a function of the Fermi energy at 300 K, and (c) on SiO2 and (d) on HfO2 as a function of temperature for EF ¼ 100 meV. n i ¼ 4  1011 cm2 are taken to obtain the curves for all figures. The values of z0 ¼ 0:335 nm and e The solid curves represent the total scattering rate τ1 ðEF Þ; the dashed curves and dotted curves 1 1 represent τ1 imp ðEF Þ þ τ ac ðEF Þ and τ imp ðEF Þ, respectively.

increasing Fermi energy and thus increasing screening, as can be seen from the dotted curve for τ 1 imp ðEF Þ, whereas for the case of the HfO2 substrate, the scattering rate is mostly contributed by SO phonon scattering, which does not strongly depend on the Fermi energy. The temperature dependence of each scattering rate for EF ¼ 100 meV can be seen in Figures 2.8(c) and (d) for graphene on SiO2 and on HfO2, respectively. The SO phonon scattering rate τ1 SO ðEF Þ increases with temperature nonlinearly at a low temperature when the thermal energy is comparable to the phonon energy, as can be seen from the solid curve in Figure 2.8(c). At a high temperature, the thermal energy is much

2.4 Surface Resistivity and Mobility

51

larger than the SO phonon energies; then τ 1 SO ðEF Þ linearly increases with temperature, as can be seen from the solid curve in Figure 2.8(d). In the temperature range considered in Figures 2.8(c) and (d), the condition kB T ≫ ℏωq is applicable to (2.41), and τ1 ac ðEF Þ linearly increases with temperature, as shown by the dashed curves. The charged-impurity scattering rate is not a function of temperature, as can be seen from (2.31) and from the dotted lines in Figures 2.8(c) and (d). However, note that at an even higher temperature such that the condition jEF j ≫ kB T is not valid, the Thomas–Fermi screening wave number is not applicable; then, τ1 imp ðEF Þ becomes a function of temperature because qs is no longer independent of temperature at such a high temperature.

2.4

Surface Resistivity and Mobility The surface conductivity of graphene on a substrate can be derived from (2.14) by following the procedure in (2.13): e σ¼

e2 πℏ2

ð∞ τðEÞ

 df0 df0 þ τðEÞ jμ → μ EdE; dE dE

ð2:62Þ

0

where the first term and the second term are respectively contributed by the carriers on the conduction band and on the valence band, the relaxation time τðEÞ is solved iteratively from (2.60), and df0 =dEjμ → μ is the same as df0 =dE except for changing the sign of μ. The surface resistivity e ρ¼e σ 1 calculated using (2.62) with the total scattering rate given in Figure 2.8 is plotted in Figure 2.9. The unit of the surface resistivity e ρ for the 2D graphene structure is Ω. Clearly, a higher scattering rate leads to a higher resistivity. Note that though the impurity scattering rate does not vary with temperature in the temperature range considered here, the resistivity due to impurity scattering is not temperature independent, as can be seen from the dotted curve in Figure 2.9(c). This temperature dependence is caused by the Fermi–Dirac distribution in the integral of (2.62). The solid curve that is numerically obtained from (2.62) and shown in Figure 2.9(c) for the surface resistivity of graphene on SiO2 is replotted in Figure 2.10 to be compared with the approximations given by (2.17) and (2.18) for the limit that μ≫ kB T. As can be seen, (2.17) is a better approximation than (2.18), which is not surprising because (2.18) is a further approximation deduced from (2.17). Both (2.17) and (2.18) converge to the exact result as the temperature approaches zero. Appreciable disparity appears between the better approximation of (2.17) and the exact surface resistivity at temperatures higher than the room temperature of 300 K. The surface conductivity of graphene is generally related to the electron density e n and hole density e p as e σ ¼ eðμee n þ μh e p Þ;

ð2:63Þ

Electronic Properties

(a)

1

HfO 2 3000

(E ) (E )+ (E ) (E ) F

2000 1500

1 imp

F

1 imp

F

1 ac

F

2500

( )

( )

2500

Resistivity,

(b)

SiO 2

3000

2000

Resistivity,

52

1500

1000

1000

500

500

0 50

100

150

0 50

200

Fermi energy, EF ( meV ) 750

150

200

Fermi energy, EF ( meV )

SiO 2

(c)

100

(d) 1000

HfO 2

( ) 700

Resistivity,

Resistivity,

( )

800

650 100

200 Temperature, T (K)

300

600 400 200 0 100

200 Temperature, T (K)

300

Figure 2.9 Surface resistivity of graphene (a) on SiO2 and (b) on HfO2 as a function of Fermi energy at 300 K, and (c) on SiO2 and (d) on HfO2 as a function of temperature for EF ¼ 100 meV. The solid curves, dashed curves, and dotted curves represent the surface resistivity associated with the scattering rates represented by the curves of the corresponding styles in Figure 2.8. Other physical parameters are the same as those used for Figure 2.8

where μe and μh are electron mobility and hole mobility, respectively. In the case of a positive Fermi energy so that e p ≈ 0, the electron mobility of graphene can be found as μe ¼

e σ 1 ¼ ; e ne e ρe ne

ð2:64Þ

whereas in the case of a negative Fermi energy so that e n ≈ 0, the hole mobility can be found as

2.4 Surface Resistivity and Mobility

53

850

800

Resistivity,

( )

Exact 750

(2.17)

700

650 100

(2.18)

200

300

400

Temperature, T (K) Figure 2.10 Surface resistivity of graphene on SiO2 calculated using (2.62) in solid curve, (2.17) in dashed curve, and (2.18) in dotted curve. The solid curve is the same as the solid curve in Figure 2.9(c), and the physical parameters are the same as those used for Figure 2.9(c).

μh ¼

e σ 1 ¼ : e pe e ρe pe

ð2:65Þ

Because of the symmetry of the conduction and valence bands with respect to the Dirac point, the electron and hole mobilities of a graphene sample are theoretically identical. In practice, however, the difference in the electron and hole scattering rates caused by impurities and defects can result in different electron and hole mobilities. Note that because e ρ is the 2D surface resistivity that has a unit of Ω and e n and e p are the 2D carrier densities that have a unit of m2 , the mobilities μe and μh of electrons and holes in graphene have the same unit of m2 V1 s1 as the mobility of carriers in a 3D material. The electron mobility μe of graphene on SiO2 at 300 K as a function of Fermi energy is calculated using the surface resistivity of graphene given by the solid curve for SiO2 in Figure 2.9(a). Although the surface resistivity decreases when the Fermi energy EF increases, the electron density e n increases with EF as given by (1.38). The net result is that the mobility is not as strong a function of EF as the surface resistivity is, as can be seen in Figure 2.11. Apparently, the lower the temperature T (resulting in a lower value of τ1 n i (resulting in a lower value of τ 1 ac ) or the lower the impurity density e imp ) is, the higher the mobility is, as can be seen by comparing the dashed and dotted curves with the solid curve in Figure 2.11. A question that easily comes to our mind is, how high can the mobility of graphene be? The most effective way to increase the mobility is to reduce the effects of all extrinsic scattering mechanisms. This can be done by increasing the separation d between the graphene sheet and the substrate. Eventually when d is large enough, the graphene sheet is “suspended” above the substrate. In this situation, both the impurity scattering rate τ1 imp and the surface phonon scattering rate τ1 SO are negligible; then, the intrinsic phonon scattering rate τ1 becomes the limiting factor for the mobility of suspended graphene. ac

Electronic Properties

15

T = 50 K, n1 = 4 1011 cm

10

T = 300 K, n1 = 4 1011 cm

2

2

(

Mobility, μ e 103 cm 2 V 1s

1

)

54

5 T = 300 K, n1 = 1.2 1012 cm 0 50

100

150

Fermi energy, EF ( meV )

2

200

Figure 2.11 Electron mobility of graphene on SiO2 at T ¼ 300 K and e n i ¼ 4  1011 cm2 (solid curve), T ¼ 50 K and e n i ¼ 4  1011 cm2 (dotted curve), and T ¼ 300 K and e n i ¼ 1:2  1012 cm2 (dashed curve). Other physical parameters are the same as those used for Figure 2.8.

Indeed, a mobility as high as 2:3  105 cm2 V1 s1 at an electron density of e n ¼ 2  1011 cm2 has been observed in suspended graphene [23]. However, this value is still far below the predicted mobility limited only by τ 1 ac . Clearly, besides 1 1 τ1 , τ , and τ , there are other scattering mechanisms that cause the reduction of ac SO imp mobility. Besides the possible defect scattering, scattering off of the carriers from the boundary edges of the graphene sample becomes important as the carrier mean free path of a high mobility reaches the micrometer order, which is usually the order of the domain size or the sample size of most graphene samples. Furthermore, without the tension provided by the supporting substrate, the suspended graphene membrane can support a new branch of phonon mode that vibrates out-of-plane, known as flexural phonons, i.e., vibrating microscopic corrugations of a graphene sheet. For suspended graphene, scattering of carriers by flexural phonons becomes the dominant phonon scattering mechanism that limits its intrinsic mobility [24,25].

2.5

Transport Experiments In this section we review some of the most important graphene transport experiments [26–28]. In an experiment, the control parameters are usually the temperature and the Fermi energy. The Fermi energy can be tuned by electrostatic carrier doping. One popular method is to apply a gate voltage Vg on the back of the substrate, as shown in Figure 2.12(a). The substrate is treated as a plane capacitor that has a capacitance of C ¼ ϵ s =ds per unit area, where ϵ s and ds are the permittivity and the thickness of the substrate, respectively. Therefore, the carrier density in graphene is e n¼

CVg e

ð2:66Þ

2.5 Transport Experiments

(a)

(b)

Graphene

Drain

Source

Vg

Vg Gate

SiO 2 Gate

55

PEO Drain

Source

Graphene

SiO 2

Figure 2.12 Schematics of two common configurations for graphene electronic devices with the gate voltage applied through (a) a bottom gate, or (b) a top solid polymer electrolyte gate.

according to elementary electrostatics. For an SiO2 substrate that has a thickness of ds ¼ 300 nm and a permittivity of ϵ s ≈ 4ϵ 0 , the unit-area capacitance C is about 1:18  108 F cm2 and the carrier density as given by (2.66) is e According to (1.38), the Fermi n ≈ 7:38  1010 Vg cm2 , where Vg has the unit of volt. pffiffiffiffiffi ffi energy can be obtained from the relation EF ¼ ℏvF πe n , which can be controlled as pffiffiffiffiffi EF ∝ Vg by varying the gate voltage Vg to vary the carrier density e n. Alternatively, a solid polymer electrolyte gate can be employed on top of the graphene sheet as a top gate, as shown in Figure 2.12(b). A solid polymer electrolyte usually consists of lithium perchlorate (LiClO4) as the electrolyte and polyethylene oxide (PEO) polymers as the electrolyte solvent. When a positive voltage is applied on the gate, the mobile Li+ ions are dissolved from LiClO4 in the PEO matrix and move toward the graphene sheet, while the ClO4– ions migrate toward the gate electrode. Debye layers formed by these ions have the thickness of Debye length dD , which is around 1 nm. Because of this extremely small thickness, the resultant capacitance per unit area is very high. According to the identity C ¼ ϵ PEO =dD, where ϵ PEO ≈ 5ϵ 0 is the permittivity of PEO, the capacitance is about 4:4  106 F cm2 , which is much higher than the capacitance of 1:18  108 F cm2 in the case of an SiO2 substrate with back gating. Therefore, when a large density of carriers in graphene is desired, a solid polymer electrolyte gate is usually considered instead of back gating. The structures shown in Figure 2.12 consist of two terminals and one gate electrode. The gate is used for controlling the carrier density in graphene, whereas the source and drain terminals are biased to create a current for measuring the resistance. The total resistance of the structure is given by Rtot ¼ V =I, where I is the current through the source and the drain, and V is the potential difference between the two terminals. However, Rtot includes not only the resistance of graphene but also the contact resistance between the electrodes and the graphene layer. To solely measure the resistance of graphene, more terminals are needed and the structures in Figure 2.12 have to be modified. A four-terminal structure with back gating is shown in Figure 2.13(a); the terminals are marked with numbers. The graphene sample has a width of W and a length large enough to span across the channel from terminal 1 to terminal 4; the length of the separation between terminals 2 and 3 is L. If the current through terminals 1 and 4 has a magnitude of I14 , and the potential difference between terminals 2 and 3 is V23 , we find

56

Electronic Properties

(a)

(b)

B W

W L

ds

L

ds

SiO 2

SiO 2

Gate

Gate

Vg

Vg

Figure 2.13 Schematics of common configurations for measuring the resistance of graphene. (a)

Four-probe measurement. (b) Hall bar measurement.

a current density e J 14 ¼ I14 =W and an electric field E23 ¼ V23 =L on the graphene sheet. Using the relation e J 14 ¼ e σ E23 ;

ð2:67Þ

we find the surface resistivity of graphene: e ρ¼e σ 1 ¼

W V23 : L I14

ð2:68Þ

The surface resistivity e ρ is solely contributed by the graphene sheet and is the desirable quantity we want to measure. The resistance of the graphene sheet is given by Rgraphene ¼

V23 L L ¼e ρ ¼ Rsgraphene ; W W I14

ð2:69Þ

where Rsgraphene is the sheet resistance of graphene. It can be seen from (2.69) that the resistance Rgraphene , the surface resistivity e ρ, and the sheet resistance Rsgraphene of graphene all have the same unit of Ω. Therefore, the terminologies are sometimes used interchangeably, particularly between the surface resistivity e ρ and the sheet resistance Rsgraphene . However, though Rsgraphene ¼ e ρ according to (2.69), surface resistivity and sheet resistance have different definitions, thus fundamentally different meanings, as discussed in Section 2.1. To distinguish between the two, the unit of the sheet resistance is sometimes written as Ω per square, or Ω=□. For example, if the sheet resistance of graphene is Rsgraphene ¼ 1 kΩ=□, then the resistance of a square graphene sample would be Rgraphene ¼ 1 kΩ. However, if there are two squares in the graphene sample, namely L=W ¼ 2, then the resistance of this sample would be Rgraphene ¼ 1 kΩ=□  2□ ¼ 2 kΩ. This technique for the measurement of resistance is called the four-probe technique, and is frequently used for measuring the resistance of graphene [28,29]. Once e ρ is known, the mobility is obtained from (2.63) with the carrier density given by (2.66).

2.5 Transport Experiments

57

With the total resistance given as Rtot ¼ V14 =I14 and the resistance of graphene given by (2.69), the contact resistance Rc is then given by Rc ¼ Rtot  Rgraphene :

ð2:70Þ

Alternatively, the mobility can be measured directly in an experiment using the structure shown in Figure 2.13(b). Such a configuration is called a Hall bar device, which has an extra terminal compared to the four-terminal structure. When a magnetic field of magnitude B is applied on the Hall bar in a upward direction normal to the graphene surface, the drifting charge carriers of the current I14 experience a Lorentz force. The trajectories of these charge carriers curve toward one side of the Hall bar, creating an electric potential of V35 across terminals 3 and 5 because of the pileup of charge carriers. The sign of V35 depends on whether the carriers are electrons or holes. Eventually the steady state is reached when the electric force created by the potential difference V35 cancels out the Lorentz force. Thus, V35 ¼ WBvd ;

ð2:71Þ

where W is the width of graphene and vd is the drift velocity of the carriers in graphene. Using (2.67), (2.71), and the relation v d ¼ μe E23 for electrons in the case of a positive Fermi energy or v d ¼ μe E23 for holes in the case of a negative Fermi energy, we obtain the electron mobility or the hole mobility as μe ¼ 

RH RH or μh ¼ ; e e ρ ρ

ð2:72Þ

where e ρ is given by (2.68), and RH is the Hall coefficient given by RH ¼

V35 : e J 14 WB

ð2:73Þ

The sign of RH indicates the type of charge carriers: RH < 0 for electrons and RH > 0 for holes. In fact, with more careful calculation, it can be shown that the Hall mobility given by (2.72) and the drift mobility given by (2.63) are not exactly the same. This is due to the fact that the mobility is a macroscopic quantity that represents the average behavior of charge carriers; the detailed scattering behavior is ignored by equating vd in (2.71) with μe E23 or μh E23 to obtain (2.72). To find the ratio of the Hall mobility to the drift mobility, one must start from (2.10) and include the Lorentz force in the F term. Nevertheless, this ratio approaches unity when jEF j ≫ kB T; under this condition the sense of “average” disappears as the mobility is predominantly determined by the carriers at the energy levels near EF. Therefore, when the temperature is low or when the graphene sample is highly doped, the drift mobility is approximately the Hall mobility. Equations (2.66), (2.68), and (2.72) are the important equations for obtaining the carrier density, surface resistivity, and mobility of graphene, respectively, in transport experiments discussed in the following.

Electronic Properties

6

=

L

+

s

L

Resistivity,

(mS) (k )

5

Conductivity,

58

4

3

2 L

1

s

0 −50

0

50

Voltage, Vg ( V ) Figure 2.14 Surface conductivity e σ and the corresponding surface resistivity e ρ¼e σ

1

of graphene on an SiO2 substrate as a function of the gate voltage Vg at T ¼ 50 K [26]. The total surface resistivity is divided into two parts of e ρ L and e ρ s , and e σL ¼ e ρ 1 L is plotted as the dashed curve. Reprinted figure with permission from S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, et al., “Giant intrinsic carrier mobilities in graphene and its bilayer,” Physical Review Letters, Vol. 100, 016602 (2008). Copyright 2008 by the American Physical Society.

The surface conductivity of graphene was measured by Morozov et al. to show the ultrahigh mobility of graphene and the dominance of long-range scattering in the carrier transport of graphene [26]. In their experiment, the graphene sample obtained from graphite by micromechanical cleavage was placed on top of an SiO2 layer of ds ¼ 300 nm thickness on a silicon wafer. The experiment was carried out using the Hall bar device at a temperature of T ¼ 50 K with back gating as shown in Figure 2.13 (b). In Figure 2.14, the experimentally measured surface conductivity e σ and the corresponding surface resistivity e ρ are plotted as functions of the gate voltage Vg . As can be seen, the minimum surface conductivity and the corresponding maximum surface resistivity appear near Vg ¼ 0, which is called the neutrality point where the total carrier density (hole density plus electron density) is minimized. It is found that if one subtracts ρ, the remaining surface a constant value of e ρ s from the total surface resistivity e resistivity e ρ e ρs ¼ e ρ L gives a surface conductivity of e σL ¼ e ρ 1 L that is linearly proportional to Vg . It is believed that e σ L is contributed by long-range scattering events, such as charged-impurity scattering, whereas e ρ s indicates the presence of short-range scattering [26]. By plugging the charged-impurity scattering rate of (2.33) into the surface conductivity formula given in (2.18), we find that e σ ∝ jEF j2 ∝ Vg , which agrees with the ρ s , which is about 100 Ω, e ρL behavior of e σ L obtained from the experiment. Compared to e is much larger; therefore, in this experiment the charged-impurity scattering is considered to be the dominant scattering mechanism. By contrast, e ρ s is believed to be

2.5 Transport Experiments

59

contributed by short-range scattering events due to phonons and point defects. By plugging the point-defect scattering rate of (2.36) or the acoustic phonon scattering rate of (2.43) into the surface conductivity formula given in (2.18), we find that the value of e σ accounting for only these scattering mechanisms is independent of Vg , which agrees with the behavior of e ρ s obtained from the experiment as a constant. However, there is still an open debate regarding what contributes to e ρ s or whether it is appropriate to divide e ρ into two parts as a linear combination of e ρ L and e ρ s in the first place. With the right condition, scattering from vacancies and cracks might also give a constant surface resistivity [3] or give a surface conductivity that increases linearly with Vg , challenging the importance of charged-impurity scattering [30]. The mobility can be deduced from the experimental data of the surface resistivity. With the carrier density given by (2.66), the mobility can be obtained using (2.64). For Vg ¼ 5 V, the surface resistivity is about e ρ ≈ 1:27 kΩ, obtained from Figure 2.14, and the carrier density is e n ≈ 3:7  1011 cm2 . Therefore, the mobility μe is about 1:3  104 cm2 V1 s1 . If we can remove the charged impurities so that e ρ L ¼ 0 and e ρ ¼e ρ s ≈ 100 Ω, the mobility can even reach 1:7  105 cm2 V1 s1 . This value can be further increased if extrinsic scattering can be eliminated. Therefore, it is concluded that ultrahigh mobility is possible for carriers in graphene if extrinsic scattering such as charged-impurity scattering can be eliminated [26]. The effect of phonon scattering cannot be clearly seen in Figure 2.14 because the temperature was kept constant in the experiment. To see the influence of phonon scattering, one can vary the temperature to measure the surface resistivity as a function of temperature [27,28]. Shown in Figure 2.15 is the temperature dependence of the surface resistivity of graphene on SiO2 measured in one experiment by Efetov and Kim [27]. In this experiment, the graphene sample was placed on the Hall bar device as shown in Figure 2.13(b). A solid polymer electrolyte gate was adopted to tune the carrier density of graphene, as shown in Figure 2.12(b). It was shown that the capacitance per unit area was as high as 3:2  106 F cm2 , enabling the carrier density to reach 1014 cm2 without a breakdown due to electrochemical reactions. As can be seen from Figure 2.15, above 100 K, the surface resistivity increases linearly with temperature, which agrees with the prediction of the theory for acoustic phonon scattering, as can be seen from the dashed curve in Figure 2.9(b). To show this analytically, we insert the acoustic phonon scattering rate of (2.43) into the surface conductivity expression given in (2.18) to obtain the resultant relation e ρ ∝ T. pffiffiffi This linear relationship is only valid when kB T ≫ ℏωq ¼ ℏv a q ≈ 2ℏva kF ∝ n2 . Therefore, we see that as the carrier density decreases, the linear region is further extended into the low-temperature region. Outside the linear region, as in the case when the temperature is sufficiently low or the carrier density is high, such that the condition kB T ≫ ℏωq is not valid, the scattering rate has to be numerically solved. When kB T ≪ ℏωq , it is found theoretically that the scattering rate, and thus the surface resistivity, is approximately proportional to T 4 [12], which is experimentally confirmed [27], as shown in Figure 2.15(b). When the temperature gradually increases toward room temperature, phonon modes of high energies, such as the SO phonons, also become important. Shown in

60

(a)

Electronic Properties

(b)

225 200

∼T

10

( )

( )

125

(

n 1013 cm

100

2

)

1.36 1

2.86

75

4.65 6.85

∼T4

50

10.8

25 0

100 T (K )

200

0.1 10

100 T (K )

Figure 2.15 Surface resistivity of graphene on SiO2 as a function of temperature for different carrier densities [27]. Different curves are associated with different carrier densities specified in the legend. In (b) Δe ρ ¼e ρ ðTÞ  e ρ 0 is plotted in log scale, where e ρ 0 is the surface resistivity in the lowtemperature limit. Reprinted figure with permission from D. K. Efetov and P. Kim, “Controlling electron–phonon interactions in graphene at ultrahigh carrier densities,” Physical Review Letters, Vol. 105, 256805 (2010). Copyright 2010 by the American Physical Society.

Figure 2.16 is the surface resistivity of graphene on SiO2 as a function of temperature for different gate voltages measured in another experiment by Chen et al. [28]. In this experiment, the graphene sample was obtained from Kish graphite by mechanical exfoliation; an SiO2 substrate of 300 nm thickness was used, and the back gate scheme was adopted using doped silicon as the back gate [28]. The resistance was measured using the standard four-probe technique as shown in Figure 2.13(a). To obtain the resistance at different temperatures, the device was placed on a cold finger cooled by liquid helium in an ultrahigh vacuum chamber, and the temperature was varied by controlling the liquid helium flow. One can see in Figure 2.16 that around and below 100 K the surface resistivity linearly increases with temperature (dashed lines), consistent with Figure 2.15 around 100 K for the acoustic phonon scattering. The T 4 dependence at very low temperatures can hardly be seen in Figure 2.16 as e ρ 0 is not subtracted from e ρ (T ) for the data shown in this figure, and the data points are limited. As the temperature increases above 100 K, the SO phonon scattering gradually becomes important. This behavior is consistent with the theory as shown in Figure 2.9(c). We can also find that for a larger gate voltage, which leads to a larger EF, the onset of SO phonon scattering is delayed until a higher temperature. This experimentally observed feature

2.6 Multilayer Graphene

61

700 10 V

( )

600

500

15V

400

20 V

30 V

300

40 V 48V 57 V

200

100 0

100

T (K )

200

300

Figure 2.16 Surface resistivity of graphene on SiO2 as a function of temperature for different gate voltages [28]. The dots are experimental measurement. The dashed lines are fits to the linear temperature dependence in the low-temperature region.

can be explained by the screening effect that suppresses the SO phonon scattering, as shown theoretically in Figure 2.7(b).

2.6

Multilayer Graphene We have regarded the surface current density and the surface conductivity of monolayer graphene as 2D quantities because the current can only flow on the graphene surface but not in the direction perpendicular to the graphene sheet. The same argument can apply to multilayer graphene. As shown in Figure 2.17(b), where the z axis is perpendicular to the graphene surface, the total current I is shared by all of the graphene layers at different values of z. The electric current is still confined on the 2D surface of each graphene layer; no current flows between two neighboring graphene layers. Therefore, the 2D surface conductivity e σ and surface current density e J can still be defined for multilayer graphene [31]. However, as the thickness increases with the number of layers, one has to view multilayer graphene as a 3D material, as in the case of graphite. When the total thickness is much larger than the spacing between neighboring layers, macroscopically the electric current can be regarded as a current that uniformly spreads in a space occupied by the graphene layers, as shown in Figure 2.17(a). When the multilayer graphene is regarded

62

Electronic Properties

(a)

(b)

E

E

E

w

w

J = E J = E =

J= E=

I

I wd

I w

I

d z y x

J Am

2

Sm

1

J Am

1

S

Figure 2.17 Models for multilayer graphene. In (a), multilayer graphene is regarded as a 3D

material with an electric current distributed among graphene layers, whereas in (b) the electric current is confined on the graphene surfaces, and 2D quantities can be defined accordingly.

as a 3D material, an electric current along the z axis is allowed, although the perpendicular conductivity σ ⊥ is very small. The ratio σ ∥ =σ⊥ has a value of about 104 for graphite [32]. If σ ⊥ is ignored due to its small magnitude, the two models for the surface conductivity of multilayer graphene shown in Figure 2.17 are related by the relation: σ∥ ¼

e σ∥ e σ ¼ ; d d

ð2:74Þ

where d is the total thickness of the multilayer graphene. In the following, we shall still regard multilayer graphene as a 2D material to facilitate a direct comparison with the 2D physical properties of monolayer graphene.

2.6.1

AB Stacking As discussed in Chapter 1, N-layer AB-stacked multilayer graphene has N conduction bands and N valence bands. If N is an even number, all 2N bands are massive bands. If N is odd, out of the 2N bands, 2N  2 bands are massive bands and the other two are linear bands defining one massless Dirac cone given by E ¼ ℏv F k. Here we shall focus on carrier energies that are much smaller than γ1 ¼ 0:37 eV, where the massive bands are approximately parabolic bands of the effective mass m ¼ jE0 j=2v2F with E0 given by (1.55). To derive the surface conductivity contributed by the massive bands that are given by E ¼ ℏ2 k 2 =2m, such as the two bands for jEj < γ1 in the band structure of AB-stacked bilayer graphene shown in Figure 1.8(a), we follow the same procedure as that taken in Section 2.2 for the monolayer graphene. In contrast to (2.13), the corresponding distribution function in this situation is given as

2.6 Multilayer Graphene

f ¼ f0  τ

eℏEx kx df0 ; m dE

63

ð2:75Þ

where the plus sign is for positive E, and the minus sign is for negative E. Substituting (2.75) into (2.14), we find e J e ¼ e

ð∞2ðπ  ℏkx fkd θdk m ð2πÞ2 g

0 0

¼ ¼

2

2e Ex πℏ

ð∞

τ

2

ð2:76Þ

0

2τeff e2 Ex πℏ2

df0 EdE dE

kB T lnðe μ=kB T þ 1Þ;

and 2

2τeff e Ex e Jh ¼ kB T lnðeμ=kB T þ 1Þ: 2 πℏ

ð2:77Þ

Combining electron and hole current densities, we obtain the total surface conductivity: e σ N¼2 ¼

4τeff e2 kB T πℏ2

  μ ln 2cosh ; 2kB T

ð2:78Þ

where the subscript N ¼ 2 denotes the surface conductivity of AB-stacked bilayer graphene. To find the relation in the surface conductivity of AB-stacked graphene among graphene of different numbers of layers, it is intuitive to set τeff in (2.78) to be the same as that in (2.17). Then we find that (2.78) is simply e σ N¼2 ¼ 2e σ N¼1, where e σ N¼1 is the surface conductivity of monolayer graphene given by (2.17). Likewise, for ABstacked trilayer graphene, as shown in Figure 1.8(b) we have two pairs of massive conduction and valence bands, which together contribute a total surface conductivity of e σ N¼2 , and one pair of massless conduction and valence bands, which contribute a total surface conductivity of e σ N¼1 . Therefore, the surface conductivity of AB-stacked trilayer graphene is e σ N¼3 ¼ e σ N¼2 þ e σ N¼1 ¼ 3e σ N¼1 . Similarly, the surface conductivity of ABstacked N-layer graphene is simply given by e σ N ¼ Ne σ N¼1, for τeff being approximately the same for different bands.

2.6.2

AA Stacking As shown in Figure 1.11, the band structure of N-layer AA-stacked multilayer graphene consists of N Dirac cones that are shifted in energy, as described by (1.66): E ¼ ℏvAA F k þ Δr ; where

ð2:79Þ

64

Electronic Properties

Δr ¼ 2γAA 1 cos

rπ ; ðN þ 1Þ

r ¼ 1; 2; . . . ; N;

ð2:80Þ

is the shift in energy for the r band. The surface conductivity can be easily calculated by shifting μ to μ  Δr in (2.15) for the electron current density and the corresponding integral for the hole current density; the result is e σr ¼

2τr;eff e2 kB T πℏ2

  μ  Δr ln 2 cosh 2kB T

ð2:81Þ

for the surface conductivity of the r band. Accordingly, the total surface conductivity contributed by all N bands is e σ AA ¼

  X 2e2 kB T X μ  Δr e σr ¼ τ ln 2 cosh : r;eff 2kB T πℏ2 r r

ð2:82Þ

Here we keep the effective scattering rate τr;eff different for different bands, as carriers on different bands have an energy difference as large as 2γAA 1 . Indeed, it has been theoretically shown that τr;eff are very different for different bands of AA-stacked multilayer graphene [33]. Therefore, the surface conductivity of AA-stacked multilayer graphene can have a very different behavior from that of AB-stacked multilayer graphene of the same number of layers.

References 1. C. Hamaguchi, Basic Semiconductor Physics (Springer, 2009). 2. J. H. Davies, The Physics of Low-dimensional Semiconductors: An Introduction (Cambridge University Press, 1998). 3. T. Stauber, N. M. R. Peres, and F. Guinea, “Electronic transport in graphene: A semiclassical approach including midgap states,” Physical Review B, Vol. 76, 205423 (2007). 4. S. Adam, E. H. Hwang, and S. Das Sarma, “Scattering mechanisms and Boltzmann transport in graphene,” Physica E, Vol. 40, pp. 1022–1025 (2008). 5. E. H. Hwang and S. Das Sarma, “Single-particle relaxation time versus transport scattering time in a two-dimensional graphene layer,” Physical Review B, Vol. 77, 195412 (2008). 6. X. Hong, K. Zou, and J. Zhu, “Quantum scattering time and its implications on scattering sources in graphene,” Physical Review B, Vol. 80, 241415 (2009). 7. N. M. R. Peres, “Colloquium: The transport properties of graphene: an introduction,” Review of Modern Physics, Vol. 82, pp. 2673–2700 (2010). 8. N. M. R. Peres, F. Guinea, and A. H. Castro Neto, “Electronic properties of disordered two-dimensional carbon,” Physical Review B, Vol. 73, 125411 (2006). 9. S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Review of Modern Physics, Vol. 83, pp. 407–470 (2011). 10. I. L. Aleiner and K. B. Efetov, “Effect of disorder on transport in graphene,” Physical Review Letters, Vol. 97, 236801 (2006). 11. M. Lundstrom, Fundamentals of Carrier Transport (Cambridge University Press, 2009).

References

65

12. E. H. Hwang and S. Das Sarma, “Acoustic phonon scattering limited carrier mobility in two-dimensional extrinsic graphene,” Physical Review B, Vol. 77, 115449 (2008). 13. S. Fratini and F. Guinea, “Substrate-limited electron dynamics in graphene,” Physical Review B, Vol. 77, 195415 (2008). 14. S. Q. Wang and G. D. Mahan, “Electron scattering from surface excitations,” Physical Review B, Vol. 6, pp. 4517–4524 (1972). 15. A. Konar, T. Fang, and D. Jena, “Effect of high-κ gate dielectrics on charge transport in graphene-based field effect transistors,” Physical Review B, Vol. 82, 115452 (2011). 16. I. T. Lin and J. M. Liu, “Surface polar optical phonon scattering of carriers in graphene on various substrates,” Applied Physics Letters, Vol. 103, 081606 (2013). 17. V. Perebeinos and P. Avouris, “Inelastic scattering and current saturation in graphene,” Physical Review B, Vol. 81, 195442 (2010). 18. M. V. Fischetti, D. A. Neumayer, and E. A. Cartier, “Effective electron mobility in Si inversion layers in metal–oxide–semiconductor systems with a high-κ insulator: The role of remote phonon scattering,” Journal of Applied Physics, Vol. 90, pp. 4587–4608 (2001). 19. D. W. Feldman, J. H. Parker, Jr., W. J. Choyke, and L. Patrick, “Phonon dispersion curves by Raman scattering in SiC, polytypes 3C, 4H, 6H, 15R, and 21R,” Physical Review, Vol. 173, pp. 787–793 (1968). 20. R. Geick, C. H. Perry, and G. Rupprecht, “Normal modes in hexagonal boron nitride,” Physical Review, Vol. 146, pp. 543–547 (1966). 21. D. L. Rode, “Electron mobility in direct-gap polar semiconductors,” Physical Review B, Vol. 2, pp. 1012–1024 (1970). 22. D. M. Basko, “Theory of resonant multiphonon Raman scattering in graphene,” Physical Review B, Vol. 78, 125418 (2008). 23. K. I. Bolotin, K. J. Sikes, Z. Jiang, et al., “Ultrahigh electron mobility in suspended graphene,” Solid State Communications, Vol. 146, pp. 351–355 (2008). 24. E. Mariani and F. von Oppen, “Temperature-dependent resistivity of suspended graphene,” Physical Review B, Vol. 82, 195403 (2010). 25. E. V. Castro, H. Ochoa, M. I. Katsnelson, et al., “Limits on charge carrier mobility in suspended graphene due to flexural phonons,” Physical Review Letters, Vol. 105, 266601 (2010). 26. S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, et al., “Giant intrinsic carrier mobilities in graphene and its bilayer,” Physical Review Letters, Vol. 100, 016602 (2008). 27. D. K. Efetov and P. Kim, “Controlling electron‒phonon interactions in graphene at ultrahigh carrier densities,” Physical Review Letters, Vol. 105, 256805 (2010). 28. J. H. Chen, C. Jang, S. Xiao, M. Ishigami, and M. S. Fuhrer, “Intrinsic and extrinsic performance limits of graphene devices on SiO2,” Nature Nanotechnology, Vol. 3, pp. 206–209 (2008). 29. K. I. Bolotin, K. J. Sikes, J. Hone, H. L. Stormer, and P. Kim, “Temperature-dependent transport in suspended graphene,” Physical Review Letters, Vol. 101, 096802 (2008). 30. L. A. Ponomarenko, R. Yang, T. M. Mohiuddin, et al., “Effect of a high-κ environment on charge carrier mobility in graphene,” Physical Review Letters, Vol. 102, 206603 (2009). 31. A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Physical Review Letters, Vol. 100, 117401 (2008). 32. D. Chung, “Review graphite,” Journal of Materials Science, Vol. 37, pp. 1475–1489 (2002). 33. I. T. Lin and J. M. Liu, “Terahertz frequency-dependent carrier scattering rate and mobility of monolayer and AA-stacked multilayer graphene,” IEEE Journal of Selected Topics in Quantum Electronics, Vol. 20, 8400108 (2014).

3

Optical Properties

3.1

Optical Fields The linear optical properties of graphene are considered in this chapter. Optoelectronic properties are considered in Chapter 4, and nonlinear optical properties are discussed in Chapter 5. All of these properties are related to the interactions between graphene and an optical field. An optical field in a medium is generally characterized by four macroscopic vectorial fields: the electric field Eðr; tÞ, the electric displacement Dðr; tÞ, the magnetic field Hðr; tÞ, and the magnetic induction Bðr; tÞ. These macroscopic optical field vectors are space- and time-varying electromagnetic fields that are governed by Maxwell’s equations: ∂B ; ∂t

ð3:1Þ

∂D þ J; ∂t

ð3:2Þ

E¼

Δ

H ¼

Δ

 D ¼ ρ;

ð3:3Þ

Δ

 B ¼ 0;

ð3:4Þ

Δ

where the current density J and the charge density ρ are constrained by the continuity equation: ∂ρ ð3:5Þ  J þ ¼ 0: ∂t

Δ

The electric displacement Dðr; tÞ and the magnetic field Hðr; tÞ are macroscopic fields generally defined as Dðr; tÞ ¼ ϵ 0 Eðr; tÞ þ Pðr; tÞ; Hðr; tÞ ¼

1 Bðr; tÞ  Mðr; tÞ; μ0

ð3:6Þ ð3:7Þ

where the polarization Pðr; tÞ and the magnetization Mðr; tÞ are macroscopically averaged densities of microscopic electric dipoles and magnetic dipoles induced by the electromagnetic field in the medium. It is generally true that at an optical frequency the

3.1 Optical Fields

67

magnetization vanishes, i.e., Mðr; tÞ ¼ 0, so that Bðr; tÞ ¼ μ0 Hðr; tÞ for an optical field. Because Mðr; tÞ ¼ 0 at an optical frequency, the optical response of a medium is fully described by the electric polarization Pðr; tÞ and the current density Jðr; tÞ in the medium that are induced by the optical field.

Optical Responses The total current density in an optical medium can be generally divided into two sources: an induced current in response to the optical field and an external current from an external current source, thus J total ¼ J ind þ J ext . Similarly, the total charge density also generally has two sources: ρtotal ¼ ρind þ ρext . Because the optical property of a material is solely determined by the response of the material to an optical field, we consider only a situation that is free of external current and charge sources such that J ext ¼ 0 and ρext ¼ 0. In this situation, J total ¼ J ind and ρtotal ¼ ρind . Induced current and charge densities are contributed by both bound charges and free conduction charge carriers: J ind ¼ J bound þ J cond and ρind ¼ ρbound þ ρcond . Optical excitation induces either displacement of bound charges or generation of free conduction charge carriers. In an optical medium, charge conservation requires that an increase of charge density at a location is always accompanied by a reduction at another location at the microscopic level. Microscopic displacement of bound charges results in induced electric dipoles, which result in a macroscopic electric polarization P; by contrast, optical generation of free charge carriers creates an equal number of electrons and holes in pairs, which results in no net macroscopic free charge density such that ρcond ¼ 0. Therefore, ρind ¼ ρbound , which is generally accounted for by the induced electric polarization P as ρind ¼ ρbound ¼   P. Therefore, the effect of charge displacement in response to an optical field can be accounted for by the electric polarization P while taking ρind ¼ 0 because it is not necessary to consider a net macroscopic induced charge density in addition to the electric polarization. In other words, there is no macroscopic free change density that is induced by an optical field and varies at an optical frequency. By contrast, an induced macroscopic current density in an optical medium is often considered such that J ind ≠ 0. Therefore, for an optical medium that is free of external current and charge sources, we have J total ¼ J ind and ρtotal ¼ 0 so that  J total ¼  J ind ¼ 0 and  D ¼ 0. The induced current resulting from the response of an optical medium to an optical field generally have two contributions: a polarization current J bound from bound charges of the medium and a conduction current J cond from free charge carriers, including electrons and holes, of the medium. Therefore, J ind ¼ J bound þ J cond . Because the polarization current J bound is a displacement current contributed by the displacement of bound charges in response to an optical field, it is generally included in the ∂D=∂t term through ∂P=∂t. By contrast, the conduction current J cond can be treated in three alternative, but equivalent, approaches: (1) it can be included in the ∂D=∂t term as part of ∂P=∂t without an explicit current term because it is an induced current; (2) it can be included in the ∂D=∂t term, because it is an induced current, but as a current term that is not a part of ∂P=∂t because it is not a displacement current; or (3) it can be treated as a

Δ

Δ

Δ

Δ

3.1.1

68

Optical Properties

current term that is completely separate from the ∂D=∂t term by considering ∂D=∂t only contributed by bound charges. In this chapter, we take the second approach by including J cond in the ∂D=∂t term, but not as part of ∂P=∂t. Then, we can write Ampère’s equation as ∂D ∂E ∂P ¼ ϵ0 þ þ J; ∂t ∂t ∂t

H ¼

ð3:8Þ

Δ

where ∂P=∂t ¼ J bound and J ¼ J cond . Then, Maxwell’s equations of an optical material that is free of external current and charge sources take the form:

Δ

E¼

∂B ; ∂t

ð3:9Þ

Δ

H ¼

∂D ; ∂t

ð3:10Þ

Δ

 D ¼ 0;

ð3:11Þ

Δ

 B ¼ 0:

ð3:12Þ

In this representation, the optical response of the bound charges is represented by the polarization P and characterized by a susceptibility tensor χ, and that of the free charge carriers is represented by the current J and characterized by a conductivity tensor σ. The total optical response is represented by the electric displacement D and characterized by a permittivity tensor ϵ.

3.1.2

Harmonic Optical Fields Optical fields are harmonic fields that vary sinusoidally with time. The field vectors defined in the preceding section are all real quantities. For harmonic fields, it is always convenient to use complex fields. We define the space- and time-dependent complex electric field, Eðr; tÞ, through its relation to the real electric field, Eðr; tÞ:1 Eðr; tÞ ¼ Eðr; tÞ þ Eðr; tÞ ¼ Eðr; tÞ þ c:c:;

ð3:13Þ

where c.c. means the complex conjugate. In our convention, Eðr; tÞ contains the complex field components that vary with time as expðiωtÞ with ω having a positive value, while Eðr; tÞ contains those varying with time as expðiωtÞ with positive ω. The complex fields of other field quantities are similarly defined. In this book, real field vectors are represented in the italic bold capital Roman font, such as Eðr; tÞ, whereas the complex field vectors of a harmonic field are represented in the regular bold capital Roman font, such as Eðr; tÞ. Real harmonic fields considered in this book include E, D, 1

In some literature, the complex field is defined through a relation with the real field as Eðr; tÞ ¼ ½Eðr; tÞ þ E ðr; tÞ=2; which differs from our definition in (3.13) by the factor 1=2. The magnitude of the complex field defined through this alternative relation is twice that of the complex field defined through (3.13).

3.1 Optical Fields

69

B, H, P, M, and J, which are discussed above, and the vector potential A to be discussed later in this chapter; the corresponding complex fields are E, D, B, H, P, M, J, and A. Complex scalar field quantities such as the complex charge density ρðr; tÞ and the complex scalar potential φðr; tÞ are also similarly defined: ρðr; tÞ ¼ ρðr; tÞ þ ρðr; tÞ and φðr; tÞ ¼ φðr; tÞ þ φðr; tÞ, where ρðr; tÞ is the real charge density and φðr; tÞ is the real scalar potential. With this definition for the complex fields, all of the linear field equations retain their forms. In terms of complex optical fields, Maxwell’s equations in the form of (3.9)–(3.12) are

Δ

E¼

∂B ; ∂t

ð3:14Þ

Δ

H¼

∂D ; ∂t

ð3:15Þ

Δ

 D ¼ 0;

ð3:16Þ

Δ

 B ¼ 0:

ð3:17Þ

The complex electric field of a harmonic optical field that has a carrier wave vector k and a carrier angular frequency ω can be further expressed as Eðr; tÞ ¼ Eðr; tÞexpðik  r  iωtÞ ¼ ^e Eðr; tÞexpðik  r  iωtÞ;

ð3:18Þ

where Eðr; tÞ is the space- and time-dependent amplitude of the field, and ^e is the unit polarization vector of the field. The vectorial field amplitude Eðr; tÞ is generally a complex vectorial quantity that has a magnitude, a phase, and a polarization. Other complex field quantities, such as Dðr; tÞ, Bðr; tÞ, and Hðr; tÞ, can be similarly expressed. For harmonic optical fields, it is often useful to consider the complex fields in the momentum space and frequency domain defined by the Fourier transform relations: ð∞ ððð Eðk; ωÞ ¼

Eðr; tÞexpðik  r þ iωtÞdrdt;

for ω > 0;

ð3:19Þ

∞ all r

Eðr; tÞ ¼

1 ð2πÞ4

ð∞ ððð Eðk; ωÞexpðik  r  iωtÞdkdω:

ð3:20Þ

0 all k

Note that Eðk; ωÞ in (3.19) is only defined for ω > 0; therefore, the integral for the time dependence of the complex field Eðr; tÞ in (3.20) only extends over positive values of ω. This is in accordance with the convention we used to define the complex field Eðr; tÞ in (3.13). Using (3.13) and (3.20) while defining the relation Eðk; ωÞ ¼ Eðk; ωÞ, we find that the real field Eðr; tÞ in the real space and time domain can be expanded as a Fourier integral of Eðk; ωÞ over the range ∞ < ω < ∞ of both positive and negative frequencies and over all values of k and k of both positive and negative wave vectors as

70

Optical Properties

Eðr; tÞ ¼ Eðr; tÞ þ Eðr; tÞ ð∞ ððð 1 ¼ Eðk; ωÞexpðik  r  iωtÞdkdω ð2πÞ4 0 all k ð∞ ððð 1 þ Eðk; ωÞexpðik  r þ iωtÞdkdω ð2πÞ4 ¼

1 ð2πÞ4

ð3:21Þ

0 all k ð∞ ððð

Eðk; ωÞexpðik  r  iωtÞdkdω: ∞ all k

All other space- and time-dependent quantities, including other field vectors and the permittivity and susceptibility tensors, are transformed in a similar manner. In the real space and time domain, Pðr; tÞ, Jðr; tÞ, and Dðr; tÞ are connected to Eðr; tÞ through convolution integrals in space and time. Through the Fourier transform, the convolution integrals in real space and time become simple products in the momentum space and frequency domain. Consequently, in the momentum space and frequency domain, Pðk; ωÞ, Jðk; ωÞ, and Dðk; ωÞ are connected to Eðk; ωÞ through direct products: Pðk; ωÞ ¼ ϵ 0 χðk; ωÞ  Eðk; ωÞ;

ð3:22Þ

Jðk; ωÞ ¼ σðk; ωÞ  Eðk; ωÞ;

ð3:23Þ

Dðk; ωÞ ¼ ϵðk; ωÞ  Eðk; ωÞ:

ð3:24Þ

and

3.2

Susceptibility and Permittivity of a Two-Dimensional Material In contrast to the free electrons that contribute to a current in the presence of an electric field, discussed in the preceding chapter, bound electrons of graphene contribute to an electric polarization P in response to an electric field, as discussed in the preceding section. A bound electron can be thought of as an electron that cannot make the transition to an empty state in an electric field because there is no available empty state, or the momentum or energy cannot be conserved in any way even through scattering by a quasiparticle. For example, for lightly n-doped graphene, free electrons are the electrons in the conduction band, and bound electrons are those in the valence band that cannot obtain sufficient momentum or energy from photons or from quasiparticles to be promoted to the conduction band. In Chapter 2, we used a model that adopts the 2D picture of graphene because both the current and the electrons are confined on the graphene sheet. The surface conductivity e σ has a unit of S, as shown in Figure 2.1. However, the field variables, such as the electric and magnetic fields, in Maxwell’s equations are 3D quantities. Therefore, in Chapter 2 we introduced another model that treats the DC conductivity σ of multilayer graphene as

3.2 Susceptibility and permittivity of 2D material

71

a 3D quantity given by (2.74), that is, σ ¼ e σ =d instead of σ ¼ e σ δðzÞ. By doing so, we assume that the current uniformly spreads “inside” the multilayer graphene sheet of an effective thickness d. In a similar manner, we can also treat the current density J, which is connected to the optical field through the optical conductivity σ, and the electric polarization P, which is connected to the optical field through the optical susceptibility χ, of monolayer graphene as 3D quantities. The current density J, which is contributed by the free electrons in graphene, and the electric polarization P, which is contributed by the bound electrons of graphene, are both confined “within” the graphene sheet that has a thickness of d. Unlike that of multilayer graphene, the thickness d of monolayer graphene is arbitrary and is usually taken as the spacing between two neighboring graphene layers in graphite [1]. This 3D picture is frequently used in the experiments conducted for the measurement of the permittivity and the refractive index of graphene. In the following discussion, we first consider the 3D model by taking the thickness of a graphene sheet to be d. In this model, the electric polarization P of a graphene sheet that is located at z ¼ 0 can be written as  Pðx; yÞ; jzj < d=2; Pðx; y; zÞ ¼ Px ðx; y; zÞ^x þ Py ðx; y; zÞ^y þ Pz ðx; y; zÞ^z ¼ 0; jzj > d=2: ð3:25Þ In (3.25), we assume that there is no variation in the z direction for d=2 < z < d=2 within the graphene sheet. Macroscopically this is justifiable as the thickness of a graphene sheet is much smaller than other physical dimensions of interest such as the length and width of the graphene sheet and the optical wavelengths. Unlike the current density J in a graphene sheet, which has no z component because a current is allowed to flow only on the plane of the graphene sheet, the electric polarization P can have a component in the z direction perpendicular to the plane of the graphene sheet, and this component can even be comparable to those in the directions parallel to the plane of the graphene sheet, as we shall see in later sections. This concept is similar to that discussed in Section 2.3.4 on the optical phonons where carbon atoms have in-plane and out-ofplane vibrations; here, instead of atoms, electrons have in-plane and out-of-plane displacements. From (3.25), the electric polarization for d=2 < z < d=2 within the graphene thickness d is thus a function of only x and y, but it has x, y, and z components, as given by Pðx; yÞ ¼ Px ðx; yÞ^x þ Py ðx; yÞ^y þ Pz ðx; yÞ^z :

ð3:26Þ

The electric susceptibility of a material, which is often also called the optical susceptibility when considering optical response, as is the case in this chapter, characterizes the induced polarization as the response of the material to an electric field. Because both the polarization and the electric field are vectors, the susceptibility is generally expressed as a second-order tensor:

72

Optical Properties

2 χxx ðx; y; zÞ χðx; y; zÞ¼4χyx ðx; y; zÞ χzx ðx; y; zÞ

χxy ðx; y; zÞ χyy ðx; y; zÞ χzy ðx; y; zÞ

3 χxz ðx; y; zÞ χyz ðx; y; zÞ 5; χzz ðx; y; zÞ

ð3:27Þ

which is explicitly expressed as a function of the space variables, x, y, and z, to account for the possible spatial inhomogeneity of the material. For undeformed graphene, an electric field that is either parallel or perpendicular to the graphene surface can only induce polarization in the direction of the field due to the symmetry of the graphene structure. Like the electric polarization given in (3.25), the susceptibility is nonzero only within the range that d=2 < z < d=2, where it is a function of only x and y. Furthermore, as a physical property, the susceptibility tensor can always be diagonalized along its principal axes, defined as ^x , ^y , and ^z , where ^x and ^y are on the graphene surface while ^z is normal to the graphene surface. Therefore, for graphene, the susceptibility in (3.27) reduces to 2 χx ðx; yÞ χðx; y; zÞ ¼ χðx; yÞ ¼4 0 0

3 0 0 χy ðx; yÞ 0 5; 0 χz ðx; yÞ

 d=2 < z < d=2; ð3:28Þ

where χx , χy , and χz are the principal susceptibilities associated with the x, y, and z principal axes, respectively. As discussed in Section 2.1, the x and y principal axes on the graphene surface cannot be arbitrarily chosen because they are determined by the hexagonal structure of the 2D graphene lattice. However, though the hexagonal lattice of graphene is structurally not isotropic, the linear electric and optical properties of undeformed graphene is isotropic on the xy plane of the graphene surface. Therefore, the principal susceptibilities associated with the x and y principal axes on an undeformed graphene surface have the same value, which is different from that of the principal susceptibility for the z principal axis: χx ¼ χy ¼ χ∥ and χz ¼ χ⊥, where ∥ and ⊥ denote the components parallel and perpendicular to the graphene surface, respectively. Therefore, for d=2 < z < d=2 within the thickness d, (3.28) can be further simplified as 2 3 χ∥ ðx; yÞ 0 0 χðx; yÞ ¼ 4 0 χ∥ ðx; yÞ 0 5: ð3:29Þ 0 0 χ⊥ðx; yÞ Using (3.26), (3.29), and the frequency-domain relation given in (3.22), P ¼ ϵ 0 χ  E;

ð3:30Þ

Px ðx; yÞ ¼ ϵ 0 χ∥ ðx; yÞEx ðx; yÞ;

ð3:31Þ

Py ðx; yÞ ¼ ϵ 0 χ∥ ðx; yÞEy ðx; yÞ;

ð3:32Þ

we obtain, for d=2 < z < d=2,

3.2 Susceptibility and permittivity of 2D material

Pz ðx; yÞ ¼ ϵ 0 χ⊥ðx; yÞEz ðx; yÞ:

73

ð3:33Þ

Note that the electric field components in the above equations do not vary with z for d=2 < z < d=2, which is generally true because of the very small thickness of graphene as compared to any optical wavelength of interest such that d ≪ λ. To find the permittivity and the refractive index of graphene, we start from Ampère’s equation. By accounting for both the optical response of free conduction electrons through the current density J and that of the bound electrons through the electric polarization P, as discussed in Section 3.1 and expressed in (3.8), Ampère’s equation in the time domain can be expressed in terms of the complex fields as H¼

∂D ∂E ∂P ¼ ϵ0 þ þ J; ∂t ∂t ∂t

ð3:34Þ

Δ

where the electric displacement D includes the optical responses from all bound and conduction electrons. By taking the Fourier transform to convert (3.34) to the frequency domain, we find iωDðωÞ ¼ iωϵ 0 EðωÞ  iωPðωÞ þ JðωÞ:

ð3:35Þ

Then, using the frequency-domain relations given in (3.22)–(3.24), PðωÞ ¼ ϵ 0 χðωÞ  EðωÞ; JðωÞ ¼ σðωÞ  EðωÞ; and DðωÞ ¼ ϵðωÞ  EðωÞ; ð3:36Þ we obtain the total permittivity ϵ as   σðωÞ : ϵðωÞ ¼ ϵ 0 1 þ χðωÞ þ i ω

ð3:37Þ

In comparison to the usual relation of ϵ ¼ ϵ 0 ð1 þ χÞ for dielectrics, (3.37) for the semiconducting graphene has an additional contribution from the conduction electrons. In experiments, it is often difficult and usually not necessary to separately measure the optical responses from bound and conduction electrons. Therefore, the total optical response ϵðωÞ is measured when it is not necessary to specifically distinguish the two contributions. So far we have taken the 3D model by assuming that the physical quantities χ and P are homogeneous in the z direction within the thickness d of the graphene sheet. For the 2D model, we can define the corresponding 2D quantities as e χ ¼ χd

and

e ¼ Pd; P

ð3:38Þ

where e χ is the surface optical susceptibility in the 2D model, P represents the 3D average e ¼ ϵ 0e polarization density per unit volume, and P χ  E represents the 2D polarization density per unit area on the graphene surface. Using these 2D quantities, we can rewrite (3.37) as

74

Optical Properties

  e e χ ðx; y; ωÞ σ ðx; y; ωÞ þi ϵðx; y; z; ωÞ ¼ ϵ 0 1 þ ;  d=2 < z < d=2; d ωd

ð3:39Þ

where e σ is the surface optical conductivity as described in Chapter 2 and given in (2.5) for the surface electric conductivity, but at an optical frequency ω. Equation (3.39) is the 3D model of the optical permittivity for a graphene sheet that has a thickness of d. Because e χ and e σ are tensors that have the principal surface susceptibilities e χx ¼ e χy ¼ e χ∥ ≠ e χz ¼ e χ ⊥ from (3.29) and the principal surface conductivities e σx ¼ e σy ¼ e σ∥ ≠ e σ z ¼ 0 from (2.5) and (2.8), the permittivity ϵ is also a tensor of the form: 2 3 ϵ∥ 0 0 ϵ ¼ 4 0 ϵ ∥ 0 5; ð3:40Þ 0 0 ϵ⊥ where

and

  e χ∥ e σ∥ ϵ∥ ¼ ϵ0 1 þ þi d ωd

ð3:41Þ

  e χ ϵ⊥ ¼ ϵ0 1 þ ⊥ : d

ð3:42Þ

χ ∥ from the bound electrons and by e σ ∥ from the free Note that ϵ ∥ is contributed both by e conduction electrons, whereas ϵ ⊥ is solely contributed by e χ ⊥ from the bound electrons because e σ ⊥ is zero due to the fact that the free electrons can only move on the plane of the graphene sheet, as discussed in Section 2.1. From the form of ϵ seen in (3.40), it is clear that a graphene sheet that is not subject to a deforming force is optically uniaxial, with the x and y principal axes being ordinary axes having an ordinary permittivity of ϵ ∥ and the z principal axis being the extraordinary axis having an extraordinary permittivity of ϵ ⊥. Accordingly, the ordinary refractive index no and the extraordinary refractive index ne are given by

and

rffiffiffiffiffi ϵ∥ no ¼ n∥ ¼ ϵ0

ð3:43Þ

rffiffiffiffiffi ϵ⊥ ne ¼ n⊥ ¼ ; ϵ0

ð3:44Þ

respectively. e We can also define the 2D surface permittivity e ϵ ¼ ϵd in the same manner as P and e χ.

3.2 Susceptibility and permittivity of 2D material

75

Then, (3.39) can be written as   e e e χ ðx; y; ωÞ ϵ ðx; y; ωÞ σ ðx; y; ωÞ ¼ ϵ0 1 þ þi ϵðx; y; z; ωÞ ¼ d d ωd

ð3:45Þ

for d=2 < z < d=2. For a finite thickness, we find from (3.45) that e ϵ ¼ ϵ 0 d þ ϵ 0e χþi

e σ : ω

ð3:46Þ

By taking the limit that d → 0 so that graphene is regarded as a 2D material, we obtain from (3.45) that e σ ðx; y; ωÞδðzÞ ; ð3:47Þ ϵðx; y; z; ωÞ ¼ e ϵ ðx; y; ωÞδðzÞ ¼ ϵ 0e χ ðx; y; ωÞδðzÞ þ i ω where e χþi ϵ ¼ ϵ 0e

e σ ; ω

ð3:48Þ

which is only meaningful on the graphene sheet at z ¼ 0 because (3.45) is only defined in the range that d=2 < z < d=2. Note that (3.48) differs from (3.46) by a constant term ϵ 0 d; nonetheless, they are consistent because (3.46) is applicable when the graphene sheet is considered to have a finite thickness d, whereas (3.48) is applicable when the graphene sheet is considered to be a true 2D material without thickness in the limit that d → 0. In practice, there is no real difference between the two seemingly different expressions because the thickness of a graphene sheet is many orders of magnitude smaller than all other physical dimensions, including the surface dimensions of the graphene sheet and the optical wavelengths. Because of the small thickness of a graphene sheet compared to the wavelength of an optical field, we shall see that the physical quantities, such as reflection and transmission through graphene, obtained using the permittivity tensor ϵ of the 3D model given by (3.39) and those obtained using ϵ of the 2D model given by (3.47) are quantitatively in agreement. As defined in (2.7) for the 2D surface current density, we have the 2D surface optical current density in the frequency domain: e Jðx; y; ωÞ ¼ e σ ðx; y; ωÞ  Eðx; y; ωÞ at z ¼ 0

ð3:49Þ

at an optical frequency ω, with J ¼ e J=d in the 3D model considering a finite thickness d for a graphene sheet or J ¼ e JδðzÞ in the 2D model assuming that the graphene sheet has e given in (3.38) in the 3D model no thickness. Similarly, we can use the relation P ¼ P=d e or the relation P ¼ PδðzÞ in the 2D model to define the 2D surface polarization density as e ðx; y; ωÞ ¼ ϵ 0 e P χ ðx; y; ωÞ  Eðx; y; ωÞ at z ¼ 0 in the frequency domain at an optical frequency ω.

(3.50)

76

Optical Properties

Table 3.1 Units of current density J, conductivity σ, polarization density P, susceptibility χ, and permittivity ϵ. A tilde is added for the 2D quantities. Physical quantities 3D quantities for the 3D model 2

J½A m  ¼ σ  E (3.36) 2

1

σ½S m 

P½C m  ¼ ϵ0 χ  E (3.36) χ½1 σ ϵ ¼ ϵ0 ð1 þ χÞ þ i (3.37) ϵ½F m1  ω

2D quantities for the 2D model

Contributed by

e J½A m  ¼ e σ  E (3.49) 1 e χ  E (3.50) P½C m  ¼ ϵ0e e σ e χ þ i (3.48) ϵ ¼ ϵ 0e ω

e σ ½S

Free electrons (Chapter 2)

e χ ½m

Bound electrons (Chapter 3)

e ϵ ½F 

Both bound and free electrons

1

The units of various physical quantities are given in Table 3.1. In the following sections, we first discuss and derive the optical conductivity e σ ∥. The susceptibilities e χ∥ and e χ ⊥ are then discussed. Finally, e ϵ and ϵ are discussed in Sections 3.6 and 3.7.

3.3

Optical Transitions in Graphene The optical properties of graphene are distinctively different from those of semiconductors and metals because intrinsic graphene has a linear band structure without a bandgap. The optical properties of graphene are frequently characterized by an optical conductivity, particularly in the far-infrared and terahertz spectral regions. As in the case of the DC conductivity discussed in the preceding chapter, the optical conductivity of graphene is also a strong function of temperature and Fermi energy. Depending on the stacking order among different layers, multilayer graphene can have an optical conductivity that differs in many characteristics from that of monolayer graphene. In this section, we start from the discussion of optical absorption to distinguish between two absorption processes that respectively govern the low-frequency and high-frequency optical properties of graphene. In Section 3.4 we consider the Hamiltonian of graphene in the presence of an external electromagnetic field, which governs the interaction of an electron in graphene with the electromagnetic field. We then use two different approaches in Section 3.5 to quantitatively characterize the optical properties of monolayer graphene in terms of the optical conductivity, followed by a derivation of the optical conductivity of multilayer graphene in Section 3.6. The permittivity of graphene is discussed in Section 3.7, and the absorbance of graphene is then calculated in Section 3.8. Before we calculate the optical response of graphene, it is important to distinguish two types of optical absorption processes that take place in graphene. When light is incident on a sheet of graphene, there is a probability that an electron absorbs a photon and makes a transition to a state of higher energy. For an intraband absorption process, the initial and final states of the electron before and after it absorbs a photon are in the same band; for an interband absorption process, the two states are in different bands. The interband and intraband absorption processes are shown in Figure 3.1 for carriers on the Dirac cone of monolayer graphene.

3.3 Optical Transitions in Graphene

(a)

77

(b)

T

0

T

0

μ = EF

Photon

Photon

μ = EF

(c)

(d) T >0

T >0

μ < EF

Photon

Photon

μ

Figure 3.1 Possible carrier transitions in monolayer graphene in different situations: (a) T → 0 with μ > 0, (b) T → 0 with μ < 0, (c) T > 0 with μ > 0, and (d) T > 0 with μ < 0. The grayscale gradient in the energy bands in (c) and (d) exaggerates the carrier distributions.

At a very low temperature such that T → 0, so that μ ≈ EF as obtained from (1.40), all states below μ are filled with electrons; in this situation, an intraband or interband optical transition of an electron can only take place from an occupied state below μ to an empty state above μ by absorbing the photon energy, as shown in Figure 3.1(a). Because the speed of light is much higher than the Fermi velocity of the carriers in graphene, i.e., c ≫ v F , and because graphene has a light-like energy–moment relationship, the momentum ℏω=c of a photon that has an angular frequency of ω is negligibly small compared to the momentum E=vF of a carrier in graphene. For an interband transition, the linear band structure of the Dirac cone allows the transition to take place with negligible change of momentum; therefore, this transition does not need the assistance

78

Optical Properties

of any scattering process for momentum conservation. For this reason, an interband transition is usually represented by a vertical line connecting a state in the valence band and a state in the conduction band. By contrast, for an intraband transition to happen, a scattering process, such as phonon scattering or impurity scattering, has to be involved to provide the necessary momentum change, as discussed in Chapter 2. In the case of n-type doping so that the chemical potential μ lies in the conduction band, as shown in Figure 3.1(a), an electron in the conduction band can be scattered from a state below μ to a state above μ by absorbing a photon to complete an intraband transition in the conduction band, whereas the electrons in the valence band are forbidden to make intraband transitions because the valence band is fully occupied. By contrast, in the case of p-type doping, intraband transitions occur in the valence band but not in the conduction band because the chemical potential μ lies in the valence band, as shown in Figure 3.1(b). At a finite temperature T > 0 in the case of n-type doping, some electrons in the valence band near the Dirac point are thermally excited to the conduction band so that the valence band is not fully occupied. As a result, intraband transitions become possible within the valence band, as shown in Figure 3.1(c). The carrier distribution is described by the Fermi–Dirac function with a chemical potential μ that is smaller than EF , μ < EF , as determined by (1.40). In a state at the energy level of μ, the probability of finding an electron is 0.5. A state above μ still has a finite probability to be occupied, though it is more likely to be empty, whereas one below μ still has a finite probably to be empty though it is more likely to be occupied. Therefore, as shown in Figure 3.1(c), it is possible for an intraband transition in the conduction band to take place above μ, and it is also possible for an electron to be promoted from the valence band to an empty state below μ in the conduction band via interband transition. These transitions are significant at high temperatures but become insignificant as the temperature approaches the lowtemperature limit that jμj ≫ k B T. In the case of p-type doping, the states in the valence band at energy levels above μ and those in the conduction band are not completely empty, though they are sparsely occupied; therefore, they can be involved in intraband transitions as well as in interband transitions. However, because of the limited number of electrons in the conduction band, most intraband transitions take place in the valence band, as shown in Figure 3.1(d). Usually, an interband transition requires a higher photon energy than an intraband transition, as can be seen in Figure 3.1. In the visible spectral region, the optical response of graphene is predominately determined by interband absorption, whereas intraband absorption is insignificant and often can be ignored. In the infrared spectral region, both interband and intraband absorption processes are important; thus neither can be ignored. In the far-infrared and terahertz spectral regions, interband absorption is inefficient because of the low photon energy; consequently, the optical response of graphene in these spectral regions is predominately determined by intraband absorption. Finally, in the DC limit that ω → 0, the response of graphene is solely contributed by the intraband scattering of charge carriers in graphene, as discussed in Chapter 2. The high-frequency optical

3.4 Hamiltonian of Graphene

79

response is discussed in this chapter by considering only the interband absorption. The low-frequency response is covered in Chapter 4.

Hamiltonian of Graphene in an Electromagnetic Field Consider an electromagnetic field with an electric field E and an accompanying magnetic field B, which are governed by Maxwell’s equations. The electric and magnetic fields can be generally expressed in terms of a vector potential A and a scalar potential φ as ∂A  φ; ∂t

Δ

E¼

B¼

Δ

 A:

ð3:51Þ

Δ

3.4

We take the Coulomb gauge such that  A ¼ 0 for the vector potential A, and ∇2 φ ¼ ρ for the scalar potential φ. The Hamiltonian of graphene under the influence of this external electromagnetic field can then be expressed as H ¼ vF jp þ eAj  eφ;

ð3:52Þ

where p is the canonical momentum of an electron carrying a charge of e in graphene. Note that because the Hamiltonian must be a real quantity, the vector potential A that appears in the Hamiltonian given in (3.52) is the real field A ¼A þ A, not the complex field A, and the scalar potential φ is the real potential φ ¼ φ þ φ, not the complex potential φ. To see the physical implication of (3.52), we resort to Hamilton’s equations: ∂H dx ¼ ; ∂px dt ∂H dpx ¼ ; ∂x dt

∂H dy ¼ ; ∂py dt

ð3:53Þ

dpy ∂H ¼ ; ∂y dt

ð3:54Þ

assuming that the graphene sheet is on the xy plane. By identifying dx kx ¼ vx ¼ vF ; dt jkj

ð3:55Þ

ky dy ¼ vy ¼ vF ; dt jkj

ð3:56Þ

we find from (3.52) and (3.53) the kinetic momentum of an electron in graphene in the presence of an electromagnetic field: ℏk ¼ p þ eA:

ð3:57Þ

By inserting the relation p ¼ ℏk  eA, obtained from (3.57), into (3.54) to take the time derivative of p while using the Hamiltonian given in (3.52), we obtain

Optical Properties

  dk dA ¼e ðv  ÞA þ φ ev ð  AÞ dt dt ∂A þ e φ  evð  AÞ ¼e ∂t ¼ eðE þ v  BÞ;

Δ

Δ

Δ

ℏ

Δ

80

ð3:58Þ

Δ

which is the classical Lorentz force, F ¼ eðE þ v  BÞ, acting on an electron. Therefore, the Hamiltonian given in (3.52) has the implication of the classical Newton’s law. The Hamiltonian operator of graphene in an electromagnetic field is obtained by replacing the relation ℏk ¼ p used to obtain (1.31) in the absence of the electromagnetic field with the relation ℏk ¼ p þ eA given in (3.57) for the kinetic momentum of an electron subject to the force of the electromagnetic field: ^ ¼ v F σ  ðp þ eAÞ  eφ; H

ð3:59Þ

where σ is the Pauli vector displayed in (1.33) but is not the conductivity tensor described in Chapter 2 and in Sections 3.1 and 3.2. Therefore, the interaction Hamiltonian of graphene describing the perturbation caused by the electromagnetic field is H^0 ¼ vF eσ  A  eφ ¼ v F eσ  ðA þ AÞ  eðφ þ φÞ;

ð3:60Þ

where A and φ are respectively the real vector and scalar potential fields, and A and φ are respectively the complex vector and scalar potential fields defined through a relation of the form given in (3.13). It is important to remember that the vector and scalar potential fields that appear in (3.59) and (3.60) have to be the real potential fields A and φ because ^ and H^0 are Hermitian operators. both H

3.5

Optical Conductivity of Monolayer Graphene Two approaches for the derivation of the optical conductivity are given in this section. The optical conductivity derived using Fermi’s golden rule is straightforward, without much mathematics involved. However, to include the finite lifetime of each energy level as well as the dispersion in the momentum space, as discussed later in Chapter 4, it is necessary to carry out the calculation using density matrix, which gives us a general result that can be used to deduce various physical quantities. In the following, we discuss both approaches in detail. We consider a plane electromagnetic field that is far away from its source such that ρ ¼ ρtotal ¼ 0, as discussed in Section 3.1; then, the scalar potential can be taken to be φ ¼ 0. In this case, the electric and magnetic fields are simply related to the vector potential A as

3.5 Optical Conductivity of Monolayer Graphene

∂A ; ∂t

B¼

Δ

E¼

 A:

81

ð3:61Þ

Then the Hamiltonian operator of graphene in an electromagnetic field given in (3.59) is reduced to the form: ^ ¼ vF σ  ðp þ eAÞ; H

ð3:62Þ

while the interaction Hamiltonian of graphene given in (3.60) is reduced to the form: H^0 ¼ v F eσ  A ¼ vF eσ  ðA þ AÞ:

ð3:63Þ

These reduced forms with φ ¼ 0 are used in the following calculations of the optical conductivity of graphene.

3.5.1

Fermi’s Golden Rule Consider a monochromatic plane optical field that is polarized in the x direction and propagates with a wave vector k ¼ k^z : Eðr; tÞ ¼ Eðr; tÞ þ Eðr; tÞ ¼ ^x Eðeikziωt þ eikzþiωt Þ;

ð3:64Þ

where the phase of the field is chosen such that it has a real amplitude E. The field seen by a graphene sheet that is located at z ¼ 0 is EðtÞ ¼ Eðx; y; 0; tÞ ¼ ^x Eðeiωt þ eiωt Þ:

ð3:65Þ

We find from (3.61) the vector potential: AðtÞ ¼ i

E iωt ðe  eiωt Þ^x : ω

ð3:66Þ

Then, from (3.63), we have ^ ðeiωt  eiωt Þ; H^ 0 ðtÞ ¼ V where

2 0

^ ¼ i vF eE σ x ¼ 6 V 4 v eE F ω i ω

i

ð3:67Þ 3 v F eE ω 7 5: 0

ð3:68Þ

Consider two states jn; k〉 and jn0 ; k0 〉 in bands n and n0 , respectively, where n and n0 are the band indices. Fermi’s golden rule is used to determine the transition rate from state jn; k〉 to state jn0 ; k0 〉 under the perturbation of a harmonically time-varying field. By 0 ðtÞ with representing the interaction Hamiltonian given in (3.67) as H^0 ðtÞ ¼ H^0 ðtÞ  H^þ

82

Optical Properties

^ 0 ðtÞ ¼ V ^ eiωt ; H

ð3:69Þ

^ þ0 ðtÞ and H ^ 0 ðtÞ are the transition rates Rþ and R that are respectively contributed by H given by R ¼

2 2π   Vk0 k  δðEk0  Ek  ℏωÞ; ℏ

ð3:70Þ

where the plus or minus sign is chosen according to the sign of iωt in (3.69), ^ jk〉 is the matrix element of operator V ^ between jn0 ; k0 〉 and jn; k〉 states, Vk0 k ¼ 〈k0 jV and Ek ¼ nℏvF jkj

and

Ek0 ¼ n0 ℏvF jk0 j:

ð3:71Þ

In (3.70), (3.71), and the following, we drop the indices n and n0 in the state notations jk〉 and jk0 〉, and in the subscripts of Vk0 k , Ek , and Ek0 to simplify the mathematical expressions. Equation (3.70) is much like (2.23), except for the fact that (3.70) addresses a time-dependent but space-independent perturbation due to the interaction with plane-wave radiation, whereas (2.23) addresses a time-independent but space-dependent perturbation arising from the scattering potential. The transition rate R associated with δðEk0  Ek  ℏωÞ in (3.70) is for the interband absorption process with Ek0 ¼ Ek þ ℏω, as shown in Figure 3.2(a), whereas Rþ associated with δðEk0  Ek þ ℏωÞ is for the stimulated emission process with Ek0 ¼ Ek  ℏω, as shown in Figure 3.2(b). Depending on the initial state of the electron, either process can take place in graphene in the presence of an optical field, which is a real harmonic field that has both eiωt and eiωt terms as seen in (3.65).

(a)

(b)

Absorption k

Photon

Simulated emission k

Photon

k

k

Figure 3.2 (a) Absorption and (b) stimulated emission of a photon through optical transitions of an

electron between energy states jk〉 and jk0 〉.

3.5 Optical Conductivity of Monolayer Graphene

83

The matrix element Vk0 k is calculated from   ^ k〉 ¼ 1 Vk0 k ¼ 〈k0 V A

ðð

^ ψk dr; ψk†0V

ð3:72Þ

^ replacing the scattering potential energy Us ðrÞ. By which is similar to (2.24) with V ^ inserting V given in (3.68) and the eigenstates given in (1.28) into (3.72), we obtain Vk0 k , which in turn gives R from (3.70) as   2π vF eE 2 0  2 j〈k jσ x k〉 δðEk0  Ek  ℏωÞ; R ¼ ℏ ω 

ð3:73Þ

where j〈k0 jσ x jk〉j2 ¼

1 þ n0 n cos 2θk δk0 ;k : 2

ð3:74Þ

Again, the plus and minus signs in the Dirac delta function correspond to the eiωt and eiωt terms in (3.67), respectively, and θk ¼ tan1 ðky =kx Þ, as defined in Chapter 1. The Dirac delta function δðEk0  Ek  ℏωÞ in (3.73) imposes energy conservation, while the Kronecker delta function δk0 ;k in (3.74) manifests momentum conservation. To satisfy both requirements, n0 ¼ n is necessarily true so that n0 n ¼ 1, which is applicable to (3.74). For this reason, carrier transitions in graphene caused by interactions with radiation are necessarily interband transitions, as schematically illustrated in Figure 3.1. The total rate of energy loss for the radiation field is h io ℏω XXn  ðR  Rþ Þ f0 ðEk Þ 1  f0 ðEk0 Þ ; A n;n0 k;k0

ð3:75Þ

where A is the area of graphene that is illuminated by the radiation. The difference of R and Rþ is calculated rather than their sum because R is the rate of photon absorption, which results in optical loss, whereas Rþ is the rate of photon emission, which results in optical gain, as shown in Figure 3.2. This difference is then multiplied by f0 ðEk Þ½1  f0 ðEk0 Þ to account for the probability of simultaneously finding state jk〉 being occupied by an electron and state jk0 〉 being empty. The energy lost from the optical field is transferred to the carriers in graphene in the form of joule heat loss per unit area given by     Re E  J  d ¼ Re E  e J ¼ Re E  e σ  E ¼ 2E 2 e σ 0 ðωÞ; ð3:76Þ where the overhead bar means time average, d is the thickness of the graphene sheet so that J ¼ e J=d and e J ¼e σ  E from (3.75), e σ ðωÞ is the surface optical conductivity e σ∥ discussed in Section 3.2, and e σ 0 is the real part of e σ . Because e σ⊥ ¼ e σ z ¼ 0, as discussed in Sections 2.1 and 3.2, the subscript “ ∥ ” of e σ ∥ is dropped in (3.76) and in the following for simple mathematical expressions.

84

Optical Properties

Equating the two expressions for the net optical loss given in (3.75) and (3.76), we have the identity 2E 2 e σ 0 ðωÞ ¼

h io ℏω XXn  ðR  Rþ Þ f0 ðEk Þ 1  f0 ðEk0 Þ : A n;n0 k;k0

ð3:77Þ

Because of the delta functions in (3.73) and (3.74), most of the terms can be eliminated from the summation in (3.77). For example, the absorption term R only exists when the initial state is in the valence band, thus n ¼ 1, and the final state is in the conduction band, thus n0 ¼ 1; likewise, the emission term Rþ only exists for n ¼ 1 and n0 ¼ 1. Intraband transitions with n ¼ n0 are not allowed and thus have no contribution to the optical conductivity expressed in (3.77). By inserting (3.73) into (3.77), followed by some simplification, we find the real part of the surface optical conductivity: ð∞ 2ðπ h i g  v F e 2 e σ ðωÞ ¼ ð1  cos 2θk Þ f0 ðEÞ  f0 ðEÞ δð2E  ℏωÞkdθk dk 2πω 2 0 0      ð3:78Þ ℏω ℏω ¼e σ 0 f0   f0 ; 2 2 0

where e σ 0 ¼ e2 =4ℏ, E ¼ ℏvF k, and g ¼ 4 is the total degeneracy due to the spin and valley degeneracies. Note that e σ 0 ¼ e2 =4ℏ ≈ 60:8 μS is the high-frequency surface optical conductivity of monolayer graphene. The imaginary part of e σ ðωÞ is obtained using the Kramers–Kronig relation: ð∞ 2ω Re e σ ðω0 Þ 0 e σ ðωÞ ¼  P 02 dω ; π ω  ω2 00

ð3:79Þ

0

where P stands for the principal value of the integral. In the limit that T → 0, (3.78) becomes a step function; then (3.79) gives   e σ 0  2jμj þ ℏω  e σ ðωÞ ¼  ln : π  2jμj  ℏω  00

Combining (3.78) and (3.80), we have, in the limit that T → 0,   i  2jμj þ ℏω  e σ ðωÞ ¼ e σ 0 Hðℏω  2jμjÞ  ln  ; π 2jμj  ℏω

ð3:80Þ

ð3:81Þ

where HðxÞ is the Heaviside step function: HðxÞ ¼ 0 for x < 0 and HðxÞ ¼ 1 for x ≥ 0. Because ln ðxÞ ¼ ln x þ iπ for x > 0, (3.81) can be reduced to a concise form: e σ ðωÞ ¼ 

ie σ 0 2jμj þ ℏω ln 2jμj  ℏω π

in the limit that T → 0:

ð3:82Þ

3.5 Optical Conductivity of Monolayer Graphene

(a)

(b)

1

1

( )

( ) 0

Optical conductivity

Optical conductivity

0

( )

( ) 0

1

( )

2 T = 100 K 3

0

1

2 EF

3

85

0

1

( )

2 T = 300 K 3

0

1

2

3

EF

Figure 3.3 Surface optical conductivity of graphene contributed by interband carrier transitions at two different temperatures: (a) T ¼ 100 K and (b) T ¼ 300 K. For both figures, the dashed curves are numerically calculated from (3.78) and (3.79), whereas the solid curves are obtained from (3.82) for the low-temperature limit.

The surface optical conductivity in the low-temperature limit given by (3.82) and the numerical solutions of (3.78) and (3.79) for the temperatures of 100 K and 300 K are plotted in Figure 3.3 for EF ¼ 100 meV. The minimum of e σ 00 ðωÞ is located at ℏω ¼ 2μ, which is lower than 2EF ¼ 200 meV because μ < EF for T ≠ 0 in the case of a positive EF ; the difference between μ and EF increases with increasing temperature, as determined by (1.40). It can be seen that the singularity of e σ 00 ðωÞ and the discontinuity of e σ 0 ðωÞ, which appear at ℏω ¼ 2EF as found from (3.82) in the limit that T → 0, are not seen in the numerical results calculated for nonzero temperatures. When the temperature is low such that jμj ≫ kB T, the analytical result given by (3.82) serves as a good approximation, as seen in Figure 3.3(a) for T ¼ 100 K, but the approximation deteriorates as the temperature increases because the condition jμj ≫ kB T is invalid, as seen in Figure 3.3(b) for T ¼ 300 K. Spectral broadening due to the finite lifetimes of the energy levels is ignored for the characteristics shown in Figure 3.3. As discussed in the following, to account for spectral broadening, we shall adopt a more general approach using the density matrix, which is a powerful tool in the calculation of various physical observables of a system.

3.5.2

Density Matrix Method As in the preceding derivation, we assume that the electric field is polarized along the x direction without loss of generality. However, instead of the monochromatic field with only one frequency component given in (3.65), we shall consider a more general form of the electric field given by

86

Optical Properties

EðtÞ ¼ EðtÞ þ EðtÞ ¼ EðtÞ^x þ EðtÞ^x ð∞ ð∞ 1 1 iωt EðωÞe dω^x þ EðωÞeiωt dω^x ¼ ; 2π 2π 0

1 ¼ 2π

ð3:83Þ

0

ð∞

EðωÞeiωt dω^x ;

∞

where EðωÞ is the Fourier component of the complex electric field EðtÞ at the frequency ω and EðωÞ ¼ EðωÞ. From (3.61), we find that i AðtÞ ¼  2π

ð∞ ∞

EðωÞ iωt e dω^x : ω

ð3:84Þ

We can multiply the integrand in (3.84) by an exponential term expðγtÞ, with γ being an infinitesimal positive number so that the electric field, i.e., the perturbation, is turned off at time t → ∞ and is adiabatically turned on as time evolves. Then, from (3.63), we have H^ 0 ðtÞ

∞ ^ ð EðωÞ V eiðωþiγÞt dω; ¼ 2π ω

ð3:85Þ

∞

where ^ ¼ ivF eσ x ¼ V



0 iv F e

 iv F e : 0

ð3:86Þ

^ 0 is ^ ¼H ^ 0 þ H^ 0 , where H For a system characterized by a Hamiltonian of the form H ^ completely solved such as in our case that H 0 ¼ v F σ  p, the interaction picture is often adopted especially in a problem involving time-dependent perturbation from an interaction Hamiltonian such as H^0 ðtÞ given in (3.85). In the interaction picture, instead of ^ ðtÞ of the problem under consideration is defined as (3.86), the operator V ^ eiH^ 0 t=ℏ ; ^ ðtÞ ¼ eiH^ 0 t=ℏ V V

ð3:87Þ

^ is that given in (3.86). In the interaction picture, the interaction Hamiltonian where V 0 ^ H ðtÞ is then given as ∞ ^ ðtÞ ð EðωÞ V eiðωþiγÞt dω ð3:88Þ H^ 0 ðtÞ ¼ 2π ω ∞

instead of that given in (3.85). Therefore, the matrix element of H^ 0 ðtÞ is

3.5 Optical Conductivity of Monolayer Graphene

V 〈k jH^0 ðtÞjk〉 ¼ k k 2π 0

0

ð∞ ∞

EðωÞ iðωk0 ωk ωiγÞt dω; e ω

87

ð3:89Þ

where ωk ¼ Ek =ℏ, ωk0 ¼ Ek0 =ℏ, and Vk0 k is obtained from (3.86):  ^ jk〉 ¼ iv F e〈k0 jσx jk〉: Vk0 k ¼ 〈k0 V

ð3:90Þ

A density matrix describes the state of a quantum system. In the interaction picture, the density matrix operator is defined as ^ρ ðtÞ ¼

XX n0 ;n

^

^

ρk0 k ðtÞeiH0 t=ℏ j k0 〉〈k j eiH 0 t=ℏ ;

ð3:91Þ

0

k ;k

where ρk0 k ðtÞ is the matrix element of ^ρ ðtÞ. As before, the subscripts n and n0 of the matrix elements and the eigenstates have been dropped to simplify the mathematical expressions. The time evolution of the density matrix in the interaction picture is governed by the Liouville–von Neumann equation as  ∂ i ^0 ^ρ ðtÞ ¼  H ðtÞ; ^ρ ðtÞ : ∂t ℏ

ð3:92Þ

^ 0ðtÞ is If we assume that the perturbation from the interaction Hamiltonian H sufficiently weak that the carrier distribution in the Dirac cone does not deviate much from its thermal equilibrium distribution in the absence of the electric field, we can approximate ^ρ ðtÞ through perturbation expansion only to the first order as the sum of the thermal equilibrium distribution ^ρ ð0Þ and a much smaller first-order timedependent perturbation term ^ρ ð1Þ ðtÞ that accounts for the deviation from ^ρ ð0Þ . Then, we can write ^ρ ðtÞ ¼ ^ρ ð0Þ þ ^ρ ð1Þ ðtÞ

ð3:93Þ

for the first-order perturbation by dropping the high-order perturbation terms ^ρ ð2Þ , ^ρ ð3Þ , and so on. Then, from (3.92), we have i ∂ ð1Þ ih ^ρ ðtÞ ¼  H^ 0 ðtÞ; ^ρ ð0Þ ∂t ℏ

ð3:94Þ

by dropping the H^ 0 ðtÞ^ρ ð1Þ ðtÞ and ^ρ ð1Þ ðtÞH^ 0 ðtÞ terms because they are of the second order and thus are much smaller than the first-order terms that are kept in (3.94). Further perturbation expansion of (3.92) is discussed in Chapter 5 for the nonlinear response of graphene. Because ^ρ ð0Þ represents the unperturbed thermal equilibrium distribution f0 , it is a diagonal matrix that has matrix elements given as

Optical Properties

〈k0 j^ρ ð0Þ jk〉 ¼ f0 ðEk Þδjk0 〉;jk〉 ¼ f0 ðEk Þδk0 ;k δn0 ;n ;

ð3:95Þ

where f0 ðEk Þ is given by (1.37) and Ek is given by (3.71). The off-diagonal matrix elements for jk〉 ≠ jk0 〉, for which k ≠ k0 and/or n ≠ n0, are zero because different states of graphene do not mix without perturbation. By using (3.89), (3.91), and (3.95), the matrix element of ^ρ ð1Þ ðtÞ is found from (3.94) as i ∂ 0 ð1Þ i X h 0 ^0 〈k j^ρ ðtÞjk〉 ¼  〈k jH ðr; tÞjk00 〉〈k00 j^ρ ð0Þ jk〉  〈k0 j^ρ ð0Þ jk00 〉〈k00 jH^ 0 ðr; tÞjk〉 ∂t ℏ n00 ;k00 i ih ¼  〈k0 jH^ 0 ðr; tÞjk〉〈kj^ρ ð0Þ jk〉  〈k0 j^ρ ð0Þ jk0 〉〈k0 jH^0 ðr; tÞjk〉 ð3:96Þ ℏ ð∞ iV 0 EðωÞ iðωk0 ωk ωiγÞt e ¼  k k f0 ðEk Þ  f0 ðEk0 Þ dω: ω 2πℏ

½



∞

By integrating (3.96) over time from ∞, when the perturbation is turned on, to t, we obtain

½

V0 〈k j^ρ ðtÞjk〉 ¼  k k f0 ðEk Þ  f0 ðEk0 Þ 2πℏ 0

ð1Þ



ð∞ ∞

EðωÞeiðωk0 ωk ωÞt dω; ð3:97Þ ωðωk0  ωk  ω  iγÞ

which clearly indicates that 〈k0 j^ρ ð1Þ ðtÞjk〉 ≠ 0 for jk0 〉 ≠ jk〉, and 〈kj^ρ ð1Þ ðtÞjk〉 ¼ 0 for jk0 〉 ¼ jk〉. Therefore, ^ρ ð1Þ ðtÞ only have nonzero off-diagonal elements; all of its diagonal elements are zero. In (3.97), the infinitesimal parameter γ is dropped in the exponential function in the numerator after serving its purpose for the integration over time to converge, but it is kept in the denominator to account for the effect of a finite relaxation rate on the spectral linewidth. The surface current density flows in the x direction because the optical field is polarized in the x direction. It is given by

½



½



e e Je ðtÞ ¼  Tr ^ρ ðtÞ^ v ðtÞ ¼  Tr ^ρ ðtÞ^v x ðtÞ ^x ; A A

ð3:98Þ

where A is the surface area of the graphene sheet, Tr½ represents the trace of the matrix in the brackets, and the velocity operator in the x direction ^v x ðtÞ is the x component of v^ðtÞ. Note that the surface current density Je as obtained in (3.98) from the density matrix calculation is a real vector field, not a complex vector field because quantum mechanical observables are real quantities. The velocity operator v^ can be found from (3.53) and (3.59):

Δ

v^ ¼

^ ¼ vF

pH

Δ

88

p ðσ

 pÞ ¼ v F σ:

Thus the x component of v^ðtÞ in the interaction picture is

ð3:99Þ

3.5 Optical Conductivity of Monolayer Graphene

^v x ðtÞ ¼ eiH0 t=ℏ vF σ x eiH0 t=ℏ :

89

ð3:100Þ

By using (3.99) and the relations 〈kjσ x jk〉 ¼ n cos θk ¼ nk x =k and 〈kjσ y jk〉 ¼ n sin θk ¼ nk y =k, it can be easily shown that 〈kj^ v jk〉 ¼ nvF k=k, as expected of the carrier velocity on the Dirac cone of graphene from the discussion in Section 1.5. From (3.97), (3.98), and (3.100), we obtain eX Je ðtÞ ¼  〈kj^ρ ðtÞ^v x ðtÞjk〉^x A n;k eX ¼ 〈kj^ρ ð1Þ ðtÞ^v x ðtÞjk〉^x A n;k ð3:101Þ e XX ð1Þ 0 0 〈kj^ρ ðtÞjk 〉〈k j^ v x ðtÞjk〉^x ¼ A n;n0 k;k0 8 9 # ð∞ " = iωt i e2 v2F XX< EðωÞe ¼ dω ^x ; j〈kjσ x jk0 〉j2 f0 ðEk0 Þf0 ðEk Þ A 2π n;n0 k;k0 : ωðEk Ek0 ℏωiℏγÞ ; ∞

where Vk0 k that appears in (3.97) has been replaced by its expression given in (3.90). The second line in (3.101) tells us that the surface current density is contributed by ^ρ ð1Þ. It can be seen that ^ρ ð0Þ does not contribute to (3.101) because there is no current in the absence of an applied electric field. By identifying Je ðtÞ as an integral of its Fourier components: 1 Je ðtÞ ¼ 2π

ð∞

e J ðωÞeiωt dω^x

ð3:102Þ

∞

in a manner similar to that done in (3.83) for EðtÞ, we obtain, from (3.101),  XX1 þ n0 n cos 2θk f0 ðEk0 Þ  f0 ðEk Þ i e2 v 2F e 0 EðωÞ δ J ðωÞ ¼  ; A ω Ek  Ek0  ℏω  iℏγ k;k 2 n;n0 k;k0 ð3:103Þ where (3.74) has been used for j〈kjσx jk0 〉j2 in (3.101). Because of the δk;k0 and f0 ðEk0 Þ  f0 ðEk Þ terms, the summation in (3.103) is meaningful only if k0 ¼ k and Ek0 ≠ Ek , i.e., only for interband transitions with n0 ¼ n and Ek0 ¼ Ek , where Ek ¼ nℏv F jkj and Ek0 ¼ n0 ℏvF jk0 j, as given in (3.71). Using the relation that e J ðωÞ ¼ e σ ðωÞEðωÞ, the surface optical conductivity can be found from (3.103):   i e2 v2F X 1  cos 2θk f0 ðEk Þ  f0 ðEk Þ e σ ðωÞ ¼  : A ω n;k 2Ek  ℏω  iℏγ 2 We first consider the limit that γ → 0. By using the identity that

ð3:104Þ

90

Optical Properties

  1 lim Im ¼ πδðxÞ; a→0 x  ia

ð3:105Þ

the real part e σ 0 ðωÞ of the surface optical conductivity e σ ðωÞ in the limit that γ → 0 can be found from (3.104) as e σ 0 ðωÞ ¼

h i o π e2 v2F Xn ð1  cos 2θk Þ f0 ðEk Þ  f0 ðEk Þ δð2Ek  ℏωÞ : ð3:106Þ A 2ω n;k

By inserting (3.73) and (3.74) into (3.77), it can be easily shown that (3.106) is identical to (3.77). Therefore, the real part e σ 0 ðωÞ can be calculated as done in (3.78), and the 00 imaginary part e σ ðωÞ can be found using (3.79). The result in the limit that γ → 0 is thus exactly the same as that obtained in the preceding subsection by using Fermi’s golden rule. Without taking any limit on γ, we can find e σ ðωÞ from (3.104) by following the same 0 procedure used in (3.78) to calculate e σ ðωÞ from (3.77): e2 v2F g X e σ ðωÞ ¼ i ω 4π2 n

0 0

1  cos 2θk f0 ðnEÞ  f0 ðnEÞ kdθk dk 2nE  ℏω  iℏγ 2

 f0 ðEÞ  f0 ðEÞ f0 ðEÞ  f0 ðEÞ þ EdE 2E  ℏω  iℏγ 2E  ℏω  iℏγ 0 " # ð∞ e2 g f0 ðEÞ  f0 ðEÞ 2 4E ¼ i 2 dE ℏ ω 4π ð2EÞ2  ðℏω þ iℏγÞ2 0 ð∞ e2 ω f0 ðEÞ  f0 ðEÞ ≈i dE; π ð2EÞ2  ðℏω þ iℏγÞ2 e

2

g ¼ i 2 4π ℏ ω

ð∞

2ðπð ∞

ð3:107Þ

0

where g ¼ 4 is the total degeneracy due to the spin and valley degeneracies, E ¼ ℏvF k > 0 in the integrand because the integral is taken over the range k > 0, and the identity that 4E2 ¼ ðℏωÞ2 for resonant transitions is used in the final step to pull 4E2 out of the integral. Therefore, ð∞ e2 ω f0 ðEÞ  f0 ðEÞ dE: e σ ðωÞ ¼ i π ð2EÞ2  ðℏω þ iℏγÞ2

ð3:108Þ

0

For γ ¼ 0, it can be shown that the principal value of (3.108) gives the same result as that given in (3.79) for e σ 00 ðωÞ, whereas e σ 0 ðωÞ given in (3.78) can be retrieved from (3.108) by utilizing the identity in (3.105). For a finite value of γ, the singularity at 2E ¼ ℏω in (3.108) is removed; then, an analytical solution of the integral in (3.108) can be found in the low-temperature limit.

3.6 Optical Conductivity of Multilayer Graphene

91

Optical conductivity

0

( )

1

( ) 0

1

( )

2

3 0

1

2

3

EF Figure 3.4 Surface optical conductivity of graphene contributed by interband carrier transitions. The dashed curves are calculated using (3.109) with γ ¼ 10 ps1 , whereas the solid curves are obtained from (3.82) for the limit γ → 0.

Following a procedure similar to that taken to obtain (3.82), we find that (3.108) reduces to e σ ðωÞ ¼ 

ie σ 0 2jμj þ ℏðω þ iγÞ ln 2jμj  ℏðω þ iγÞ π

in the limit that T → 0;

ð3:109Þ

where e σ 0 ¼ e2 =4ℏ, as defined in (3.78). The expression in (3.109) gives both the real and imaginary parts of e σ ðωÞ. Apparently, (3.109) is identical to (3.82) in the limit that γ → 0. The surface optical conductivities obtained from (3.109) with γ ¼ 10 ps1 and from (3.82) for the limit γ → 0 are plotted in Figure 3.4. As can be seen, the spectrum of interband carrier transitions at a photon energy of ℏω ¼ 2EF is broadened by a finite value of γ. In Chapter 4 we show that γ has the physical meaning of the infinitesimal scattering rate of carriers that accounts for the spectral broadening; the inverse of γ gives the finite lifetime of the carriers, which can be related to the relaxation time τðEÞ discussed in Chapter 2.

3.6

Optical Conductivity of Multilayer Graphene Monolayer graphene has only two bands: one conduction band and one valence band. The optical conductivity in the high-frequency region considered in this chapter is derived by considering the interband carrier transitions between these two bands while ignoring intraband transitions. By comparison, the optical conductivity of multilayer graphene is more difficult to calculate because interband transitions can take place not only between conduction and valence bands, but also between different conduction bands or between different valence bands. Therefore, the characteristics of the optical conductivity of multilayer graphene are more complicated than those of monolayer graphene shown in Figure 3.3.

92

Optical Properties

In principle, the optical conductivity of multilayer graphene can be derived by following the same procedure taken in the preceding section as long as the interaction Hamiltonian can be identified. In the following, the optical conductivities of AB-stacked bilayer graphene and AA-stacked bilayer graphene are discussed.

3.6.1

AB-Stacked Bilayer Graphene AB-stacked bilayer graphene has two conduction bands and two valence bands, as shown in Figure 3.5. To distinguish different bands, the bottom valence band is labeled n ¼ 2 with energy E2;k for the state j  2; k〉 of wave vector k in this band, the top valence band n ¼ 1 with energy E1;k for the state j  1; k〉 of wave vector k in this band, the bottom conduction n ¼ 1 with energy E1;k for the state j1; k〉 of wave vector k in this band, and the top conduction band n ¼ 2 with energy E2;k for the state j2; k〉 of wave vector k in this band. Because of the symmetry of the conduction and valence bands with respect to the Dirac point, we have E2;k ¼ E2;k and E1;k ¼ E1;k by taking E ¼ 0 at the Dirac point. To calculate the optical conductivity contributed by interband transitions, all possible transitions among these bands have to be considered. Due to the complexity of the derivation for this optical conductivity, the result is given below directly without detailed derivation. The real part of the surface optical conductivity in the limit T → 0 is [2]  ℏω þ 2γ1 0 e σ ðωÞ ¼ e σ0 Hðℏω  2jμjÞ ℏω þ γ1 ℏω  2γ1 þ Hðℏω  2γ1 ÞHðℏω  2jμjÞ ℏω  γ1 γ2 ð3:110Þ þ 1 2 Hðℏω  2jμj  γ1 Þ ðℏωÞ γ2 þ 1 2 Hðℏω  2jμj þ γ1 ÞHðℏω  γ1 Þ ðℏωÞ   2jμj þ γ1 2jμj  γ1 þγ1 ln  ln Hðjμj  γ1 Þ δðℏω  γ1 Þ ; γ1 γ1

g

where e σ 0 ¼ e2 =4ℏ, as defined in (3.78), and γ1 is the interlayer interaction energy of ABstacked graphene. The imaginary part, e σ 00 ðωÞ, can be found by using the Kramers– Kronig relation expressed in (3.79) [3]. Because there are five different interband transitions, (3.110) has five terms, which in the case of n-type doping correspond to the transitions of E1;k → E1;k , E2;k → E2;k , E2;k → E1;k , E1;k → E2;k , and E1;k → E2;k in the sequence from the first to the last term, and in the case of p-type doping correspond to the transitions of E1;k → E1;k , E2;k → E2;k , E1;k → E2;k , E2;k → E1;k , and E2;k → E1;k . Examples of these transitions are marked from 1 to 5, as shown in the left panels of Figure 3.5. In the case of undoped intrinsic AB-stacked graphene with μ ¼ 0, as shown in Figure 3.5(a), the valence band is fully occupied while the conduction band is empty. The E1;k → E1;k transition is always possible for any frequency. Therefore, the surface

3.6 Optical Conductivity of Multilayer Graphene

4

(a)

93

4

E2

E

0

E1

0

E

1

2 0

1

−2 E −4 −0.2

−0.1

4

(b)

0 ka0

0.1

2

−3

0.2

0

1

2

3

4

3

4

3

4

1

4

0

0

0

E

1

2

−2 −4 −0.2

−0.1

4

(c)

0 ka0

0.1

−3

0.2

0

1

2 1

4

0

0

0

E

1

2

−2 −4 −0.2

−0.1

0 ka0

0.1

0.2

−3

0

1

2 1

Figure 3.5 Possible optical transitions (left panels) and surface optical conductivity (right panels) of AB-stacked bilayer graphene at a temperature T → 0 for five different chemical potentials: (a) μ ¼ 0, (b) μ ¼ 0:2γ1 , (c) μ ¼ 1:2γ1 , (d) μ ¼ 0:2γ1 , and (e) μ ¼ 1:2γ1 . The carrier filling is indicated by the gray area below the chemical potential. The optical conductivity is normalized to e σ 0 ¼ e2 =4ℏ. Different numbers indicate the interband transitions between different pairs of bands; the onset of each transition is marked by the corresponding number in the spectrum of the surface optical conductivity. In each right panel, the real part e σ 0 and the imaginary part e σ 00 of the surface optical conductivity are plotted as the solid and dashed curves, respectively, and the arrow represents the Dirac delta function corresponding to the last term in (3.110).

optical conductivity is nonzero across the whole spectrum. As the frequency increases such that ℏω > γ1 , the E2;k → E1;k and E1;k → E2;k transitions are now possible, represented by a kink at ℏω ¼ γ1 in the spectrum of the surface optical conductivity. For ℏω > 2γ1, the E2;k → E2;k transition also becomes possible, represented by another

Optical Properties

4

(d)

4

0

0

0

E

1

2

−2 −4 −0.2

−0.1

4

(e)

0 ka0

0.1

−3

0.2

0

1

2

3

4

3

4

1

4

1

2 0

0

0

E

94

−2 −4 −0.2

−0.1

0 ka0

0.1

0.2

−3

0

1

2 1

Figure 3.5 (cont.)

kink at ℏω ¼ 2γ1 . The E1;k → E2;k transition is forbidden because the E1 band is empty; the E2;k → E1;k transition is also forbidden because the E1 band is fully occupied. In the case of light n doping with μ ¼ 0:2γ1 as shown in Figure 3.5(b), all the abovementioned transitions are still possible but are restricted by some conditions. For example, because of Pauli blocking, the E1;k → E1;k transition is only possible for a sufficiently large photon energy such that ℏω > 2μ, below which this interband transition is not possible and the optical conductivity is zero if intraband transitions are not considered. The kink in the spectrum of the surface optical conductivity at ℏω ¼ γ1 for the case of μ ¼ 0 is now split into two kinks located at ℏω ¼ γ1 and ℏω ¼ 2μ þ γ1 . This is due to the fact that the onset of the E2;k → E1;k transition is now moved from ℏω ¼ γ1 to ℏω ¼ 2μ þ γ1 because of Pauli blocking. For μ > 0, the E1;k → E2;k transition is also possible at ℏω ¼ γ1 because the conduction band E1 is now partially filled. The contribution of the E1;k → E2;k transition is given by the last term in (3.110) that contains a Dirac delta function, which is represented by a vertical arrow in each right panel of Figure 3.5 at ℏω ¼ γ1 . This delta function can be replaced by a Lorentzian function if the finite lifetimes of the energy states are considered. In the case of heavy n doping with μ ¼ 1:2γ1 , which is larger than the interlayer interaction energy, instead of the E1;k → E1;k transition, the E1;k → E2;k transition has the lowest possible transition energy at ℏω ¼ γ1 , as marked in Figure 3.5(c) with a vertical arrow. The spectral weight of the E1;k → E2;k transition is smaller than that in the case of μ ¼ 0:2γ1 because the conduction band E2 is now partially filled and some

3.6 Optical Conductivity of Multilayer Graphene

95

transitions are thus Pauli blocked. The onset of the E1;k → E2;k transition is shifted from ℏω ¼ γ1 to ℏω ¼ 2ðμ  γ1 Þ þ γ1 ¼ 2μ  γ1 . The kink in the spectrum of the surface optical conductivity at ℏω ¼ 2μ is due to the onset of the E1;k → E1;k and E2;k → E2;k transitions, and the kink at ℏω ¼ 2μ þ γ1 is due to the onset of the E2;k → E1;k transition. Because of the symmetric band structure of AB-stacked bilayer graphene, the surface optical conductivity in the case of p doping is the same as that of n doping, as shown in the right panels of Figures 3.5(d) and (e) for the corresponding negative μ. The possible transitions are visualized by vertically flipping the diagrams in the left panels of Figures 3.5(b) and (c) while reversing the directions of arrows, as shown in the left panels of Figures 3.5(d) and (e). In the high-frequency spectral region where ℏω ≫ μ and ℏω ≫ γ1 ≈ 0:37 eV, the surface optical conductivity of AB-stacked bilayer graphene is double that of monolayer graphene such that e σ ¼ 2e σ 0 , as can be seen in Figure 3.5 in the high-frequency region. In fact, although not derived here, in the high-frequency region the surface optical conductivity of N-layer AB-stacked multilayer graphene is N times that of monolayer graphene, e σ ¼ Ne σ 0 , much like the case of the DC conductivity discussed in Chapter 2.

3.6.2

AA-Stacked Bilayer Graphene There are two Dirac cones in AA-stacked bilayer graphene, as shown in Figure 3.6. As in the case of AB-stacked bilayer graphene, we name the bottom valence band n ¼ 2 with energy E2;k for the state j  2; k〉 of wave vector k in this band, the top valence band n ¼ 1 with energy E1;k for the state j  1; k〉 of wave vector k in this band, the bottom conduction n ¼ 1 with energy E1;k for the state j1; k〉 of wave vector k in this band, and the top conduction band n ¼ 2 with energy E2;k for the state j2; k〉 of wave vector k in this band. Different from the situation in AB-stacked bilayer graphene, for which the two pairs of massive conduction and valence bands have the same out-ofplane momentum, k z , as shown in Figure 1.9, however, the two Dirac cones shown in Figure 3.6 for AA-stacked bilayer graphene have different k z values, as discussed in Chapter 1. Because of the requirement for the conservation of k z momentum, transitions from the bottom Dirac cone (n ¼ 2 and 1) to the top Dirac cone (n ¼ 1 and 2) are forbidden; therefore, only the E2;k → E1;k and E1;k → E2;k transitions within each Dirac cone are allowed. Consequently, we can express the surface optical conductivity of AA-stacked bilayer graphene in terms of the surface optical conductivity of monolayer graphene as e σ AA ðωÞ ¼ e σ ðωÞjμ1 ¼μΔ1 þ e σ ðωÞjμ2 ¼μΔ2 ;

ð3:111Þ

where e σ ðωÞ is the surface optical conductivity of monolayer graphene given in (3.78) and (3.79) with the chemical potential μ replaced by the effective chemical potential μ1 for the first term and by μ2 for the second term. The effective chemical potentials are

Optical Properties

respectively related to the shifts of the Dirac cones in energies of Δ1 and Δ2 , given in AA (2.80); for N ¼ 2, we find that μ1 ¼ μ  γAA 1 and μ2 ¼ μ þ γ1 . At a temperature T → 0, (3.111) reduces to

0

E1 E

0

1

E

2

0 ka0

0.5

−3

0

2

4

6

AA 1

4

0

3

0

0

E

AA 1

ð3:112Þ

3

E2

−4 −0.5 (b)

  ie σ0 2jμ  γAA 2jμ þ γAA 1 j þ ℏω 1 j þ ℏω ln þ ln ; π 2jμ  γAA 2jμ þ γAA 1 j  ℏω 1 j  ℏω

0

4

E

(a)

AA 1

e σ ðωÞ ¼ 

−4 −0.5

0.5

−3

0

2

4

6

AA 1

4

3

0

AA 1

(c)

0 ka0

0

0

E

96

−4 −0.5

0 ka0

0.5

−3

0

2

4

6

AA 1

Figure 3.6 Possible optical transitions (left panels) and surface optical conductivity (right panels) of

AA-stacked bilayer graphene at a temperature T → 0 for different chemical potentials: (a) μ ¼ 0, (b) AA AA AA AA μ ¼ 0:2γAA 1 , (c) μ ¼ 1:2γ1 , and μ ¼ γ1 (gray curves), (d) μ ¼ 0:2γ1 , and (e) μ ¼ 1:2γ1 , and μ ¼ γAA (gray curves). The carrier filling of the bands is shown by the gray area below the 1 chemical potential. The optical conductivity is normalized to e σ 0 ¼ e2 =4ℏ. In each right panel, the real 0 part e σ and the imaginary part e σ 00 of the surface optical conductivity are plotted as the solid and dashed curves, respectively. The onsets of the transitions E1 → E2 and E2 → E1 are marked by numbers 1 and 2, respectively. The corresponding transitions are shown in the left panel.

3.6 Optical Conductivity of Multilayer Graphene

4

0

0

0

3

0

E

AA 1

(d)

97

1 −4 −0.5

0.5

−3

0

2

4

6

AA 1

4

0

3

0

0

E

AA 1

(e)

0 ka0

−4 −0.5

0 ka0

0.5

−3

0

2

4

6

AA 1

Figure 3.6 (cont.)

AA which can be found using (3.82) by replacing μ with μ  γAA 1 and μ þ γ1 for the first and second terms in (3.111), respectively. The possible interband transitions and the surface optical conductivity of AA-stacked bilayer graphene are shown in Figure 3.6 for different chemical potentials. As can be seen, the onset of the E1;k → E2;k transition AA is located at 2jμ  γAA 1 j, and that of the E2;k → E1;k transition at 2jμ þ γ1 j. For μ ¼ 0, AA both transitions have the same onset at ℏω ¼ 2γ1 , and the surface optical conductivity AA is twice that of monolayer graphene of μ ¼ γAA 1 for ℏω > 2γ1 , as shown in the right panel of Figure 3.6(a). As μ increases, the onset points of the two transitions E1;k → E2;k and E2;k → E1;k are separated, moving away in opposite directions from the onset photon energy ℏω ¼ 2γAA 1 for μ ¼ 0, as can be seen in the right panel of Figure 3.6(b) for μ ¼ 0:2γAA . When μ further increases to the point that μ ¼ γAA 1 1 , the onset of the E1;k → E2;k transition is at zero photon energy because jμ  γAA j ¼ 0. The optical 1 AA conductivity is now characterized by only one jump at 2jμ þ γ1 j ¼ 4γAA 1 , shown as the gray curves in the right panel of Figure 3.6(c). For μ > γAA , the optical conductivity 1 has two discontinuities, but now the threshold photon energy for the onset of the E1;k → E2;k transition and that for the E2;k → E1;k transition both increase with increasing μ. Because of the symmetric band structure of AA-stacked bilayer graphene, the surface optical conductivity in the case of p doping is the same as that of n doping as shown in the right panels of Figures 3.6(d) and (e) for the corresponding negative μ. The possible transitions are visualized by vertically flipping the diagrams in the left panels of

98

Optical Properties

Figures 3.6(b) and (c) while reversing the directions of arrows, as shown in the left panels of Figures 3.6(d) and (e). As in the case of AB-stacked bilayer graphene, in the high-frequency spectral region where ℏω ≫ μ and ℏω ≫ γAA 1 ≈ 0:22 eV, the surface optical conductivity of AA-stacked bilayer graphene is given by e σ ¼ 2e σ 0, which is double that of monolayer graphene. For N-layer AA-stacked multilayer graphene, we have N terms in (3.112) instead of only two terms. In the high-frequency spectral region, the surface optical conductivity is N times that of monolayer graphene. This result is not surprising because there are N Dirac cones in N-layer AA-stacked multilayer graphene; each of the Dirac cones contributes e σ 0 to the total surface optical conductivity so that e σ ¼ Ne σ 0.

3.7

Permittivity of Monolayer and Multilayer Graphene In the preceding sections, we have derived for monolayer and multilayer graphene the theoretical values of the surface optical conductivity e σ ðωÞ, or e σ ∥ ðωÞ with the subscript restored. In the optical region, the surface optical conductivity is contributed by electrons that make transitions to the conduction band through optical absorption. In addition to these free electrons generated through absorption of photons, there are also bound electrons that can interact with the optical field. Their response is encoded in the surface optical susceptibility e χ ∥ for the electric field parallel to the graphene surface, and in the surface optical susceptibility e χ ⊥ for the electric field perpendicular to the graphene surface. Therefore, the total optical response of graphene, represented by the permittivity tensor ϵ, is contributed by both the nonzero surface optical conductivity element e σ∥ and the two surface optical susceptibility elements e χ ∥ and e χ ⊥, as shown in (3.41) and (3.42), which are given again here:

and

  e χ∥ e σ∥ ϵ∥ ¼ ϵ0 1 þ þi d ωd

ð3:113Þ

  e χ ϵ⊥ ¼ ϵ0 1 þ ⊥ : d

ð3:114Þ

The calculation of e χ ∥ and e χ ⊥ can be carried out by using the density function theory (DFT), which involves the calculation of the charge density distribution and the full band structure. The simulation code for such calculation is readily available in the literature; here we only give the calculated results [1,4]. According to the selection rules, only transitions from π to π bands or from σ to σ bands are allowed when the electric field is parallel to the graphene surface, whereas only transitions from π to σ bands or from σ to π bands are allowed when the electric field is perpendicular to the graphene surface. It can be seen from the band structure shown in Figure 1.2(c) that below ℏω ≈ 9 eV only transitions from π to π bands are possible. Therefore, in the spectral range for photons of

3.7 Permittivity of Monolayer and Multilayer

(b) 30

20

20

0

0

(a) 30

99

10

10

2.5 0

0

0

10

10 0

1

2

(eV)

3

4

5

only

=0

0

1

2

3

4

(eV)

Density of states (a.u.)

(c)

4 eV

5

0 Electron enegery (eV)

5

Figure 3.7 (a) Real and (b) imaginary parts of the permittivities ϵ ∥ (solid curves) and ϵ ⊥ (dashed curves) calculated using DFT for intrinsic graphene in the zero-temperature limit that T → 0, so that μ ¼ EF ¼ 0. The dotted curve is given by the analytical solution ie σ ∥ =ωd, where d ¼ 0:335 nm. (c) Density of states of graphene without the Dirac cone approximation, which shows the Van Hove singularities [1].

a few electron volts, ϵ ∥ is mostly contributed by the π electrons while the σ electrons contribute only a constant background permittivity. The spectral dependence of the permittivity ϵ ∥ is shown in Figure 3.7 for intrinsic graphene in the limit that T → 0, thus μ ¼ EF ¼ 0 as given by (1.40). The thickness d in (3.113) and (3.114) is taken to be 0.335 nm, which is the distance between two neighboring graphene layers in graphite. For the real part ϵ 0∥ shown in Figure 3.7(a), there is a background permittivity of 2:5ϵ 0 , which consists of 1:5ϵ 0 contributed by the σ electrons and ϵ 0 contributed by the free space, respectively [5]. The frequency dependence of ϵ 0∥ is contributed by the π

5

100

Optical Properties

electrons. The imaginary part ϵ∥00 is contributed by e σ∥ ¼ e σ 0 for ℏω > 2μ ¼ 2EF ¼ 0, as given by (3.109). However, by plotting the term e σ ∥ =ωd of (3.113) and comparing it with ϵ∥00 calculated using DFT, as shown in Figure 3.7(b), we find that there is an absorption peak around ℏω ≈ 4 eV, which is absent in e σ ∥ . This discrepancy is due to the fact that our calculation of e σ ∥ in the preceding section is carried out under the Dirac cone approximation, which takes the band to be linear and is valid only near the Dirac point. However, away from the Dirac point, the band is not linear anymore, as shown in Figure 1.2(c). The absorption at the photon energy of ℏω ≈ 4 eV corresponds to the transition from π to π bands at the H point. If one calculates the density of states by using the full band structure for (1.34), the density of states thus obtained shows two peaks, known as Van Hove singularities [1]. The energy difference of these two peaks gives the absorption peak at ℏω ≈ 4 eV, as shown in Figure 3.7(c). The permittivity ϵ ⊥ is also shown in Figure 3.7. Below ℏω ¼ 9 eV, the surface susceptibility e χ ⊥ is real because the bound electrons do not absorb photons. These bound electrons give a core background permittivity of ϵ 0e χ ⊥=d ≈ 1:5ϵ 0 [1]. Therefore, ϵ ⊥ is real, and from (3.114) we obtain ϵ⊥0 ≈ 2:5ϵ 0 and ϵ⊥00 ¼ 0. For multilayer graphene, (3.113) and (3.114) are still valid with the thickness d larger than that of monolayer graphene. It is shown using DFT that the permittivities given in Figure 3.7 are still accurate for multilayer graphene and even for graphite. Therefore, for (3.113) and (3.114) to remain approximately constant regardless of the number N of layers, we have e σ ∥ ¼ Ne σ 0 , as obtained in the preceding section, e χ ∥ ðN ¼ 1Þ, and e χ ⊥ðNÞ ≈ Ne χ ⊥ðN ¼ 1Þ because the thickness of N-layer graχ ∥ ðNÞ ≈ Ne phene is d ≈ 0:335N nm. Much effort has been made to measure the refractive index n∥ of both monolayer and multilayer graphene using reflection spectroscopy [6,7], spectroscopic ellipsometry [8,9], and other techniques [4]. Interested readers can refer to Reference [4] for a review of the measured and calculated refractive index values of various graphene samples. The measured values of n∥ are not consistent in the literature. For both monolayer and fewlayer multilayer graphene, the majority of values are found in the range of n0∥ = 2.6–2.9 for the real part of n∥ , and in the range of n∥00 = 1.3–1.5 for the imaginary part of n∥ [4]. The discrepancy is due to the variations in the sample quality and the uncertainties from different measurement techniques. The wavelength used for the measurement can also affect the results. Although small compared to the dispersion in other parts of the spectrum, the dispersion of ϵ ∥ can still be seen around ℏω ≈ 2 eV. To compare the DFT results with the experimental data, we obtain the theoretical value of ϵ ∥ ¼ ϵ 0 ð5:36 þ 7:64iÞ at ℏω ¼ 1:96 eV from Figure 3.7. Then we find that pffiffiffiffiffiffiffiffiffiffiffi n∥ ¼ ϵ ∥ =ϵ 0 ≈ 2:7 þ 1:4i, which falls in the range of the experimentally measured results. To date, there has not been any direct measurement of n⊥ for monolayer or multilayer graphene. However, we can refer to graphite for the experimental value of n⊥ because the value of n⊥ for graphite is predicted to be close to that of monolayer and multilayer graphene [1,10]. For graphite, experiments found that the real part of n⊥ is about 00 00 n0⊥ ¼ 1:5–2:0 and the imaginary part n⊥ is either approximately zero or n⊥ ≈ 0:65 [9].

3.8 Absorbance of Monolayer and Multilayer

101

00 From the DFT calculation, we find that ϵ 0⊥ ≈ 2:9ϵ 0 and ϵ⊥ ≈ 0:4ϵ 0 for graphite [1], which in turn give n⊥ ¼ 1:7 þ 0:1i. Apparently, there is a large discrepancy among experimental results as well as between the experiment and theoretical values of n⊥. In the case 00 of graphene, we have ϵ 0⊥ ≈ 2:5ϵ 0 and ϵ⊥ ¼ 0 according to Figure 3.7; thus n⊥ ≈ 1:58. Clearly, there is a need for more measurements of the experimental data of n⊥ for both graphene and graphite.

3.8

Absorbance of Monolayer and Multilayer Graphene Once the permittivity of graphene is obtained, it is straightforward to find the transmittance T, the reflectance R, and the absorbance A of monolayer graphene for a plane wave at normal incidence. We introduced two models of graphene at the beginning of this chapter. When the 2D model is used, graphene is assumed to be strictly a 2D material. All the physical quantities such as permittivity and conductivity are only defined on the graphene sheet. The surface permittivity of graphene in this 2D model is given by (3.48): e ϵ ¼ ϵ 0e χþi

e σ : ω

ð3:115Þ

Another model regards graphene as a 3D material that is characterized by a thickness of d. All 3D physical quantities have the same units as those that are normally used for other 3D materials. The permittivity of graphene in this 3D model is given by (3.39):   e e χ σ ϵ ¼ ϵ0 1 þ ; ð3:116Þ þi d ωd where the thickness of monolayer graphene is usually taken to be d ¼ 0:335 nm, the same as the thickness of each graphene layer in graphite. In a situation in which the kr

(a) Ei

x z

n1 n2

Er

ki

kr

(b)

Ei n1

Er

ki

}

n Et

d

n2 Et

kt kt Figure 3.8 Schematics of an optical field Ei that is normally incident on a sheet of monolayer or multilayer graphene using (a) the 2D model, and (b) the 3D model. The reflected field Er and the transmitted field Et are also shown. In (a), the graphene sheet is located at z ¼ 0, separating two dielectric media of refractive indices n1 and n2 . In (b), a thickness of d and a refractive index of n∥ are considered for the monolayer or multilayer graphene.

102

Optical Properties

dimensions of all other physical parameters of the graphene sheet and the optical wavelength are much larger than the graphene thickness d, these two models converge and are essentially the same. For example, considering a graphene sheet sandwiched between two semi-infinite dielectrics as shown in Figure 3.8, the same values of T, R, and A have to be obtained by using either the 2D or 3D model in the limit that d≪λ, where λ is the wavelength of the optical field; in this limit, even if the 3D model is used, graphene is essentially 2D seen by the optical field. For visible light, λ is between four and seven hundred nanometers, which is much larger than the thickness of graphene. Therefore, the assumption that d ≪ λ is generally justified in the optical region even for multilayer graphene of a few layers. To demonstrate this concept, in the following we derive the transmittance T using both the 2D and 3D models. Then, we show that the derived results of T are identical in the limit that d ≪ λ. We consider a monochromatic incident electromagnetic wave of a frequency ω, which propagates in the z direction and is polarized in the x direction, as shown in Figure 3.8. The electromagnetic wave is normally incident on the N-layer graphene sheet located at z ¼ 0, which absorbs, reflects, and transmits the incident wave. The refractive indices of the dielectric media above and below the graphene sheet are n1 and n2 , respectively. The total electromagnetic field in the n1 medium can be expressed as E1 ¼ Ei þ Er ¼ ^x E i expðik i z  iωtÞ þ ^x E r expðik r z  iωtÞ;

ð3:117Þ

where E i and E r are the electric field amplitudes of the incident and reflected fields, respectively. The electromagnetic field in the n2 medium is E2 ¼ Et ¼ ^x E t expðik t z  iωtÞ;

ð3:118Þ

where E t is the electric field amplitude of the transmitted field. The wave numbers in (3.117) and (3.118) are ki ¼ k r ¼ n1 ω=c and k t ¼ n2 ω=c. From Maxwell’s equations, the corresponding magnetic fields H1 and H2 can be found. Considering the 2D model given by (3.115), the boundary conditions require that E1 jz¼0 ¼ E2 jz¼0 ; ^z  ðH2  H1 Þjz¼0 ¼

∂e ϵ E : ∂t

ð3:119Þ ð3:120Þ

The first condition requires that Ei þ Er ¼ Et;

ð3:121Þ

and the second condition requires that n1 ðE i  E r Þ  n2 E t ¼e σ ∥;eff E t ; Z0 where

ð3:122Þ

3.8 Absorbance of Monolayer and Multilayer

e σ ∥;eff ¼ iωϵ 0e σ∥ χ∥ þ e

103

ð3:123Þ

is the effective surface optical conductivity pffiffiffiffiffiffiffiffiffiffiffi that we define to simplify the mathematical expressions; e σ ∥ ¼ Ne σ 0 ; and Z0 ¼ μ0 =ϵ 0 is the impedance of the free space. By using the relation in (3.121) together with that in (3.122), we find the reflectance and the transmittance:  2   E r   n1  n2  Z 0 e σ ∥;eff 2 R ¼   ¼  ; ð3:124Þ Ei n1 þ n2 þ Z 0 e σ ∥;eff   2 n2  E t  4n1 n2 T ¼   ¼ : ð3:125Þ n1 E i jn1 þ n2 þ Z0 e σ ∥;eff j2 By using the relation A ¼ 1  T  R from energy conservation, the absorbance of graphene can be obtained: A¼

4n1 Z0 e σ 0∥;eff jn1 þ n2 þ Z0 e σ ∥;eff j2

:

ð3:126Þ

Now we show that the optical reflectance, transmittance, and absorbance derived using the 3D model in the limit that d ≪λ have the same values as those obtained above using the 2D model. As shown in Figure 3.8(b), there are two boundaries in the 3D model: one is the top interface between the dielectric of a refractive index n1 and the graphene sheet; the other one is the bottom interface between the dielectric of a refractive index n2 and the graphene sheet. Therefore, to solve for the transmittance of the system, we have to consider the internal reflection inside the graphene sheet. The transmittance can be solved in a way similar to that taken for finding the transmittance of a Fabry–Pérot cavity. The transmittance is thus given by T¼

n2  t1g tg2 eiδ 2   ; n1 1  rg1 rg2 e2iδ

ð3:127Þ

where t1g ¼ 2n1 =ðn∥ þ n1 Þ is the transmission coefficient from the medium of n1 into graphene, tg2 ¼ 2n∥ =ðn∥ þ n2 Þ is the transmission coefficient from graphene into the medium of n2 , rg1 ¼ ðn∥  n1 Þ=ðn∥ þ n1 Þ is the reflection coefficient on the graphene side at the top interface, rg2 ¼ ðn∥  n2 Þ=ðn∥ þ n2 Þ is the reflection coefficient on the graphene side at the bottom interface, δ ¼ ωn∥ d=c, and n∥ is the refractive index of graphene given by (3.43), with ϵ ∥ given by (3.113). The refractive index n∥ is used rather than n⊥ because the electric field is parallel to the graphene surface. Under the condition pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that d ≪ λ, we have n∥ ≈ ie σ ∥;eff =ωϵ 0 d and expðiδÞ≈ 1 þ iδ. Then (3.127) reduces to T¼

2 n2  2n1    ; n1 n1 þ n2  iδn∥

ð3:128Þ

which is identical to (3.125). Other physical quantities such as A and R can also be shown to be identical in the limit that d ≪λ. It is important to check the consistency

104

Optical Properties

(b)

(a)

0.65

1 0.975

0.6 Transmittance, T

Transmittance, T

0.9 0.8 0.7 0.6

0.55 0.5 0.45 0.4

0.5 0.4 0

5 10 15 20 Number of layers, N

25

0.35 25

30 35 40 45 Number of layers, N

50

Figure 3.9 Transmittance of multilayer graphene as a function of the number N of layers for N in the range (a) from N ¼ 0 to N ¼ 25 and (b) from N ¼ 25 to N ¼ 50. The dielectrics on the two sides of the graphene sheet are taken to be the free space with n1 ¼ n2 ¼ 1, and the thickness of the graphene sheet is given by d ≈ 0:335N nm. The transmittance is calculated using ϵ∥ ¼ ϵ0 ð5:36 þ 7:64iÞ given by Figure 3.7 for ℏω ¼ 1:96 eV. The open circles are obtained using the 2D model of (3.125), the solid dots are obtained using the 3D model of (3.127), the dashed curve is given by the 2D model of (3.125) using e σ ∥;eff ¼ Ne σ 0 in the Dirac cone approximation, and the solid curve is obtained from (3.131) assuming that NZ0 e σ 0 ≪1.

between the 2D and 3D models as done above because in the literature they are interchangeably used. Without taking the limit that d ≪λ, numerical comparison can be done between (3.125) and (3.127), which are respectively obtained by using the 2D and 3D models. The unknown variable e σ ∥;eff is linked to ϵ ∥ through the relation ϵ∥ ¼ ϵ0 þ i

e σ ∥;eff ωd

ð3:129Þ

using (3.41) and (3.123), and the values of ϵ ∥ can be found from Figure 3.7. For ℏω ¼ 1:96 eV, ϵ ∥ ¼ ϵ 0 ð5:36 þ 7:64iÞ given by Figure 3.7; thus, from (3.129) we obtain e σ ∥;eff ¼ iωdϵ 0 ð4:36 þ 7:64iÞ. Note that d ≈ 0:335N nm is a function of the number N of layers. By inserting the above value of e σ ∥;eff into (3.125) and (3.127), we obtain the transmittance as a function of N using the 2D and 3D models, respectively. The transmittance curves are plotted in Figure 3.9 in open circles for the 2D model and in solid dots for the 3D model. As can be seen, the transmittance decreases because of increasing absorption as the number of layers increases. The difference in transmittance between the two models is negligibly small all the way up to 25 layers. However, above 25 layers the multilayer graphene has to be treated as a 3D material because the condition d ≪ λ is not satisfied anymore.

3.8 Absorbance of Monolayer and Multilayer

105

Because ϵ ∥ ¼ ϵ 0 ð5:36 þ 7:64iÞ and Z0 e σ 0∥;eff ≈ 0:025 for monolayer graphene at the optical photon energy of ℏω ¼ 1:96 eV, we can further simplify these models by assuming that Z0 e σ 0∥;eff ≪1. Furthermore, we can ignore the imaginary part of the effective surface optical conductivity e σ ∥;eff given by (3.123) because the absorption is σ ∥;eff ¼ N e σ 0 as we obtained in determined by the real part of e σ ∥;eff . Then, we can write e the preceding sections for multilayer graphene in the zero-temperature limit that T → 0. For a sheet of multilayer graphene suspended in the free space with n1 ¼ n2 ¼ 1, we find from (3.124)–(3.126) using these assumptions that   Z0 Ne σ0 R ¼  2 þ Z Ne σ 0

T¼

1 σ 0 =2j2 j1 þ Z0 Ne A¼

0

2   ≈ 0; 

ð3:130Þ

≈ 1  NZ0 e σ 0 ¼ 1  Nπα;

ð3:131Þ

≈ NZ0 e σ 0 ¼ Nπα;

ð3:132Þ

Z0 Ne σ0 j1 þ Z0 Ne σ 0 =2j2

where α ¼ e2 =4πϵ 0 ℏc ¼ e2 Z0 =4πℏ ≈ 1=137 is the fine-structure constant and Z0 e σ 0 ¼ πα. As can be seen, for each additional graphene layer, the transmittance decreases by an amount of Z0 e σ 0 , while the absorbance increases by Z0 e σ 0. Using the pffiffiffiffiffiffiffiffiffiffiffi relations that Z0 ¼ μ0 =ϵ 0 ≈ 120π Ω and e σ 0 ¼ e2 =4ℏ ≈ 60:8 μS, we find that Z0 e σ 0 ¼ πα ≈ 0:023. The transmittance given by the 2D model (3.125) using e σ ∥;eff ¼ Ne σ 0 is plotted as the dashed curves in Figures 3.9(a) and (b), and that obtained from (3.131) assuming NZ0 e σ 0 ≪1 is plotted as the solid curve in Figure 3.9(a). We can see that the transmittance is overestimated if e σ ∥;eff ¼ Ne σ 0 is assumed. This is due to the fact that by taking σ 0 we effectively ignore the absorption peak at ℏω ≈ 4 eV shown in Figure e σ ∥;eff ¼ Ne σ 0 is less than the true value of the real part of 3.7(b). Therefore, the value of e σ ∥;eff ¼ Ne e σ ∥;eff , and the absorbance is underestimated. It also can be seen that the transmittance is σ 0 ≪1 is assumed. This is because when N is large, NZ0 e σ 0 ≪1 is underestimated if NZ0 e not valid, and therefore the result given by (3.131) cannot accurately approximate the exact result given by (3.127) using the true values of e σ ∥;eff when N is large. Nevertheless, when N is small, these approximations are all justified and accurately give the transmittance of T ≈ 0:977 for N ¼ 1, as marked in Figure 3.9(a). The transmittance of T ≈ 0:977 for a monolayer graphene was first observed in an experiment done by Nair et al. [11]. In this experiment, it was found that for each additional graphene layer, the transmittance dropped by 2.3 percent in the visible spectral region, consistent with the results given in Figure 3.9(a). The model given by (3.131) was used, and they found that Z0 e σ 0 ¼ πα ¼ 0:023, which is frequently used to characterize the interaction between light and electrons. Therefore, they concluded that in the visible spectral region, it is appropriate to take the approximation that e σ ∥;eff ≈ Z0 e σ 0 ¼ πα; the modification due to the absorption peak at ℏω ≈ 4 eV is negligible, and it is not necessary to use the exact e σ ∥;eff given by (3.123).

106

Optical Properties

References 1. M. Klintenberg, S. Lebègue, C. Ortiz, et al., “Evolving properties of two-dimensional materials: from graphene to graphite,” Journal of Physics: Condensed Matter, Vol. 21, 335502 (2009). 2. E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Physical Review B, Vol. 77, 155409 (2008). 3. M. Jablan, H. Buljan, and M. Soljačić, “Transverse electric plasmons in bilayer graphene,” Optics Express, Vol. 19, pp. 11236–11241 (2011). 4. S. Cheon, K. D. Kihm, H. G. Kim, et al., “How to reliably determine the complex refractive index (RI) of graphene by using two independent measurement constraints,” Scientific Reports, Vol. 4, 6364 (2014). 5. L. G. Johnson and G. Dresselhaus, “Optical properties of graphite,” Physical Review B, Vol. 7, pp. 2275–2285 (1973). 6. Z. H. Ni, H. M. Wang, J. Kasim, et al., “Graphene thickness determination using reflection and contrast spectroscopy,” Nano Letters, Vol. 7, pp. 2758–2763 (2007). 7. M. Bruna and S. Borini, “Optical constants of graphene layers in the visible range,” Applied Physics Letters, Vol. 94, 031901 (2009). 8. F. J. Nelson, V. K. Kamineni, T. Zhang, et al., “Optical properties of large-area polycrystalline chemical vapor deposited graphene by spectroscopic ellipsometry,” Applied Physics Letters, Vol. 97, 253110 (2010). 9. G. E. Jellison, Jr., J. D. Hunn, and H. N. Lee, “Measurement of optical functions of highly oriented pyrolytic graphite in the visible,” Physical Review B, Vol. 76, 085125 (2007). 10. A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Physical Review Letters, Vol. 100, 117401 (2008). 11. R. R. Nair, P. Blake, A. N. Grigorenko, et al., “Fine structure constant defines visual transparency of graphene,” Science, Vol. 320, p. 1308 (2008).

4

Optoelectronic Properties

4.1

Dispersion in Frequency and Momentum In the preceding chapter, the surface optical conductivity e σ ðωÞ of graphene contributed by interband transitions was considered, which characterizes the optical response of graphene in the high-frequency region and has a real part given by (3.106) when the relaxation rate is taken to be zero by ignoring the finite relaxation times of the states involved in the transitions. We find by using this surface optical conductivity that monolayer graphene has an approximately constant absorbance of about 2.3 percent in the optical spectral region. However, in the low-frequency region, the surface optical conductivity e σ ðωÞ obtained in the preceding chapter is not valid. For example, if we set ω ¼ 0 for e σ ð0Þ to find the DC surface conductivity, we find that e σ ð0Þ is also zero, which is inconsistent with the DC surface conductivity obtained in Chapter 2. The discrepancy comes from the fact that e σ ðωÞ given in the preceding chapter only accounts for interband absorption of photons with electrons making transitions from a valence band to a conduction band, whereas the DC surface conductivity given in Chapter 2 is solely contributed by the intraband transitions of charge carriers within a valence band or within a conduction band. The goal of this chapter is to give a comprehensive model that can describe the optoelectronic properties of graphene in a broad frequency range, from the zero frequency for the electronic properties discussed in Chapter 2 to the optical frequencies for the optical properties discussed in Chapter 3. Furthermore, the momentum dispersion due to spatially nonlocal response of graphene to optical excitation is considered in this chapter. As we shall see in later chapters, spatial nonlocality is important when the external electric field involved in the interaction is characterized by a wavelength comparable to the physical size of the graphene sheet or the graphenebased device. In this situation, the electron density in graphene cannot be approximated by a spatially homogeneous density. As a consequence, the response of graphene to an electric field is not only a function of the frequency ω of the electric field but also a function of a wave vector q that accounts for the spatial nonlocality of the response. In the following, we start from the discussion of the Drude model. The Drude model gives the optical conductivity arising from the intraband transitions of carriers. Because the photon energy required to make an intraband transition is usually smaller than that for an interband transition, the Drude model is suitable for the description of the optical

108

Optoelectronic Properties

response of graphene in the terahertz and far-infrared spectral regions. By combining the optical conductivity obtained from the Drude model with the optical conductivity contributed by interband transitions, derived in the preceding chapter, a comprehensive model valid in a broad spectral range is obtained. Then we extend our discussion beyond the Drude model. Microscopically the response of graphene can be characterized by a polarizability function, from which we obtain the nonlocal response and dynamic screening of graphene, which cannot be derived using the Drude model. Finally, the polarizability function and the dynamic screening of AB-stacked bilayer graphene are discussed, which are compared with those of monolayer graphene and 2D electron gas.

4.2

Drude Model The Drude model is frequently used to describe the transport properties of conducting materials. In the derivation of the model, the kinetic theory of ideal gases is used, and electrons are treated classically as free particles accelerated by an external electric field. The electrons collide randomly with immobile positive ions; collectively, damping of the average electron velocity due to the collisions is characterized by an average relaxation time τ. From quantum mechanics, however, we know that electrons have to be treated as waves, which cannot be scattered by a perfectly periodic potential in an ideal crystal. Instead of collisions from ion to ion in the periodic crystal structure, the sources of the collisions that scatter electrons are lattice imperfections, impurities, and other quasiparticles, particularly phonons, as discussed in Chapter 2. The derivation of the Drude model can be found in most solid-state physics books and is not repeated here. In the Drude model, the conductivity of a material at a frequency of ω is σðωÞ ¼

σð0Þ ; 1  i ωτ

ð4:1Þ

where σð0Þ is the DC conductivity and τ is the average momentum relaxation time of the charge carriers. Equation (4.1) applies to all materials as long as the Drude model is applicable. The specific properties of the material are characterized by the DC conductivity σð0Þ and the relaxation time τ. By applying the relation given in (4.1) for the conductivity of a 3D material to the surface conductivity for a 2D material, the surface optical conductivity determined by the Drude model is given directly here as e σ ðωÞ ¼

e σ ð0Þ ; 1  i ωτ

ð4:2Þ

where τ is the average momentum relaxation time of all the free electrons contributing to the surface conductivity. In Chapter 2, this effective relaxation time is formally defined as τeff ; here the subscript “eff” is dropped for simplicity.

4.2 Drude Model

(a)

(b)

( )

2

( )

1

0

1

Surface optical conductivity,

Surface optical conductivity,

( )

0

3

0

3

109

( )

0

1

2

3

2

( ) 1

0

( ) 1

0

1

2

3

EF

EF

Figure 4.1 Surface optical conductivity of graphene contributed by (a) intraband transitions and (b) both intraband and interband transitions, as a function of optical frequency. The dashed and solid curves are calculated at temperatures of 100 K and 300 K, respectively, in the limit that γ → 0. The Fermi energy is assumed to be EF ¼ 100 meV, and the scattering rate is taken to be τ1 ¼ 2:5  1011 s1 . The surface optical conductivity is normalized to e σ 0 ¼ e2 =4ℏ.

For graphene, e σ ð0Þ is given by (2.17). By inserting (2.17) into (4.2) we obtain e σ ðωÞ ¼

2e2 kB T πℏ2 ðτ1  i ωÞ

 ln 2 cosh

 μ ; 2kB T

ð4:3Þ

which is plotted in Figure 4.1(a) for T ¼ 100 K and T ¼ 300 K using the parameters τ1 ¼ 2:5  1011 s1 (for τ ¼ 4 ps) and EF ¼ 100 meV with the corresponding chemical potential μ obtained from (1.40). As can be seen from (4.3), the Drude surface optical conductivity decreases with frequency and approaches zero in the high-frequency region. This behavior manifests the fact that when ω ≫ τ 1 , the electrons driven by a high-frequency field oscillate so fast that they behave much like bound electrons because they do not have enough time to travel a sufficiently long distance to collide with scattering centers before changing the direction of motion. As a consequence, the surface optical conductivity contributed by intraband transitions diminishes as the frequency increases; the real part decreases faster than the imaginary part because e σ 0ðωÞ ∝ ω2 while e σ 00 ðωÞ ∝ ω1 , as seen in Figure 4.1(a). However, interband carrier transitions become possible as the frequency increases. By combining the surface optical conductivity contributed by the intraband transitions given in (4.3) and that contributed by the interband transitions given in (3.108), we obtain the overall surface optical conductivity of graphene:

Optoelectronic Properties

e σ ðωÞ ¼

2e2 kB T πℏ2 ðτ1  iωÞ

 ln 2 cosh

μ 2kB T



ð∞ e2 ω f0 ðEÞ  f0 ðEÞ dE; i π ð2EÞ2  ðℏω þ iℏγÞ2 0

ð4:4Þ

which is plotted in Figure 4.1(b) for T ¼ 100 K and T ¼ 300 K. In the low-frequency region, e σ 0ðωÞ first decreases monotonically with increasing frequency because of decreasing contribution from intraband transitions as the frequency increases, but it starts to increase with frequency around ℏω ≈ 2EF because of the onset of interband transitions. Therefore, e σ 0ðωÞ has a minimum value at an intermediate frequency, the value of which depends on the temperature and the Fermi energy. We also note that e σ 00 ðωÞ becomes negative around ℏω ≈ 2EF due to the resonant dip in the spectrum of e σ 00 ðωÞ from interband transitions, as discussed in Section 3.5 and shown in Figure 3.3. The trough of the e σ 0ðωÞ curve in a relatively broad spectral region and the negative value of e σ 00 ðωÞ in a spectral region around ℏω ≈ 2EF are notable spectral characteristics of the surface optical conductivity of graphene. As we shall see in Chapters 6 and 7, each feature leads to interesting physical phenomena and useful applications. For example, by tuning the Fermi energy of graphene, the surface optical conductivity changes accordingly in the terahertz and far-infrared regions, as shown in Figure 4.2. Therefore, the absorption of light can be modulated through tuning the

3

( ) EF = 100 meV

0

( ) Surface optical conductivity,

110

EF = 200 meV

2

1

0

1 0

( ) 100

200

300

(meV) Figure 4.2 Surface optical conductivity of graphene as a function of optical frequency at 300 K in the limit that γ → 0. The Fermi energy is assumed to be EF ¼ 100 meV and 200 meV for the gray curves and black curves, respectively, and the scattering rate is taken to be τ1 ¼ 2:5  1011 s1 . The surface optical conductivity is normalized to e σ 0 ¼ e2 =4ℏ.

4.3 Polarizability Function

111

optical conductivity; this phenomenon can be utilized in the design of a terahertz optical modulator. In the low-temperature limit that jμj ≫ kB T, the surface optical conductivity given by (4.4) can be simplified by using (2.18) and (3.109): e σ ðωÞ ¼

e2 jμj πℏ

2

ðτ1

 iωÞ



ie σ 0 2jμj þ ℏðω þ iγÞ ; ln 2jμj  ℏðω þ iγÞ π

for jμj ≫ kB T:

ð4:5Þ

Strictly speaking, in (4.5) τ1 ðωÞ is equal to γðωÞ and both are functions of the frequency ω because they are both the total scattering rate. However, as we can see in Figure 4.1, in the low-frequency region, the second term in (4.5), which is contributed by interband transitions, can be ignored; then, only the intraband scattering rate τ1 is important, which is contributed by the scattering mechanisms discussed in Section 2.3. By contrast, in the high-frequency region, the first term contributed by intraband transitions can be ignored because at high frequencies only the interband scattering rate γ is important, which is related to the electron–hole recombination rate. Therefore, as an approximation, τ1 and γ are assigned two different constant values representing the scattering rates contributed by different mechanisms at low and high frequencies, respectively, as seen in (4.5). For monolayer graphene, τ1 has an approximately constant value that is independent of frequency [1]. In the literature, τ 1 is sometimes assumed to have the same value as γ to simplify the model. So far we have ignored the spatial nonlocality of the system. We assume that the response of graphene to the external excitation is independent of the location; physical quantities such as the carrier density, the current density, and the electric field on the graphene surface are considered to be uniformly distributed everywhere. This assumption is justified if the external field is characterized by a wavelength much larger than the physical size of the structure and that of the graphene sheet under consideration. If this condition is not true, e σ ðωÞ described above has to be modified, and a new model is needed, as discussed below.

4.3

Polarizability Function Consider an external electric field Eext ðr; tÞ that is a function of both spatial location r and time t. Note that Eext ðr; tÞ is the complex field of the real field Eext ðr; tÞ, as defined in (3.13). If Eext ðr; tÞ has a continuous spectrum, it can be expressed according to (3.20) as Eext ðr; tÞ ¼

1 ð2πÞ4

ð∞ ððð

Eext ðq; ωÞeiq  riωt dqdω:

ð4:6Þ

0 all q

If Eext ðr; tÞ is composed of discrete spectral components, then it can be expanded as

112

Optoelectronic Properties

Eext ðr; tÞ ¼

XX

Eext ðq; ωÞeiq  riωt :

ð4:7Þ

ω>0 q

Note that, as discussed in Section 3.1, a complex field is expanded only over positive frequencies, though a real field is expanded over both positive and negative frequencies. Because of the spatial and temporal variations of Eext ðr; tÞ, the surface optical conductivity that characterizes the response of a graphene sample to this field is a function of not only the frequency ω but also the wave vector q, thus e σ ðq; ωÞ. By comparison, the surface optical conductivity e σ ðωÞ given in (4.4) is a function of only frequency but not wave vector because it is obtained under the assumption that the electric field has the spatially uniform form of (3.83), which is a function of only time and not space. In the following, we first carry out a general discussion in a 3D model. A polarizability function Πðq; ωÞ that is a function of wave vector and frequency is defined to quantify the response of a material to a spatially and temporally varying field. Then, the optical conductivity σðq; ωÞ and the electric susceptibility χðq; ωÞ can be found in terms of the polarizability function. With appropriate modifications the results are applied to the 2D properties of graphene. The relation between the electric permite tivity ϵðq; ωÞ of graphene and the surface polarizability function Πðq; ωÞ of graphene in e the 2D model is then derived at the end of this section. Therefore, once Πðq; ωÞ is

known, all other physical quantities, including e σ ðq; ωÞ, e χ ðq; ωÞ, and ϵðq; ωÞ, that characterize the response of a graphene sample to a spatially and temporally varying e field can be found. The surface polarizability function Πðq; ωÞ is then found in Section 4.4 for monolayer graphene under various conditions, and in Section 4.5 for ABstacked bilayer graphene. For simplicity, we consider an isotropic medium so that the conductivity σ and the susceptibility χ are scalars. For graphene, which is isotropic on its plane, the scalar surface conductivity e σ and the scalar surface susceptibility e χ considered below are e σ ∥ and e χ ∥ , respectively. Due to the spatial variation of the external electric field Eext ðr; tÞ, the free carriers in the originally homogeneous graphene are redistributed, creating an induced charge density ρind ðr; tÞ and a corresponding induced electric field Eind ðr; tÞ in response to the external field. The problem becomes much easier if the response is small compared to the external perturbation. Then what is left to be solved is only the charge density ρext ðr; tÞ that is necessary to support the external electric field Eext ðr; tÞ. However, this is usually not true, and in most cases ρind ðr; tÞ is comparable to ρext ðr; tÞ. Therefore the primary goal in this section is to find the induced charge density ρind ðr; tÞ of graphene in response to the external field Eext ðr; tÞ, or in response to the total electric field Eðr; tÞ ¼ Eext ðr; tÞ þ Eind ðr; tÞ. Once the induced charge density ρind ðr; tÞ is found, the induced current density Jind ðr; tÞ and the corresponding optical conductivity can be obtained. Note that ρðr; tÞ is the complex charge density defined in Section 3.1 in relation to the real charge density ρðr; tÞ as ρðr; tÞ ¼ ρðr; tÞ þ ρðr; tÞ. According to the fundamental electrostatic theory, the complex scalar potential φðr; tÞ, which is defined in Section 3.1 in relation to the real scalar potential φðr; tÞ as φðr; tÞ ¼ φðr; tÞ þ φðr; tÞ, is related to the total complex electric field as

4.3 Polarizability Function

Δ

Eðr; tÞ ¼  φðr; tÞ:

113

ð4:8Þ

By taking the Fourier transform on (4.8), we obtain Eðq; ωÞ ¼ iqφðq; ωÞ:

ð4:9Þ

From the continuity equation, the induced charge density ρind ðr; tÞ and the induced current density Jind ðr; tÞ have the relation

Δ

∂ρind ðr; tÞ þ ∂t

 Jind ðr; tÞ ¼ 0;

ð4:10Þ

which gives iωρind ðq; ωÞ þ iq  Jind ðq; ωÞ ¼ 0 through Fourier transform. By inserting (4.9) and the relation Jind ðq; ωÞ ¼ σðq; ωÞEðq; ωÞ into (4.11), we obtain the optical conductivity: σðq; ωÞ ¼

iω ρind ðq; ωÞ : q2 φðq; ωÞ

ð4:11Þ that

ð4:12Þ

Clearly, the response can be expressed as σðq; ωÞ. A strong response measured by a large value of σðq; ωÞ is obtained if a high charge density ρind ðq; ωÞ is induced by a small scalar potential φðq; ωÞ. Alternatively, one can measure the response by the ratio of the induced electron density nind ðq; ωÞ ¼ ρind ðq; ωÞ=ðeÞ to the total electric potential energy V ðq; ωÞ ¼ eφðq; ωÞ. This ratio is defined as the response function: Πðq; ωÞ ¼

nind ðq; ωÞ ρind ðq; ωÞ ¼ 2 : V ðq; ωÞ e φðq; ωÞ

ð4:13Þ

From (4.12) and (4.13), the optical conductivity is related to the response function as σðq; ωÞ ¼

iωe2 Πðq; ωÞ: q2

ð4:14Þ

The response function Πðq; ωÞ is also called the polarizability function, or simply the polarizability, because it also determines the tendency of the charges in a material to be polarized by an electric field. The induced polarization density Pind ðr; tÞ is related to ρind ðr; tÞ as

Δ

ρind ðr; tÞ ¼   Pind ðr; tÞ

ð4:15Þ

or, in the momentum space and frequency domain, ρind ðq; ωÞ ¼ iq  Pind ðq; ωÞ: Using (4.9), (4.13), and (4.16), it can be shown that

ð4:16Þ

Optoelectronic Properties

Pind ðq; ωÞ ¼ 

e2 Πðq; ωÞEðq; ωÞ; q2

ð4:17Þ

or simply Pind ðq; ωÞ ¼ ϵ 0 χðq; ωÞEðq; ωÞ;

ð4:18Þ

where χðq; ωÞ ¼ 

e2 Πðq; ωÞ ϵ 0 q2

ð4:19Þ

is defined as the susceptibility. Equations (4.14) and (4.19) lead to the relation that σðq; ωÞ ¼ iωϵ 0 χðq; ωÞ. This relation seems strange at first look because in preceding chapters we have made the division that the conductivity σ is contributed by free electrons through interband and intraband transitions, whereas the susceptibility χ is contributed by bound electrons. To avoid confusion, some clarification is necessary. The induced charge density includes the contributions from the induced spatial and temporal displacement of the bound charges and the induced spatial and temporal redistribution of the free conduction charge carriers; thus, ρind ðr; tÞ ¼ ρbound ðr; tÞ þ ρcond ðr; tÞ. Correspondingly, the induced current density in a medium has two contributions: the polarization current from the bound charges of the medium and the conduction current from free charge carriers; thus Jind ðr; tÞ ¼ Jbound ðr; tÞ þ Jcond ðr; tÞ. The polarization current Jbound ðr; tÞ of bound electrons is a displacement current, whereas the conduction current Jcond ðr; tÞ is carried by free charge carriers in the medium. Regardless of the sources of contributions, ρind ðr; tÞ and Jind ðr; tÞ are always related to each other through the continuity equation given in (4.10), as required by conservation of charges. Therefore, ρind ðr; tÞ and Jind ðr; tÞ are not independent of each other but dictate each other. Usually, Jbound being a displacement current is included in the ∂D=∂t term through ∂P=∂t as a polarization term Jbound ¼ ∂P=∂t, whereas the conduction current Jcond can be treated in three different alternative approaches, as discussed in Section 3.1. In Chapter 3, the polarization current is accounted for as Jbound ¼ ∂P=∂t, and the conduction current is considered as a separate current term Jcond , both of which are included in the ∂D=∂t term so that Ampère’s equation takes the form:

Δ

114

H¼

∂D ∂E ∂P ¼ ϵ0 þ þ Jcond : ∂t ∂t ∂t

ð4:20Þ

By explicitly dividing Jind into Jbound and Jcond as above, the response of the bound electrons to the electric field is generally characterized by a susceptibility χ through the polarization P, whereas the response of the conduction electrons is characterized by a conductivity σ through the current Jcond . This is the approach taken in the preceding chapters; in this approach, the total permittivity is expressed as ϵðωÞ ¼ ϵ 0 ½1 þ χðωÞ þ iσðωÞ=ω, as given in (3.37). Therefore, χ and σ are not related to each other in the preceding chapters because they account for responses from different sources.

4.3 Polarizability Function

115

In this section, however, we take a different approach by considering the total induced charge density ρind ðr; tÞ and the corresponding induced current density Jind ðr; tÞ without explicitly dividing each of them into bound and conduction parts. In this approach, we have two alternatives. First, being a current in general, Jind can be treated in a manner similar to that done for a conduction current by connecting it to the electric field through a conductivity σ. In this treatment, Ampère’s equation takes the form without a polarization term:  H ¼ ϵ0

∂E þ Jind : ∂t

ð4:21Þ

Δ

Then, the total response is completely characterized by the conductivity σ given in (4.14) without the need to use a susceptibility. Alternatively, being an induced current, Jind can be treated as a displacement current by connecting it to the electric field through a susceptibility χ. In this treatment, Ampère’s equation takes the form without a current term: H¼

∂D ∂E ∂Pind ¼ ϵ0 þ : ∂t ∂t ∂t

ð4:22Þ

Δ

Then, the total response is completely characterized by the susceptibility χ given in (4.19) without the need to use a conductivity. Because ρind ¼   Pind as expressed in (4.15), we find from the continuity equation given in (4.10) that

Δ

Jind ¼

∂Pind : ∂t

ð4:23Þ

This relation clearly indicates that (4.21) and (4.22) are identical to each other. Consequently, the conductivity σ given in (4.14) is equivalent to the susceptibility χ given in (4.19); either of them completely characterizes the response of the material to the electric field. The relation σðq; ωÞ ¼ iωϵ 0 χðq; ωÞ makes the connection between the two. In this approach, instead of the expression given in (3.37) and stated above, the permittivity tensor can be expressed as   σðωÞ ϵðωÞ ¼ ϵ 0 1 þ χðωÞ ¼ ϵ 0 þ i : ð4:24Þ ω In a real experiment it is difficult, and generally not necessary, to distinguish whether the measured response of graphene is contributed by free or bound electrons. Furthermore, a bound electron in the valence band can become a free electron in the conduction band by absorbing a photon when the photon energy is high enough. Therefore, it becomes unnecessary to distinguish between free and bound electrons, especially when the response we intend to find is applicable across a wide spectral region. For this reason, in the literature the experimental measurement is sometimes expressed in terms of σ and sometimes in terms of χ, or frequently in terms of the refractive index n. In fact, we have used the relationship σ ¼ iωϵ 0 χ in (3.123) in

116

Optoelectronic Properties

Table 4.1 Units of the 2D and the corresponding 3D physical quantities.

2D 3D

Optical conductivity e σ (2D), σ (3D)

e Polarizability function Π (2D), Π (3D)

Susceptibility e χ (2D), χ (3D)

S S m1

C1 V1 m2 C1 V1 m3

m 1

Chapter 3, where we defined an “effective optical conductivity” as the sum of the optical conductivity from free electrons and the contribution from the bound electrons. When we only consider the free electrons, we can arguably write χðq; ωÞ ¼ iσðq; ωÞ=ωϵ 0 as the effective susceptibility contributed by free electrons. The physical relations listed in the equations above can be applied to 2D or 3D quantities, which have different units as listed in Table 4.1. For 2D quantities, a tilde is added on top of the variables. The surface current density e J ind and the surface optical e are discussed in Chapter 2, and other quantities such as the surface conductivity σ e ind are discussed in electric susceptibility e χ and the surface electric polarization P Chapter 3. Specifically, for graphene, the 2D surface polarizability function is defined as e ρ ðq; ωÞ n ind ðq; ωÞ e e Πðq; ωÞ ¼ ¼ 2ind ; V ðq; ωÞ e φðq; ωÞ

ð4:25Þ

where e n ind and e ρ ind are induced surface electron density and surface charge density, respectively, on the graphene sheet. Note that V ðq; ωÞ and φðq; ωÞ remain 3D quantities. The surface optical conductivity of graphene is related to the surface polarizability function as e σ ðq; ωÞ ¼

iωe2 e Πðq; ωÞ; q2

ð4:26Þ

and the surface electric susceptibility of graphene is related to the surface polarizability function as e χ ðq; ωÞ ¼ 

e2 e Πðq; ωÞ: ϵ 0 q2

ð4:27Þ

Similar to the relationship between their 3D counterparts as discussed above, the surface optical conductivity e σ ðq; ωÞ and the surface electric susceptibility e χ ðq; ωÞ are related as e χ ðq; ωÞ ¼ i

e σ ðq; ωÞ ; ϵ0 ω

ð4:28Þ

which can be found from (4.26) and (4.27). In the case when only 2D quantities are considered, the relations derived above are only meaningful with q ¼ qx^x þ qy^y and r ¼ x^x þ y^y on the graphene sheet.

4.3 Polarizability Function

Graphene

117

1 2

Figure 4.3 Graphene sheet sandwiched between two dielectrics of permittivities ϵ 1 and ϵ 2 .

Although in the following we solely focus on the 2D physical properties, some quantities are inherently three-dimensional. For example, though the charge density e ρ on the graphene sheet is a 2D quantity because the charges are confined on the 2D surface, the scalar potential φ is a 3D quantity because the Coulomb force produced by the surface charges on graphene necessarily extends into the surrounding environment. For this e reason, the surface polarizability function Πðq; ωÞ as defined in (4.25) is expressed in e terms of a ratio between the 2D quantity ρ ind ðq; ωÞ and the 3D quantity φðq; ωÞ. Similarly, the electric field E is necessarily a 3D quantity, though the surface current density e J is confined on the 2D graphene sheet. Therefore, to derive the electric permittivity ϵðq; ωÞ or to establish the relation between φðq; ωÞ and e ρ ðq; ωÞ, we have to take into account the surrounding dielectrics and also the normal component of the electric field. Consider a graphene sheet sandwiched between two dielectrics of permittivities ϵ 1 and ϵ 2 , which occupy the semi-infinite spaces above and below the graphene sheet, respectively, as shown in Figure 4.3. The electric permittivity ϵðq; ωÞ is defined as [2] ϵðq; ωÞ ¼ ϵ ave

φext ðq; ωÞ ; φðq; ωÞ

ð4:29Þ

where ϵ ave ¼ ðϵ 1 þ ϵ 2 Þ=2. In Chapter 3, the permittivity is expressed in a tensor form, as given by (3.40). Because in this chapter we only consider free electrons and the conductivity that only exists in the direction parallel to the graphene sheet, the permittivity given by (4.29) is the parallel component contributed by free electrons. The physical meaning is clearer if we rearrange (4.29) in the form: φðq; ωÞ ¼ ϵ ave

φext ðq; ωÞ : ϵðq; ωÞ

ð4:30Þ

When the external potential φext ðq; ωÞ is applied on the graphene sheet, the free electrons on the graphene sheet are rearranged to produce an induced potential φind ðq; ωÞ to screen the external field. Unless ϵðq; ωÞ ¼ ϵ ave , the effective potential felt by the electrons is the screened potential given by ϵ ave φext ðq; ωÞ=ϵðq; ωÞ instead of φext ðq; ωÞ. From (4.9), we also find that the screened electric field is given by ϵ ave Eext ðq; ωÞ=ϵðq; ωÞ. e To establish the relation between ϵðq; ωÞ and Πðq; ωÞ, we consider a situation in which the induced charge density on the graphene sheet gives rise to a scalar potential that decays away from the graphene sheet. Assuming that the graphene sheet is located at

118

Optoelectronic Properties

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼ 0, the decay constants are γ1 ¼ q2  k12 ¼ q2  n21 ω2 =c2 for z > 0, where the dielectric medium has a refractive index of n1 corresponding to the permittivity ϵ 1, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ2 ¼ q2  k22 ¼ q2  n22 ω2 =c2 for z < 0, where the dielectric medium has a refractive index of n2 corresponding to the permittivity ϵ 2. In the following, we take the electrostatic limit that the frequency is low such that ω → 0 and q ≫ ω=c; a more general case is considered in Chapter 6. In the electrostatic limit, we have q ≫ ω=c so that both decay constants in the z direction have approximately the same magnitude as the wave number on the xy plane: γ1 ≈ q and γ2 ≈ q. Then, the induced scalar potential has the form: φind ðr; z; tÞ ¼ φind ðq; ωÞeqjzj eiq  riωt ;

ð4:31Þ

where only one Fourier component φind ðq; ωÞ is considered. From (4.31) and (4.8) we have Ez ðr; z ¼ 0þ ; tÞ ¼ qφind ðq; ωÞeiq  riωt

ð4:32Þ

Ez ðr; z ¼ 0 ; tÞ ¼ qφind ðq; ωÞeiq  riωt :

ð4:33Þ

and

The Dz fields just above and below the graphene sheet are given by the relations Dz ðr; z ¼ 0þ ; tÞ ¼ ϵ 1 Ez ðr; z ¼ 0þ ; tÞ and Dz ðr; z ¼ 0 ; tÞ ¼ ϵ 2 Ez ðr; z ¼ 0 ; tÞ, respectively. We can obtain the surface charge density on the graphene sheet using the boundary condition: e ρ ind ðr; 0; tÞ ¼ Dz ðr; 0þ ; tÞ  Dz ðr; 0 ; tÞ ¼ ðϵ 1 þ ϵ 2 Þqφind ðq; ωÞeiq  riωt ;

ð4:34Þ

where a tilde is added for e ρ ind because it is a 2D quantity. Thus, we find the relation between the scalar potential and the surface charge density: e ρ ind ðq; ωÞ ¼ ðϵ 1 þ ϵ 2 Þqφind ðq; ωÞ:

ð4:35Þ

From (4.13) and (4.35), we find that the surface polarizability function, or simply the surface polarizability, of the graphene sheet can be expressed as e ρ ðq; ωÞ e Πðq; ωÞ ¼ 2ind e φðq; ωÞ 2ϵ ave q φind ðq; ωÞ e2 φðq; ωÞ   2ϵ ave q φext ðq; ωÞ 1  ¼ : e2 φðq; ωÞ

¼

We can then obtain ϵðq; ωÞ from (4.29) and (4.36) as

ð4:36Þ

4.3 Polarizability Function

ϵðq; ωÞ e2 e Πðq; ωÞ: ¼1 ϵ ave 2ϵ ave q

119

ð4:37Þ

e By using the relation between e χ ðq; ωÞ and Πðq; ωÞ, given in (4.27), and that between e e σ ðq; ωÞ and Πðq; ωÞ, given in (4.26), we find that the electric permittivity given in (4.37) can be expressed as ϵðq; ωÞ ¼ ϵ ave þ ϵ 0

e e σ ðq; ωÞ χ ðq; ωÞ ¼ ϵ ave þ i 1 : 2q ω 2q1

ð4:38Þ

By comparing this relation to that given in (3.39), where the relation between ϵ and e χ and that between ϵ and e σ are defined for a graphene sheet of a thickness d that is suspended in free space, we find that (4.38), and thus (4.37), indeed gives the permittivity of a graphene sheet that is sandwiched between two dielectrics of an average permittivity of ϵ ave if we identity 2q1 as the effective thickness of the graphene sheet. This effective thickness of 2q1 results from the fact that, by taking the decaying constants to be γ1 ≈ q and γ2 ≈ q, we have assumed that the electric potential field created by the charge density on the graphene sheet penetrates a distance of q1 each above and below the graphene sheet into the upper and lower dielectrics. Note that different from that in (3.39), ϵðq; ωÞ in (4.38) is not expressed as a sum of the contributions from e χ and e σ , but is expressed in terms of either e χ or e σ because in this chapter we treat e χ and e σ as equivalents, which are related through (4.28), as discussed above. It is worth mentioning that instead of (4.36) the surface polarizability function is sometimes defined as [3] e ðq; ωÞ ¼ 2qϵ ave φind ðq; ωÞ : Π e2 φext ðq; ωÞ

ð4:39Þ

In such a case, we can still obtain ϵðq; ωÞ from (4.29) and (4.39) in a different form: ϵ ave e2 e ¼1þ Π ðq; ωÞ: ϵðq; ωÞ 2ϵ ave q

ð4:40Þ

To compare (4.37) with (4.40), we rewrite (4.37) as ϵ ave e s ðq; ωÞΠðq; e ¼ 1þU ωÞ; ϵðq; ωÞ

ð4:41Þ

where e s ðq; ωÞ ¼ U

1 e ωÞ 2ϵ ave q=e2  Πðq;

¼

e2 2ϵðq; ωÞq

ð4:42Þ

is the 2D screened Coulomb potential energy. In fact, we have used (4.42) for the calculation of the scattering rate in (2.30). As can be seen, (4.40) describes the full response to the external field via the bare Coulomb interaction, whereas (4.41) describes

120

Optoelectronic Properties

the response to the screened field via the screened Coulomb interaction. Usually we prefer the relation given in (4.41) to that given in (4.40) for the reason that the screened Coulomb interaction is short-ranged and is free from singularity in the limit that q → 0.

4.4

Random-Phase Approximation From the above discussion, we find that once the relation between e ρ ind ðq; ωÞ and φðq; ωÞ e is found, the surface polarizability function Πðq; ωÞ can be obtained from (4.25), which in turn gives the surface optical conductivity e σ ðq; ωÞ from (4.26), the surface electric susceptibility e χ ðq; ωÞ from (4.27), and the electric permittivity ϵðq; ωÞ from (4.37). However, it is very difficult, if not impossible, to solve for the polarizability function by considering the response of every single electron to an external field while accounting for the interactions among the electrons. This difficulty can be circumvented by adopting several approximations. First, the magnitude of the external field is assumed to be sufficiently small so that only the linear response to φðq; ωÞ is important while highorder nonlinear responses can be ignored. By adopting this linear approximation, the surface polarizability function can be expressed in the form of a ratio of e ρ ind ðq; ωÞ to φðq; ωÞ, as given by (4.13). Second, the electron–electron interaction is ignored. This approximation can be justified by the fact that the electron–electron interaction is in part accounted for by the screening effect and the fact that the screened Coulomb interaction is much weaker than the bare Coulomb interaction. Finally, we only consider the inphase response of electrons to the external perturbation; thus the phase difference between the external field and the induced field is independent of the locations of the elections. By contrast, the out-of-phase response of electrons is assumed to average out to zero due to the random locations of the electrons in a large quantity. Therefore, this approximation is also called the random-phase approximation (RPA) [4]. In the followe ing we derive the surface polarizability function Πðq; ωÞ using the density matrix [5].

Electron doping with EF > 0 and the aforementioned approximations are assumed. The derived results with a positive EF should also apply to the case of hole doping with EF < 0 in the Dirac cone approximation due to the symmetry of the band structure. The Hamiltonian can be written as ^ ¼H ^0 þH ^ 0ðr; tÞ; H

ð4:43Þ

^ 0 jψk 〉 ¼ Ek jψk 〉. The interaction Hamiltonian of graphene in ^ 0 ¼ vF σ  p and H where H an electromagnetic field is given in (3.60): ^ 0 ¼ vF eσ  A  eφ; H

ð4:44Þ

where A is the real vector potential and φ is the real scalar field. In Chapter 3 we considered the optical properties of graphene, for which the scalar potential in the absence of an external charge source at the optical frequency being considered can be taken to be φ ¼ 0 because there is no net free charge density at the optical frequency, as discussed in Section 3.1. Therefore, the interaction Hamiltonian has the form

4.4 Random-Phase Approximation

121

^ 0 ¼ v F eσ  A for the discussions in Chapter 3. This approach is equivalent to including H all the effects of optical response in an induced current density J ind, as is done in Chapter 3. This approach is commonly taken for the consideration of the optical properties of a material when the effect of the spatial variation of the optical field is negligible and thus ignored, but only that of the temporal variation is considered for the frequency dependence of the optical properties. When the effect of the spatial variation of the optical field is not negligible, as is the case for the optoelectronic properties considered in this chapter, it is no longer valid to set φ ¼ 0. In this situation, alternatively, the effects of optoelectronic response can be included in a spatially varying induced charge density ρind ðr; tÞ, thus a spatially varying scalar potential φðr; tÞ, as seen in the preceding sections. In this approach, we consider an interaction Hamiltonian of the form   ^ 0ðr; tÞ ¼ eφðr; tÞ ¼ e φðr; tÞ þ φðr; tÞ ¼ V ðr; tÞ þ V ðr; tÞ; ð4:45Þ H where φ is the real scalar potential and φ is the complex scalar potential, and V ðr; tÞ ¼ eφðr; tÞ:

ð4:46Þ

For the following discussions in this chapter, we consider the interaction of graphene with a spatially varying monochromatic optical field at a frequency ω such that the ^ 0ðr; tÞ is expressed in the interaction picture as perturbation Hamiltonian H ^ 0ðr; tÞ ¼ H

X

^ ðq0; ωÞeiq0  r eiωtþγt þ c:c:; V

ð4:47Þ

q0

^ ðq0; ωÞ is related to φðq0; ωÞ through the space and time domain relation where V V ðr; tÞ ¼ eφðr; tÞ given in (4.46) as ^ ðq0; ωÞ ¼ eiH^ 0 t=ℏ eφðq0; ωÞeiH^ 0 t=ℏ V

ð4:48Þ

^ 0ðr; tÞ expressed in in the interaction picture. Note that the interaction Hamiltonian H (4.47) includes a complex conjugate term because it has to be a Hermitian operator. As in Chapter 3, exp ðγtÞ is added in (4.47) so that the perturbation is adiabatically turned on as time evolves and is switched off at time t →  ∞. Normally, (4.47) is integrated over the frequency spectrum in analogy to (3.88) if the field has a broad frequency spectrum. Here we consider a monochromatic perturbation field for simplicity so that only one frequency component exits. Once the perturbation is defined, the matrix component can be found: ^ 0ðr; tÞjk〉 ¼ eiðωkþq ωk ωÞtþγt 〈k þ qjH

X

0

eφðq0; ωÞ〈k þ qjeiq  r jk〉

q0

eiðωkþq ωk þωÞtþγt

X

0

eφðq0; ωÞ〈k þ qjeiq  r jk〉;

ð4:49Þ

q0

where ωk ¼ Ek =ℏ ¼ nvF jkj, ωkþq ¼ Ekþq =ℏ ¼ n0v F jk þ qj, and n and n0 are the band indices of states jk〉 and jk þ q〉, respectively. As in Chapter 3, to simplify the notations,

122

Optoelectronic Properties

the n and n0 indices are omitted in the expressions of jk〉, jk þ q〉, Ek , and Ekþq . 0 The matrix components 〈k þ qjeiq  r jk〉 are found by using (1.28): ðð 1 0 0 ψ†kþq eiq  r ψk dr 〈k þ qjeiq  r jk〉¼ A ¼

1 þ n0neiðθk θkþq Þ δq;q0 2

ð4:50Þ

¼ Fðk; k þ qÞδq;q0; where A is the area of the graphene sheet under consideration and F is a form factor defined as Fðk; k0 Þ ¼

1 þ n0neiðθk θk0Þ ¼ F ðk0; kÞ: 2

ð4:51Þ

By inserting (4.50) into (4.49), we obtain ^ 0ðr; tÞjk〉 ¼ eφðq; ωÞFðk; k þ qÞeiðωkþq ωk ωÞtþγt 〈k þ qjH

ð4:52Þ

eφðq; ωÞFðk; k þ qÞeiðωkþq ωk þωÞtþγt : With the density operator ^ρ ðtÞ defined in (3.91), the time evolution of the density matrix element can be obtained from (3.92) by following the procedure similar to that used to obtain (3.96):   ∂ i 〈k þ qj^ρ ðtÞjk〉 ¼ f0 ðEk Þ  f0 ðEkþq Þ Fðk; k þ qÞ ∂t ℏ h i  eφðq; ωÞeiðωkþq ωk ωÞtþγt þ eφðq; ωÞeiðωkþq ωk þωÞtþγt ; ð4:53Þ where (4.52) is used. Note that ^ρ ð0Þ is independent of time so that ∂^ρ ð1Þ =∂t ¼ ∂^ρ =∂t, which is used in obtaining (4.53). Integrating (4.53) over time, we obtain   〈k þ qj^ρ ðtÞjk〉 ¼ f0 ðEk Þ  f0 ðEkþq Þ Fðk; k þ qÞ  

eφðq; ωÞeiðωkþq ωk ωÞtþγt eφðq; ωÞeiðωkþq ωk þωÞtþγt þ Ekþq  Ek  ℏðω þ iγÞ Ekþq  Ek þ ℏðω þ iγÞ



ð4:54Þ in analogy to (3.97). To find the carrier density, we introduce the carrier-density operator in the interaction picture as ^

^

^ n ðr; tÞ ¼ eiH 0 t=ℏ δð^r  rÞeiH 0 t=ℏ ;

ð4:55Þ

4.4 Random-Phase Approximation

123

where ^r is the position operator and r is a specific location in space. The carrier density can be found by finding the trace of ^ρ ðtÞ^n ðr; tÞ. For example, in the steady state without perturbation, we have 〈k0j^ρ ðtÞjk〉 ¼ f0 ðkÞδk;k0 and ðð 1 ^ ^ ψ†k0eiH 0 t=ℏ δð^r  rÞeiH 0 t=ℏ ψk d^r 〈k0 j^ n ðr; tÞjk〉 ¼ A ð4:56Þ 1 0 iðkk0 Þ  r iðωk ωk0Þt ¼ Fðk; k Þe e A using (1.28). Then the total electron density without perturbation is   e n ðrÞ ¼ Tr ^ρ ðtÞ^ n ðr; tÞ ¼g

X

〈kj^ρ ðtÞ^n ðr; tÞjk〉

n;k

¼g

XX 〈kj^ρ ðtÞjk0〉〈k0j^n ðr; tÞjk〉 n;k n0;k0

¼

ð4:57Þ

gX f0 ðkÞ A n;k

g ¼ 2 4π

ðð f0 ðkÞdk;

which gives the total electron density n0, consistent with (1.37) by changing the variable from k to energy E. In (4.57), g is the total degeneracy, and the off-diagonal elements have no contribution because 〈kj^ρ ðtÞjk0〉 ¼ 0 for k ≠ k0 when no perturbation is present. In the perturbed situation, 〈k þ qj^ρ ðtÞjk〉 is given by (4.54) so that 〈k0j^ρ ðtÞjk〉 ≠ f0 ðkÞδk;k0. Then, because the surface charge density is given by the carrier density through the relation e ρ ðr; tÞ ¼ ee n ðr; tÞ, the induced charge density can be found as   e ρ ind ðr; tÞ ¼ e Tr ^ρ ðtÞ^ n ðr; tÞ ¼ eg

X 〈k0j^ρ ðtÞ^ n ðr; tÞjk0〉 n0;k0

¼ eg

XX

ð4:58Þ 0

0

〈k j^ρ ðtÞjk〉〈kj^n ðr; tÞjk 〉

n0;k0 n;k

¼ e

g XX 0 0 〈k j^ρ ðtÞjk〉Fðk0; kÞeiðk kÞ  r eiðωk0ωk Þt ; A n;n0 k;k0

where (4.57) is used. Note that the carrier density as calculated in (4.58) is a real quantity such that e ρ ind ðr; tÞ ¼ e ρ ind ðr; tÞ þ e ρ ind ðr; tÞ, where e ρ ind ðr; tÞ is the complex carrier

124

Optoelectronic Properties

density. We can replace the dummy variable k0 with k þ q in (4.58) to express (4.58) in the form: e ρ ind ðr; tÞ ¼ e

g XX 〈k þ qj^ρ ðtÞjk〉Fðk þ q; kÞeiq  r eiðωkþq ωk Þt ; A n;n0 k;q

ð4:59Þ

which has to be equal to the sum of all the Fourier components e ρ ind ðq; ωÞ: X X  e e e ρ ind ðr; tÞ ¼ ρ ind ðq; ωÞeiq  r eiωtþγt þ ρ ind ðq; ωÞeiq  r eiωtþγt q

¼

X q

q

e ρ ind ðq; ωÞe

iq  r iωtþγt

e

þ

X

ð4:60Þ e ρ ind ðq; ωÞeiq  r eiωtþγt

q

in analogy to (4.47). By using (4.54), (4.59), and (4.60) we obtain e ρ ind ðq; ωÞ ¼ e2 φðq; ωÞ

g X X 2 f0 ðEk Þ  f0 ðEkþq Þ ; jFj A n;n0 k Ekþq  Ek  ℏðω þ iγÞ

ð4:61Þ

where jFj2 ¼ Fðk; k þ qÞ  Fðk þ q; kÞ ¼ jFðk; k þ qÞj2 because Fðk þ q; kÞ ¼ F ðk; k þ qÞ: e e From the relation e ρ ind ðq; ωÞ ¼ e2 φðq; ωÞΠðq; ωÞ given in (4.25) to define Πðq; ωÞ,we finally obtain the desired surface polarizability function: g XX 1 þ n0n cos ðθkþq  θk Þ f0 ðEk Þ  f0 ðEkþq Þ e Πðq; ωÞ ¼ ; A n0;n k 2 ℏðω þ iγÞ þ Ek  Ekþq

ð4:62Þ

where (4.51) is used to explicitly carry out jFj2 . Note that in our convention we use expðiωtÞ for the complex time-varying field; an extra minus sign has to be added in front of the summation in (4.62) if the convention expðiωtÞ is used instead. The physical meaning of the term f0 ðEk Þ  f0 ðEkþq Þ in (4.62) is clearer if we rewrite the term as f0 ðEk Þ½1  f0 ðEkþq Þ  f0 ðEkþq Þ½1  f0 ðEk Þ. Therefore, f0 ðEk Þ in (4.62) is associated with the probability of a transition from the state jk〉 to the state jk þ q〉, whereas f0 ðEkþq Þ in (4.62) is associated with the probability of a transition from the state jk þ q〉 to the state jk〉. Because each of these two transitions is the reverse process of the other, we calculate the difference of these probabilities instead of calculating their sum. If there is an equal probability of the transition from jk þ q〉 to jk〉 and the transition from jk〉 to jk þ q〉, the net effect is zero; in this case, the transitions between the two states jk þ q〉 and jk〉 do not contribute to the surface polarizability function e Πðq; ωÞ.

4.4.1

Beyond Random-Phase Approximation Strictly speaking, the preceding derivation leading to (4.62) is valid only in the limit that γ → 0. For a nonzero value of γ, the derivation for the polarizability function has to be carried out within the relaxation-time approximation, as discussed in Chapter 2. However, instead of relaxing toward the thermal equilibrium f0 as dictated by (2.12),

4.4 Random-Phase Approximation

125

the elements of the density matrix relax toward local equilibrium characterized by a local chemical potential, which is determined by a balance equation so that the number of carriers is locally conserved. Without going into the details of the calculation, the surface polarizability function for γ ≠ 0 is given as [6] e γ ðq; ωÞ ¼ Π

ω þ iγ

e e ω=Πðq; ωÞ þ iγ=Πðq; 0Þ

;

ð4:63Þ

e e where Πðq; ωÞ is that given in (4.62), with γ ≠ 0, and Πðq; 0Þ is obtained by setting ω þ iγ to zero in (4.62). The surface polarizability function given in (4.63) is derived within the framework of RPA in the relaxation-time (RT) approximation; e γ ðq; ωÞ is sometimes called the therefore the surface polarizability function Π e γ ðq; ωÞ approaches RPA–RT surface polarizability function. In the limit that γ → 0, Π e Πðq; ωÞ given in (4.62).

4.4.2

Low-Temperature Approximation In the following, we derive the explicit form of the surface polarizability function given by (4.62) in the limit that T → 0. The surface polarizability function given by (4.62) is contributed by both intraband transitions, for which n0n ¼ 1, and interband transitions, for which n0n ¼ 1. For intrinsic graphene, which has a zero Fermi energy, in the limit that the temperature T → 0, which leads to a zero chemical potential μ ¼ EF ¼ 0, the only possible carrier transitions are interband transitions from the valence band to the conduction band because the valence band is completely occupied and the conduction band is completely empty, as shown in Figure 4.4(a). Therefore, most transitions can be eliminated from the summation in (4.62). After the elimination, we obtain the following surface polarizability function [7]: e 0 ðq; ωÞ ¼ g Π 4π2

ð∞ 2ðπ 0 0

 1  cos ðθkþq  θk Þ f0 ðjEk jÞ ℏðω þ iγÞ  jEk j  jEkþq j 2

 f0 ðjEkþq jÞ  kdθk dk; ℏðω þ iγÞ þ jEk j þ jEkþq j

ð4:64Þ

e 0 indicates where jEk j ¼ ℏvF jkj, jEkþq j ¼ ℏvF jk þ qj, and the zero in the subscript of Π zero Fermi energy EF ¼ 0 for intrinsic graphene. Because only interband transitions are allowed in this case, n0n ¼ 1. The integral is carried out from k ¼ 0 to k → ∞ so that all possible interband transitions on the Dirac cone are considered. In (4.64), the first term in the brackets of the integrand represents a transition from the valence band to the conduction band by absorbing a momentum of ℏq, as illustrated by the solid arrows in Figure 4.4. The second term represents a transition from the valence band to the conduction band by emitting a momentum of ℏq, as illustrated by the dashed arrows in Figure 4.4. As we shall see shortly, the latter type of transition does not follow the law of conservation of energy, and it does not contribute to the polarizability function in the limit that γ → 0.

126

Optoelectronic Properties

(a)

k +q

(b)

k+q

k

q k +q

EF

k k+q

k

q k +q

k

k

Figure 4.4 (a) Examples of a transition from jk〉 to jk þ q〉, shown by a solid arrow, and a transition

from jk þ q〉 to jk〉, shown by a dashed arrow, in the limit that T → 0. For EF ¼ 0, the valence band is fully occupied, indicated by the dark gray area. In (b), the transitions in (a) are shown in the k space. The angle θkþq  θk between k þ q and k and the angle θ between k and q are indicated.

If we rotate the dashed arrows in Figure 4.4(b) around the Dirac point by 180°, the dashed k vector turns into the solid k þ q vector, and the dashed k þ q vector turns into the solid k vector. Therefore, we can swap the subscripts k and k þ q for the second term in the brackets of (4.64) without changing the value of the integral. This process is tantamount to the transformation of an integral over k into an integral over k0 ¼ k þ q. This technique is valid because of the fact that for each k there is always a corresponding k0 if q is predetermined and the fact that the Dirac cone has rotational symmetry; therefore, rotation of the axis does not affect the value of the integral. Following this process, we transform (4.64) into the form:  ð∞ ðπ  1 jkj þ jqj cos θ e 0 ðq; ωÞ ¼ Π 1  π2 ℏ jk þ qj 0 0 " # 1 1   kdθdk; ðω þ iγÞ  vF jkj  v F jk þ qj ðω þ iγÞ þ vF jkj þ vF jk þ qj ð4:65Þ where g ¼ 4, f0 ðjEk jÞ ¼ f0 ðjEkþq jÞ ¼ 1 for the fact that μ ¼ EF ¼ 0, and θ is the angle between k and q as shown in Figure 4.4(b). The term in the parentheses in (4.65) is obtained by replacing 1  cos ðθkþq  θk Þ in (4.64) with

4.4 Random-Phase Approximation

1  cos ðθkþq  θk Þ ¼ 1 

k  ðk þ qÞ jkj þ jqj cos θ ¼1 : jkjjk þ qj jk þ qj

127

ð4:66Þ

Because cos θ is an even function of θ, the integral over the interval ½0; π has the same value as that over ½π; 2π. Therefore, in (4.65) we write the integral over the interval ½0; π and multiply it by 2 to cover the whole 2π angular range. The integral in (4.65) can be simplified by a change of variables from integration over pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi θ to that over k 0 ¼ jk þ qj ¼ k 2 þ 2kq cos θ þ q2. Then, k þ q ≥ k 0 ≥ jk  qj for 0 ≤ θ ≤ π over the range of the integration, 2k 0 dθ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dk 0; ðk 0 þ kÞ2  q2 q2  ðk 0  kÞ2

ð4:67Þ

and (4.66) becomes 1

jkj þ jqj cos θ q2  ðk 0  kÞ2 ¼ : jk þ qj 2kk 0

ð4:68Þ

By inserting (4.67) and (4.68) into (4.65), we obtain e 0 ðq; ωÞ ¼  1 Π π2 ℏ

ð ð∞ jkqj 0 kþq

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2  ðk 0  kÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk 0 þ kÞ2  q2



 1 1   dk 0dk: ðω þ iγÞ  v F k  vF k 0 ðω þ iγÞ þ vF k þ v F k 0

ð4:69Þ

00

e 0 ðq; ωÞ can be easily found e 0 ðq; ωÞ of Π In the limit that γ → 0, the imaginary part Π from (4.69). By using (3.105), we find from (4.69) that e 000 ðq; ωÞ ¼ 1 Π πℏ

ð∞ jkqj ð 0 kþq

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i q2  ðk 0  kÞ2 h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δðω  vF k  vF k 0Þ  δðω þ vF k þ v F k 0 Þ dk 0dk: ðk 0 þ kÞ2  q2 ð4:70Þ

The first Delta function in the brackets selects Ek0 to have the value of ℏω  ℏvF k with Ek ¼ ℏvF k < 0 for the jk〉 state to be in the valence band and Ek0 ¼ ℏvF k 0 ¼ ℏω  ℏvF k > 0 for the jk0〉 state to be in the conduction band, which manifests the law of energy conservation, whereas the second Dirac function does not contribute to the integral. Therefore, for the integral to be nonzero, ω has to be larger than vF k because v F k 0 ¼ ω  v F k has a positive value between vF jk  qj and v F ðk þ qÞ. With these conditions, it can be shown that the inequality ω=vF þ q > 2k > ω=vF  q is true for both cases of q > k and k > q. Therefore, in (4.70) we only need to integrate k from ðω=v F  qÞ=2 to ðω=vF þ qÞ=2; we have

128

Optoelectronic Properties

e 000 ðq; ωÞ Π

Hðω  v F qÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ πℏvF ω2  v 2F q2

ðω=v FðþqÞ=2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2F q2  ðω  2vF kÞ2 dk

ðω=vF qÞ=2

  ðω=vF þqÞ=2 Hðω  v F qÞ iv F q2 2vF k  ω pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G ¼ vF q πℏv F ω2  v 2F q2 4 ðω=v F qÞ=2 2 q ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hðω  vF qÞ; 4ℏ ω2  v2F q2

j

ð4:71Þ

where HðxÞ is the Heaviside step function that has a value of 0 for x < 0 and a value of 1 for x ≥ 0, and pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi GðxÞ ¼ x x2  1  In ðx þ x2  1Þ:

ð4:72Þ

e 0 ðq; ωÞ can be found by using the Kramers–Kronig relation: e 00 ðq; ωÞ of Π The real part Π ð∞ 0 2 ω0 e e  ðq; ωÞdω0 Π0ðq; ωÞ ¼ P 0 2 Im Π 0 π ω  ω2 0

ð4:73Þ

q2 ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HðvF q  ωÞ: 4ℏ v2F q2  ω2 Therefore, we have, from (4.71) and (4.73), " # 2 q 1 i e 0 ðq; ωÞ ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HðvF q  ωÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hðω  v F qÞ : ð4:74Þ Π 4ℏ v 2F q2  ω2 ω2  v2F q2 It is tempting to combine the two terms in the brackets in (4.74) to write pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= v 2F q2  ω2 , which gives the real part in (4.74) for v F q > ω and the imaginary part in (4.74) for ω > vF q. However, by doing so, an unphysical singularity at vF q ¼ ω appears. The singularity can be removed if we restore the nonzero γ by writing ω as ω þ iγ in (4.74); thus, we can simplify (4.74) as e 0 ðq; ωÞ ¼  Π

iq2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4ℏ ðω þ iγÞ2  v 2F q2

ð4:75Þ

which can be numerically shown to be the same as (4.69) for any positive γ. In the limit that γ → 0, (4.75) approaches (4.74), as shown in Figure 4.4. When the phase velocity of the field, ω=q, is larger than the Fermi velocity of electrons in graphene, i.e., ω=vF q > 1, interband transitions are allowed, as shown in Figure 4.4. In this case, the surface e 0 ðq; ωÞ given by (4.74) is purely imaginary, and therefore the polarizability function Π surface optical conductivity given by (4.26) is real, signifying a joule loss due to interband absorption. When ω=q < vF , interband transitions of electrons are not allowed. In this case, the surface optical conductivity is purely imaginary, signifying a π=2 phase shift between the oscillation of electrons and the driving electric field.

4.4 Random-Phase Approximation

3

(b)

F

)

2

2

(q

4

0

(q 0

3

4

F

)

(a)

129

1

0

1

0 0

1

2

0

q F

1

2 q F

e 0 ðq; ωÞ normalized to Figure 4.5 (a) Real part and (b) imaginary part of the polarizability function Π q=4ℏvF in the limit that T → 0. The solid curves are obtained from (4.74) in the limit that γ → 0. The dashed and dotted curves are obtained from (4.75) using γ ¼ 0:05vF q and γ ¼ 0:1vF q, respectively.

e 00 extends below ω=v F q ¼ 1 and Π e 000 extends above For a nonzero value of γ, Π ω=v F q ¼ 1, as shown in Figure 4.5, signifying the broadening of the energy level dictated by the uncertainty principle due to the finite relaxation time given by γ1. In most cases, synthesized graphene samples are doped either intentionally or unintentionally so that EF ≠ 0, as shown in Figure 4.6 in the case that EF > 0 for n-type doping. The interband transitions that are originally allowed in Figure 4.4 are now Pauli blocked if the final states of these transitions in the conduction band are filled with electrons; only interband transitions to final states in the conduction band above EF are allowed. Furthermore, it is now possible for an electron in the conduction band to make an intraband transition to a higher energy state by absorbing a momentum of ℏq or to a lower energy state by releasing a momentum of ℏq, as shown in Figure 4.6. The polarizability function in the limit that T → 0 for EF ≠ 0 can be written as e e 0 ðq; ωÞ  Π e  ðq; ωÞ þ Π e þ ðq; ωÞ; Πðq; ωÞ ¼ Π EF EF

ð4:76Þ

where the first term is given by (4.75), the second term accounts for the subtraction of the blocked interband transitions (with a superscript minus sign for n0n ¼ 1) that have final states below the Fermi energy, and the last term is the contribution from the intraband transitions (with a superscript plus sign for n0n ¼ 1). The second term in (4.76) has e 0 ðq; ωÞ: a form similar to that of Π e  ðq; ωÞ Π EF

1 ¼ 2 π ℏ

kðF ð π

jkj þ jqj cos θ jk þ qj



1 ðω þ iγÞ  v F jkj  vF jk þ qj 0 0  ð4:77Þ 1  kdθdk: ðω þ iγÞ þ vF jkj þ v F jk þ qj 1

130

Optoelectronic Properties

k +q

k +q

EF k

k +q

k

k

Figure 4.6 Examples of transitions between states jk〉 and jk þ q〉 in the limit that T → 0 for EF > 0. The interband transitions shown in Figure 4.4 are now Pauli blocked if the final states in the conduction band are filled with electrons. Only interband transitions to final states in the conduction band above EF are allowed.

Note that the above integral is taken over the interval ½0; kF  instead of ½0; ∞Þ for e þ ðq; ωÞ of (4.76) can be found from (4.62) by e 0 ðq; ωÞ in (4.65). The last term Π Π EF setting n0 ¼ n ¼ 1 for the intraband transition in the conduction band: e þ ðq; ωÞ Π EF

g ¼ 2 4π

kðF 2ðπ

0 0

1 þ cos ðθkþq  θk Þ f0 ðEk Þ  f0 ðEkþq Þ kdθdk: 2 ℏðω þ iγÞ þ Ek  Ekþq ð4:78Þ

By following the same procedure as that taken for (4.65), we can transform (4.78) to e þ ðq; ωÞ ¼ 1 Π EF π2 ℏ 

kðF ð π

jkj þ jqj cos θ 1þ jk þ qj



0 0

1 ðω þ iγÞ  vF jkj þ v F jk þ qj

1 ðω þ iγÞ þ vF jkj  vF jk þ qj  ð4:79Þ kdθdk:

4.4 Random-Phase Approximation

131

Equations (4.77) and (4.79) can be combined: e þ ðq; ωÞ Π EF

By expressing becomes



e  ðq; ωÞ Π EF

jk þ qj ¼

2 ¼ 2 π ℏ

kðF ð π"

ω þ iγ þ 2vF jkj þ v F jqj cos θ

ðω þ iγ þ v F jkjÞ2  ðvF jk þ qjÞ2 00 # ω þ iγ  2vF jkj  v F jqj cos θ  kdθdk: ðω þ iγ  v F jkjÞ2  ðvF jk þ qjÞ2

ð4:80Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 þ 2kq cos θ þ q2 , jqj ¼ q, and jkj ¼ k, (4.80)

kðF ð π þ  2 X ω þ 2vF k þ v F q cos θ e e Π EF ðq; ωÞ  Π EF ðq; ωÞ ¼ 2 kdθdk; 2 π ℏ α¼1 ω þ 2ω v F k  2v2F kq cos θ  v2F q2 0 0

ð4:81Þ where ω ¼ ðω þ iγÞ. Using the integral identity that ðπ 0

a þ cos θ ab dθ ¼ π 1 þ b þ cos θ b1

rffiffiffiffiffiffiffiffiffiffiffi! b1 ; bþ1

ð4:82Þ

where a and b are complex numbers, the integration on θ in (4.81) can be carried out: ðπ

ω þ 2v F k þ vF q cos θ

dθ 2ω vF k  2v 2F kq cos θ  v 2F q2 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π π ω þ v F q þ 2vF k ω  vF q þ 2vF k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ¼ : 2vF k 2vF k ðω þ iγÞ2  v 2F q2 ω2 þ

ð4:83Þ

By plugging (4.83) into (4.81), we obtain e  ðq; ωÞ e þ ðq; ωÞ  Π Π EF EF kðF pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X 2kF ω þ vF q þ 2v F k ω  v F q þ 2vF k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dk  ; α ¼ πℏvF α¼1 ℏv 2 Fπ 2 q2 ðω þ iγÞ  v F 0

ð4:84Þ

where the summation is taken over α ¼ 1 and α ¼ 1, ωþ is taken for ω when α ¼ 1, and ω is taken for ω when α ¼ 1. The integral in (4.84) is similar to the one in (4.71). The difference is that we are now dealing with complex numbers instead of real numbers. Although the derivation is similar, special care has to be taken for the sign change due to the crossing of the branch cut. For example,

132

Optoelectronic Properties

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω þ vF q þ 2vF k ω  v F q þ 2vF k 8qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < ðω þ 2vF kÞ2  v2 q2 ; α ¼ 1 or 2vF k > ω; F qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ : ðω þ 2v kÞ2  v 2 q2 ; otherwise: 

F

ð4:85Þ

F

Using (4.85), the integral over k in (4.84) can be found as kð2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 v F ω þ v F q þ 2v F k ω  vF q þ 2v F k dk ¼ 4 k1 8     2v F k2 þ ω þ iγ 2v F k1 þ ω þ iγ > > G ; for α ¼ 1; ð4:86Þ >

2v F k2  ω  iγ 2v F k1  ω  iγ > >  G sgnð2v F k1  ωÞ ; :G sgn ð2v F k2  ωÞ vF q vF q for α ¼ 1; where k1 and k2 are real positive numbers, sgnðxÞ ¼ 1 for x > 0 and sgnðxÞ ¼ 1 otherwise, and GðxÞ is given by (4.72). Using (4.86), we can reduce (4.84) to e þ ðq; ωÞ  Π e  ðq; ωÞ Π EF EF 2kF þ ¼ πℏvF

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  πℏv F ðω þ iγÞ2  v2F q2 kðF

Hðω  2v F kF Þ

f

kðF

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωþ þ vF q þ 2vF k ωþ  v F q þ 2vF k dk

0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω þ vF q þ 2vF k ω  v F q þ 2v F k dk

0 ω=2v ð F

Hð2vF kF  ωÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω þ vF q þ 2vF k ω  v F q þ 2v F k dk

0 kðF

Hð2v F kF  ωÞ ω=2v F þδ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω þ vF q þ 2vF k ω  v F q þ 2v F k dk

g

ð4:87Þ

f

    2vF kF þ ω þ iγ ω þ iγ q2 G qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  G vF q vF q 4πℏ ðω þ iγÞ2  v 2F q2 "    # 2vF kF  ω  iγ ω þ iγ Hðω  2v F kF Þ G  G vF q vF q "        # iγ ω þ iγ 2kF  ω  iγ iγ þ δ Hð2vF kF  ωÞ G G þG G ; vF q vF q vF q vF q

2kF ¼ þ πℏv F

g

4.4 Random-Phase Approximation

133

where δ is an infinitesimal positive number. Equation (4.87) can be further reduced by recognizing the identities GðxÞ þ GðxÞ ¼ iπ and Gðδ  ixÞ ¼ GðixÞ; thus we have "   þ  2kF q2 2v F kF þ ω þ iγ e e qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  G Π EF ðq; ωÞ  Π EF ðq; ωÞ ¼  þ vF q πℏvF 4πℏ ðω þ iγÞ2  v2F q2 #   2v F kF  ω  iγ sgnð2v F kF  ωÞG þ iπ : ð4:88Þ vF q e Combining (4.88) with (4.75), we finally obtain the polarizability function Πðq; ωÞ from (4.76): "   2 2k q 2v F kF þ ω þ iγ F e qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  G þ Πðq; ωÞ ¼  vF q πℏv F 4πℏ ðω þ iγÞ2  v 2F q2  # ð4:89Þ 2vF kF  ω  iγ sgnð2v F kF  ωÞG : vF q e γ ðq; ωÞ is obtained by plugging (4.89) into The corresponding RPA–RT polarizability Π (4.63). e e γ ðq; ωÞ for γ ¼ 10 ps1 ¼ 1  1013 s1 The RPA Πðq; ωÞ for γ → 0 and the RPA–RT Π are plotted in Figure 4.7 and Figure 4.8, respectively. The interband and intraband scattering regions, which are also called Landau damping regions, are also shown in Figures 4.7(b) and 4.8(b). Inside the interband scattering region, the imaginary part of e Πðq; ωÞ is dominantly contributed by interband transitions given by e 0 ðq; ωÞ  Π e  ðq; ωÞ in (4.76). Inside the intraband scattering region, Π e 00 ðq; ωÞ is Π EF þ

e ðq; ωÞ in (4.76) because the intraband transitions are allowed. mostly contributed by Π EF

e 00 ðq; ωÞ are shown in Figure 4.9. In the Examples of the transitions that contribute to Π e 00 ðq; ωÞ is zero outside the Landau damping regions, as shown in limit that γ → 0, Π Figure 4.7(b). For γ ≠ 0, as seen in Figure 4.8(b), the boundaries become blurred, and the damping regions expand into the originally scattering-free region, similar to the case shown in Figure 4.5.

4.4.3

Low-Frequency Approximation In Chapter 2, we discuss the screening effect by introducing the screening wave number qs , which is responsible for the reduction of the scattering rate by screening charged impurities and polar phonons in the substrate. The screening wave number qs is related to the dielectric function ϵðqÞ as

134

Optoelectronic Properties

(a)

(q, ) ( E

2 2 F

(b)

)

0

1

(q, ) ( E

1

2.5

2.5

2

2

1.5

1

0.5

0.5

1

0

0

2

2 2 F

) 0

Interband

1.5

1

0

F

0.5

1

EF

EF

2

F

Intraband

0

1

q kF

2 q kF

e Figure 4.7 (a) Real part and (b) imaginary part of the polarizability function Πðq; ωÞ normalized to EF =ℏ2 v2F in the limits that T → 0 and γ → 0. In (b), the boundaries of the interband and intraband 0 00 e ðq; ωÞ and Π e ðq; ωÞ, normalized scattering (Landau damping) regions are shown. The values of Π to EF =ℏ2 v2F , are shown in grayscale.

(a)

(q, ) ( E

2 2 F

(b)

)

0

1

(q, ) ( E

1

2.5

2.5

2

2

1.5

F

2 2 F

)

0.5

1

EF

EF

2

F

0

Interband

1.5

1

1

0.5

0.5

Intraband

0

0 0

1

2 q kF

0

1

2 q kF

e γ ðq; ωÞ normalized to Figure 4.8 (a) Real part and (b) imaginary part of the polarizability function Π EF =ℏ2 v2F in the limit that T → 0. A scattering rate of γ ¼ 10 ps1 ¼ 1  1013 s1 is used. In (b), the boundaries of the0 interband and intraband scattering (Landau damping) regions are shown. e γ and Π e 00γ , normalized to EF =ℏ2 v2 , are shown in grayscale. The values of Π F

4.4 Random-Phase Approximation

kF

(a)

135

2kF

2EF

EF

EF

2.5 (b)

EF

2

Interband

1.5

1 Intraband 0.5

0 0

1

2 q kF 00

e ðq; ωÞ. The scale Figure 4.9 (a) Examples of the transitions on the Dirac cone that contribute to Π of the Dirac cone shown on the right-hand side is half that of the Dirac cone shown on the left-hand side. In (b), the corresponding transitions on the ω versus q diagram are marked with the same numbers as those marking the transitions in (a).

 ϵðqÞ ¼ ϵ avg

 qs 1þ : q

ð4:90Þ

Therefore, a large value of qs gives a large value of ϵðqÞ, signifying a strong screening effect. By comparing (4.90) with (4.37), we can see that qs is related to the polarizability function as

Optoelectronic Properties

qs ¼ 

e2 e Πðq; ω → 0Þ: 2ϵ ave

ð4:91Þ

By setting ω ¼ 0 in (4.89), we have       iq 2kF 2kF e Πðq; ω → 0Þ ¼ e ρ ðEF Þ 1 þ þ iδ  G  iδ ; ð4:92Þ G 8kF q q ρ ðEF Þ ¼ 2EF =πℏ2 v 2F is where δ ¼ γ=vF q is an infinitesimal positive number and e the density ofpstates energy given by (1.34). Then from (4.92) using ffiffiffiffiffiffiffiffiffiffiffiffi at thepFermi ffiffiffiffiffiffi the identity x  iδ ¼ i x, where x is a negative real number, it can be shown that 8 ρ ðEF Þ; q ≤ 2kF ; >

ρ ðEF Þ 1 þ G ; q > 2kF : :e 4kF q Note that iGð2kF =qÞ in (4.93) is a positive real number. In the limit that q → ∞, e Πðq; ω → 0Þ ¼ e ρ ðEF Þð1 þ πq=8kF Þ. Equation (4.93) is plotted in Figure 4.10 e þ ðq; ωÞ and for q in the range between 0 and 4kF . The polarizabilities Π EF



e ðq; ωÞ, which are contributed by intraband and interband transitions, e 0 ðq; ωÞ þ Π Π EF respectively, exactly add up to a constant value for q ≤ 2kF. For q > 2kF, the polarizability is mostly contributed by the interband transitions.

0)

F

(E )

2

1.5

1

(q,

136

0.5

0

0

2 q kF

4

e Figure 4.10 Negative polarizability function Πðq; ω → 0Þ normalized to e ρ ðEF Þ in the lowfrequency limit, shown as the solid curve. The dashed and dotted curves are numerically obtained, þ  e ðq; ωÞ and Π e 0 ðq; ωÞ þ Π e ðq; ωÞ contributed by intrarepresenting the polarizabilities Π EF EF band and interband transitions, respectively.

4.4 Random-Phase Approximation

137

The Thomas–Fermi screening wave number of graphene is found from (4.91) in the limit that q → 0 as qs ¼ 

e2 e Πðq → 0; 0Þ; 2ϵ ave

ð4:94Þ

which has been discussed in (2.32). From (4.94) and (4.93), we find that the Thomas–Fermi screening wave number of graphene is qs ¼

e2 e ρ ðEF Þ; 2ϵ ave

ð4:95Þ

which is consistent with (2.32) for μ ¼ EF in the limit that T → 0 as considered here.

4.4.4

Long-Wavelength Approximation In the long-wavelength limit (q → 0 so that q → 0), the first term in the brackets in (4.89) has the form of GðxÞ ¼ x2  lnð2xÞ, where GðxÞ is given by (4.72) and x ¼ ð2v F kF þ ω þ iγÞ=vF q, and the second term in the brackets in (4.89) has the form of GðxÞ ¼ x2  lnð2xÞ for 2v F kF > ω and GðxÞ ¼ x2 þ lnð2xÞ for 2vF kF ≤ ω, where x ¼ ð2v F kF  ω  iγÞ=vF q. Then, (4.89) becomes ! 2 2 2 2k q v q F F e → 0; ωÞ ¼  1þ Πðq þ πℏv F 4πℏðω þ iγÞ 2ðω þ iγÞ2  

¼

   8kF ω þ iγ 2vF kF  ω  iγ þ ln vF q 2vF kF þ ω þ iγ q

q2

EF

2

πℏ ðω þ iγÞ

2

þ

ð4:96Þ

q2 2EF  ℏðω þ iγÞ ln ; 4πℏðω þ iγÞ 2EF þ ℏðω þ iγÞ

where the high-order terms of q have been ignored. For a nonzero γ, (4.96) has to be e → 0; ωÞ ∝ q2 modified in accordance to (4.63). As can be seen in (4.93) and (4.96), Πðq 1 e e and Πðq → 0; 0Þ ∝ q. Therefore, Π ðq → 0; ωÞ is the dominant term in (4.63). By plugging (4.96) into (4.63), we obtain e → 0; ωÞ: e γ ðq → 0; ωÞ ¼ ω þ iγ Πðq Π ω

ð4:97Þ

Then the surface optical conductivity can be found from (4.14) using (4.97): iωe2 e Π γ ðq → 0; ωÞ q2 2 e EF ie σ 0 2EF þ ℏðω þ iγÞ ¼ 2 ln :  π 2EF  ℏðω þ iγÞ πℏ ðγ  iωÞ

e σ ðq → 0; ωÞ ¼

ð4:98Þ

138

Optoelectronic Properties

Because both (4.5) and (4.98) are obtained by ignoring the spatial dispersion, i.e., by taking q → 0, we find that these two equations are identical if jμj ¼ EF and τ1 ¼ γ. Equation (4.98) is not valid for μ < 0 because we have assumed electron doping in the derivation of the polarizability function. Strictly speaking, γ is a function of ω and the values of γ at high and low frequencies can be very different. Therefore, as expressed in (4.5), it is practical to reserve γ for the scattering rate at a high frequency but to use τ1 for the scattering rate at a low frequency; then, both γ and τ 1 are assumed to be independent of ω as an approximation. This frequency-independent τ1 is the effective 1 scattering rate τ1 for relaxation eff in Chapter 2. In fact, the interband relaxation time γ transitions from the conduction band to the valence band is much longer than the intraband scattering time τeff [8].

4.4.5

Polarizability Function at a Nonzero Temperature In the preceding sections, it is assumed that the temperature T → 0. Without going into the details of the calculation, in the limit that γ → 0 the real part of the polarizability function at a nonzero temperature T is given as [9] e ʹðq; ω; TÞ Π i q2 h ¼ Gðq; ω; TÞ f ðω; vF qÞHðω  v F qÞ  G ðq; ω; TÞ f ðv F q; ωÞHðvF q  ωÞ πℏ   4kB T μ ð4:99Þ  2 2 ln 2 cosh ; 2kB T πℏ vF and the imaginary part is e Πʺðq; ω; TÞ i q2 h ¼ Gðq; ω; TÞ f ðvF q; ωÞHðv F q  ωÞ þ Gþ ðq; ω; TÞ f ðω; v F qÞHðω  vF qÞ ; πℏ ð4:100Þ where HðxÞ is the Heaviside step function that has a value of 0 for x < 0 and a value of 1 for x ≥ 0, 1 f ðx; yÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 x  y2 Gðq; ω; TÞ ¼

and

pffiffiffiffiffiffiffiffiffiffiffiffiffi β x2  1   dx; ℏjvF qx þ βωj  2αμ α;β¼1 exp þ1 1 2kB T ∞ X ð

pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2   dx: ℏjvF qx  ωj  2αμ þ1 1 exp 2kB T

ð π X G ðq; ω; TÞ ¼  þ 2 α¼1

ð4:101Þ

ð4:102Þ

1

ð4:103Þ

4.4 Random-Phase Approximation

(a)

( q, ) ( E

2 2 F

(b)

)

0

1

( q, ) ( E

1

2.5

2.5

2

2

1.5

1

0.5

0.5

0

1

0

2 q kF

2 2 F

) 0

Interband

1.5

1

0

F

0.5

1

EF

EF

2

F

139

Intraband

0

1

2 q kF

e Figure 4.11 (a) Real part and (b) imaginary part of the polarizability function Πðq; ω; TÞ normal-

ized to EF =ℏ2 v2F in the limit that γ → 0. For these plots, the temperature is taken to be T ¼ 300 K and the Fermi energy is EF ¼ 100 meV. In (b), the boundaries of the interband and intraband scattering (Landau damping) regions are shown.

The chemical potential μ is obtained from (1.41) for a given Fermi energy EF. At a nonzero temperature, the average energy required for an interband transition is reduced because some of the originally occupied states in the conduction band become available as electrons are thermally excited to higher energy states. Therefore, the Landau damping region extends into the low-energy region on the ωq plane near the boundaries of the interband and intraband scattering regions, as shown in Figure 4.11. The expanding of the Landau damping regions is clearly seen by comparing Figure 4.11 (b) with Figure 4.7(b). The static screening wave number can be found by setting ω → 0 in (4.99) and e 00 ðq; 0; TÞ ¼ 0; thus, the polariz(4.100). In this limit, we find that Gðq; 0; TÞ ¼ 0 and Π ability function is reduced to e Πðq; 0; TÞ ¼

  q 4kB T μ G ðq; 0; TÞ  2 2 ln 2 cosh : 2πℏv F 2kB T πℏ v F

In the limit that kB T ≫ EF , (4.104) further reduces to [10] "  # kB T q2 EF 2 e Πðq; 0; TÞ ¼ e ρ ðEF Þ ln 4 þ ; EF 24kF 2 kB T whereas in the limit that kB T ≪ EF ,

ð4:104Þ

ð4:105Þ

Optoelectronic Properties

e Πðq; 0; TÞ e ρ ðEF Þ 8   π 2 kB T 2 > > > ; for ℏvF q < 2μ; 1þ > < 6 EF sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼   > > μ μ 2μ q 2μ πq 2π2 EF μ kB T 2 > 1 > þ 1 þ   sin ; for ℏvF q > 2μ; : EF 2EF ℏvF q 2kF ℏv F q 8kF 3ðℏv F qÞ2 EF ð4:106Þ where e ρ ðEF Þ ¼ 2EF =πℏ2 v 2F . Therefore, the Thomas–Fermi screening wave number can be found from (4.94), (4.105), and (4.106) as 8 2 ee ρ ðEF Þ kB T > > ln 4; > > < 2ϵ ave EF " qs ðTÞ ¼  2 # 2 2 > e e Þ π k T ρ ðE > F B > > 1 ; : 2ϵ 6 EF ave

for kB T ≫ EF ;

ð4:107Þ

for kB T ≪ EF :

ð4:108Þ

The Thomas–Fermi screening wave number is plotted in Figure 4.12 as a function of temperature. The exact qs is plotted by using (4.94) and (4.104), whereas the high- and low-temperature approximations are plotted by using (4.107) and (4.108), respectively. As can be seen in Figure 4.12, qs first decreases with temperature in the low-temperature region and then linearly increases with temperature in the high-temperature region, which is consistent with the approximations in (4.107) and (4.108). For kB T=EF < 0:25, qs is well approximated by the low-temperature approximation given by (4.108). Therefore, for graphene samples of EF > 100 meV, qs at room temperature (kB T ≈ 26 meV) can be well estimated with (4.108).

1.2

(4.94), (4.104)

F

( (E )

2

ave

)

1.4

qs (T ) e2

140

1

0.8

(4.108) 0

(4.107) 0.5

1

kBT EF Figure 4.12 Thomas–Fermi screening wave number as a function of temperature. The solid curve is the exact result obtained from (4.94) and (4.104); the dashed curves are high- and low-temperature approximations obtained from (4.107) and (4.108), respectively.

4.5 Polarizability Function of Bilayer Graphene

4.5

141

Polarizability Function of Bilayer Graphene e For AB-stacked bilayer graphene, the polarizability function Πðq; ωÞ can be calculated from (4.62) by replacing cos ðθkþq  θk Þ with cos 2ðθkþq  θk Þ due to the different chirality of AB-stacked bilayer graphene from that of monolayer graphene. By identifying Ek ¼ njkj2 =2m and Ekþq ¼ n0jk þ qj2 =2m, i.e., low-energy parabolic band approximation, for the parabolic bands of AB-stacked bilayer graphene, an analye tical form of the polarizability function Πðq; ωÞ for AB-stacked bilayer graphene can be obtained in the limit that T → 0 and γ → 0. However, the expression is very complicated; interested readers are referred to Reference [11] for the complete expression. In the limit that q → 0, the surface optical conductivity e σ ðωÞ of AB-stacked e → 0; ωÞ bilayer graphene is obtained from (4.26) using the polarizability function Πðq 0 of AB-stacked bilayer graphene. The expression for the real part e σ ðωÞ of e σ ðωÞ is given by (3.110) with an extra term of aðEF ÞδðℏωÞ contributed by intraband transitions given by [12]   4jμje2 jμj þ γ1 jμj  γ1 aðEF ÞδðℏωÞ ¼ þ Hðjμj  γ1 Þ δðℏωÞ; 2ℏ 2jμj þ γ1 2jμj  γ1

ð4:109Þ

where γ1 is the interlayer interaction energy of AB-stacked bilayer graphene. As we have seen in Figure 1.8, AB-stacked bilayer graphene has two conduction bands that are separated by an energy gap of γ1 . The first and second terms in (4.109) are contributed by the intraband transitions in the lower and upper conduction bands, respectively. Intraband transitions in the upper conduction band are possible only when jμj is larger than γ1 so that there are free electrons in the upper conduction band. To be precise, in finding the surface optical conductivity, the polarizability function e Π γ ðq; ωÞ given in (4.63) for γ ≠ 0 has to be used to take the nonzero scattering rate γ into e account. However, (4.63) requires former knowledge of Πðq; ωÞ derived by assuming nonzero γ, in analogy to (4.89) for monolayer graphene. An easy yet sufficiently accurate way is to first replace the delta function δðxÞ with the Lorentzian function ðγ=πÞ=ðx2 þ γ2 Þ for both δðℏω  γ1 Þ in (3.110) contributed by interband transitions and δðℏωÞ in (4.109) contributed by intraband transitions, and then to take the limit that γ → 0. The result is shown to be a good approximation compared to the full numerical calculation [12]. The static screening wave number of AB-stacked bilayer graphene can be found by e setting the limit ω → 0 in Πðq; ωÞ; the expression is given as [13] h i e Πðq; ω → 0Þ ¼ e ρ ðEF Þ gðqÞ  f ðqÞHðq  2kF Þ ;

ð4:110Þ

where the factor e ρ ðEF Þ ¼ 2m=πℏ2 is the density of states of AB-stacked bilayer graphene in the parabolic band approximation discussed in Chapter 1. The functions gðqÞ and f ðqÞ are given as

142

(a)

Optoelectronic Properties

(b)

F

(E )

2

1.5

0)

1.5

0)

F

(E )

2

1

(q ,

(q ,

1

0.5

0.5

0

0 4

2 q kF

0

0

2 q kF

4

e Figure 4.13 Negative polarizability function Πðq; ω → 0Þ, shown in both (a) and (b) as solid curves, of AB-stacked bilayer graphene in the low-frequency limit normalized to the density of states at the Fermi energy. (a) The dashed and dotted curves represent the negative polarizability functions of monolayer and 2DEG, respectively. (b) The dashed and dotted curves represent the negative polarizability functions of AB-stacked bilayer graphene contributed by intraband and interband transitions, respectively.

1 gðqÞ ¼ 2 2kF

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kF2 þ 4kF4 þ q4 4 4 4kF þ q  ln 4kF2

and 2k 2 þ q2 f ðqÞ ¼ F 2 2kF q

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q  q2  4k 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF : q2  4kF þ ln q þ q2  4kF2

ð4:111Þ

ð4:112Þ

The polarizability function Πðq; ω → 0Þ given by (4.110) for AB-stacked bilayer graphene is plotted as solid curves in Figures 4.13(a) and (b). The polarizability functions of monolayer graphene and 2D electron gas (2DEG) are also plotted in Figure 4.13(a) for comparison. The polarizability function of 2DEG is given by [14] 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2k F Πðq; ω → 0Þ ¼ e ρ ðEF Þ41  1  Hðq  2kF Þ5: ð4:113Þ q Because the screening wave number is linearly proportional to Πðq; ω → 0Þ according to (4.91), the relative strength of the screening is also compared in Figure 4.13(a) for monolayer graphene, bilayer graphene, and 2DEG. As can be seen, around q ¼ 2kF the screening is stronger for bilayer graphene than for monolayer graphene. The weaker screening of monolayer graphene is due to the suppression of backward scattering, as

4.5 Polarizability Function of Bilayer Graphene

143

manifested by the form factor jFj2 ¼ ½1 þ cos ðθkþq  θk Þ=2 in (4.62) for intraband transitions: For q ¼ 2kF, we have θkþq  θk ¼ π found from Figure 4.4 and thus jFj2 ¼ 0 for monolayer graphene. By contrast, for AB-stacked bilayer graphene the form factor is jFj2 ¼ ½1 þ cos 2ðθkþq  θk Þ=2, which has the maximum value of jFj2 ¼ 1 at q ¼ 2kF . The polarizability functions of AB-stacked bilayer graphene contributed by intraband and interband transitions are plotted separately in Figure 4.13(b). As can be seen, the backward scattering is enhanced at q ¼ 2kF for intraband transitions, plotted as the dashed curve. This peak is absent for monolayer graphene, as can be seen in Figure 4.10. The polarizability function given in (4.110) for AB-stacked graphene is obtained in the limit that T → 0. In the high-temperature limit, (4.110) is modified as [15,16]   q2 EF e Πðq; 0; TÞ ¼ e ρ ðEF Þ 1 þ 2 ; 6kF kB T

for kB T ≫ EF ;

and in the low-temperature limit for small values of q, it becomes [15,16] ( "  2 #) 4 q k T B 2 e 1þπ Πðq; 0; TÞ ≈  e ρ ðEF Þ 1 þ ; for kB T ≪ EF : EF 16kF4

ð4:114Þ

ð4:115Þ

e From (4.114) and (4.115), we find that Πðq; 0; TÞ ≈  e ρ ðEF Þ in the limit that q → 0, e → 0; 0; TÞ ¼ e which is independent of temperature. By plugging Πðq ρ ðEF Þ in (4.94), we obtain the Thomas–Fermi screening wave number of AB-stacked bilayer graphene: qs ¼

e2 e ρ ðEF Þ; 2ϵ ave

ð4:116Þ

which can be compared with (4.107) and (4.108) for monolayer graphene. The comparison of the screening wave number can be made for monolayer graphene, AB-stacked bilayer graphene, and 2DEG, as listed in Table 4.2. The screening wave number is normalized to e2 e ρ ðEF Þ is 2EF =πℏ2 v2F for monolayer graphene, 2m=πℏ2 for ABρ ðEF Þ=2ϵ ave , where e stacked bilayer graphene, and gm=2πℏ2 for 2DEG of degeneracy g. Table 4.2 Screening wave number of monolayer graphene, AB-stacked bilayer graphene, and 2DEG. The screening ρ ðEF Þ=2ϵave . wave numbers are normalized to e2e

Monolayer graphene AB-stacked bilayer graphene 2DEG

High temperature, kB T ≫ EF

Low temperature, kB T ≪ EF

All wave number q   kB T q2 EF 2 ln 4 þ (4.105) EF 24kF2 kB T

q¼0

q2 EF (4.114) 6kF2 kB T   EF q 2 EF 1 2 kB T 6kF kB T

1þ

1

  π2 kB T 2 (4.106) 6 EF

1 (4.115) 1  eEF =kB T

144

Optoelectronic Properties

References 1. I. T. Lin and J. M. Liu, “Terahertz frequency-dependent carrier scattering rate and mobility of monolayer and AA-stacked multilayer graphene,” IEEE Journal of Selected Topics in Quantum Electronics, Vol. 20, 8400108 (2014). 2. C. Kittel, Introduction to Solid State Physics (Wiley, 2004). 3. J. Sólyom, Fundamentals of the Physics of Solids: Volume 3—Normal, Broken-Symmetry, and Correlated Systems (Springer, 2010). 4. D. Bohm and D. Pines, “A collective description of electron interactions I. Magnetic interactions,”Physical Review, Vol. 82, pp. 625‒634 (1951). 5. H. Ehrenreich and M. H. Cohen, “Self-consistent field approach to the many-electron problem,” Physical Review, Vol. 115, pp. 786‒790 (1959). 6. N. D. Mermin, “Lindhard dielectric function in the relaxation-time approximation,” Physical Review B, Vol. 1, pp. 2362–2363 (1970). 7. B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New Journal of Physics, Vol. 8, 318 (2006). 8. S. Winnerl, M. Orlita, P. Plochocka, et al., “Carrier relaxation in epitaxial graphene photoexcited near the Dirac point,” Physical Review Letters, Vol. 107, 237401 (2011). 9. M. R. Ramezanali, M. M. Vazifeh, A. Reza, P. Marco, and A. H. MacDonald, “Finitetemperature screening and the specific heat of doped graphene sheets,” Journal of Physics A: Mathematical and Theoretical, Vol. 42, 214015 (2009). 10. E. H. Hwang and S. Das Sarma, “Screening-induced temperature-dependent transport in two-dimensional graphene,” Physical Review B, Vol. 79, 165404 (2009). 11. R. Sensarma, E. H. Hwang, and S. Das Sarma, “Dynamic screening and low-energy collective modes in bilayer graphene,” Physical Review B, Vol. 82, 195428 (2010). 12. E. J. Nicol and J. P. Carbotte, “Optical conductivity of bilayer graphene with and without an asymmetry gap,” Physical Review B, Vol. 77, 155409 (2008). 13. E. H. Hwang and S. Das Sarma, “Screening, Kohn anomaly, Friedel oscillation, and RKKY interaction in bilayer graphene,” Physical Review Letters, Vol. 101, 156802 (2008). 14. F. Stern, “Polarizability of a two-dimensional electron gas,” Physical Review Letters, Vol. 18, pp. 546‒548 (1967). 15. S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Review of Modern Physics, Vol. 83, pp. 407‒470 (2011). 16. M. Lv and S. Wan, “Screening-induced transport at finite temperature in bilayer graphene,” Physical Review B, Vol. 81, 195409 (2010).

5

Nonlinear Optical Properties

5.1

Nonlinear Susceptibility and Nonlinear Conductivity In the preceding two chapters, we consider only the linear optical response of graphene; fields of different frequencies do not mix, and the amplitude of the response is linearly proportional to the strength of the excitation optical field. In this chapter, the nonlinear optical response of graphene is discussed. In general, the nonlinear optical response of a material is described by a polarization that is a nonlinear function of the optical field; such a nonlinear polarization is characterized by a nonlinear susceptibility. Because graphene can be treated as a conductor, its nonlinear optical response can alternatively be described by an optical current that is a nonlinear function of the optical field; such a nonlinear current is characterized by a nonlinear conductivity. As in the case discussed in the preceding two chapters for the relation between linear susceptibility and linear conductivity, these two alternative descriptions of nonlinear optical properties of graphene are equivalent. In this section, we first define the nonlinear susceptibility and then make the connection between the nonlinear susceptibility and the nonlinear conductivity. Except for strong optical saturation and some extremely nonlinear interactions, the perturbation method can be used to describe nonlinear optical response by expanding the total optical polarization in terms of a perturbation series of linear and nonlinear optical polarizations: Pðr; tÞ ¼ Pð1Þ ðr; tÞ þ Pð2Þ ðr; tÞ þ Pð3Þ ðr; tÞ þ . . . ;

ð5:1Þ

where Pð1Þ ðr; tÞ is the linear polarization that is linearly proportional to the optical field Eðr; tÞ, and Pð2Þ ðr; tÞ and Pð3Þ ðr; tÞ are the second- and third-order nonlinear polarizations that respectively depend quadratically and cubically on the optical field [1]. In the real space and time domain, the linear and nonlinear optical polarizations are related to the optical field through convolution integrals: ðt ð1Þ P ðr; tÞ ¼ ϵ 0 χð1Þ ðt  t0 Þ  Eðr; t0 Þdt0; ð5:2Þ ∞

146

Nonlinear Optical Properties

ðt ðt

ð2Þ

P ðr; tÞ ¼ ϵ 0

χð2Þ ðt  t1 ; t  t2 Þ:Eðr; t1 ÞEðr; t2 Þdt1 dt2;

ð5:3Þ

∞ ∞ ð3Þ

ðt ðt ðt

P ðr; tÞ ¼ ϵ 0

. χð3Þ ðt  t1 ; t  t2 ; t  t3 Þ .. Eðr; t1 ÞEðr; t2 ÞEðr; t3 Þdt1 dt2 dt3;

∞ ∞ ∞

ð5:4Þ where χð1Þ is the linear susceptibility discussed in the preceding two chapters, χð2Þ is the second-order nonlinear susceptibility, and χð3Þ is the third-order nonlinear susceptibility. For simplicity, we have assumed local responses for the susceptibilities to be expressed as response functions only in time but not in space so that the spatial convolution for the linear and nonlinear polarizations can be ignored [1]. The complex linear and nonlinear polarizations are defined with respect to the corresponding real linear and nonlinear optical polarizations expressed in (5.1) in the same manner as the general definition given in (3.13) for the complex field: PðnÞ ðr; tÞ ¼ PðnÞ ðr; tÞ þ PðnÞðr; tÞ ¼ PðnÞ ðr; tÞ þ c:c:;

ð5:5Þ

where PðnÞ is the nth-order real optical polarization and PðnÞ is the corresponding nthorder complex optical polarization. The optical field involved in a nonlinear interaction usually contains multiple, discrete frequency components. Such a field can be expanded in terms of its frequency components as X X Eðr; tÞ ¼ Eq ðrÞ exp ðiωq tÞ ¼ E q ðrÞ exp ðikq  r  iωq tÞ; ð5:6Þ q>0

q>0

where E q ðrÞ is the slowly varying amplitude and kq is the wave vector of the ωq frequency component, as defined in (3.18) for a harmonic optical field. The linear and nonlinear optical polarizations also contain multiple frequency components and can be expanded as X PðnÞ ðr; tÞ ¼ PðnÞ ð5:7Þ q ðrÞ exp ðiωq tÞ: q>0

Note that except for the linear optical polarization Pð1Þ q ðrÞ, we do not attempt to further ðnÞ express any nonlinear optical polarization Pq ðrÞ for n ≠ 1 in terms of a slowly varying polarization amplitude multiplied by a fast-varying spatial phase term, as is done for Eq ðrÞ. The reason is that the wave vector that characterizes the fast-varying spatial phase of a nonlinear optical polarization PðnÞ q ðrÞ for n ≠ 1 is not simply determined by the frequency ωq but is dictated by the fields that generate the nonlinear optical polarization. In the discussions of nonlinear optical polarizations, we also use the notations Eðωq Þ and PðnÞ ðωq Þ that are respectively defined as

5.1 Nonlinear Susceptibility and Nonlinear Conductivity

Eðωq Þ ¼ Eq ðrÞ

and PðnÞ ðωq Þ ¼ PðnÞ q ðrÞ:

147

ð5:8Þ

Therefore, the real optical field and real optical polarizations can be respectively expressed as X X Eðr;tÞ ¼ Eðωq Þeiωq t þ Eðωq Þeiωq t ¼ Eðωq Þeiωq t ð5:9Þ

½



q

q>0

and PðnÞ ðr; tÞ ¼

Xh

i X PðnÞ ðωq Þeiωq t þ PðnÞðωq Þeiωq t ¼ PðnÞ ðωq Þeiωq t : ð5:10Þ q

q>0

Note that in the last terms of (5.9) and (5.10) for the real field and real polarization, the summation runs through both positive and negative values for the index q; that is, if the field has n frequency components, the summation runs through both 1; 2; . . . ; n and 1; 2; . . . ; n. By comparison, in (5.6) and (5.7) for the complex field and complex polarization, the summation runs only through the positive values of 1; 2; . . . ; n for the index q if the field has n frequency components. A frequency with a negative index is interpreted as a negative frequency: ωq ¼ ωq :

ð5:11Þ

Field and polarization components of negative frequencies are interpreted as Eðωq Þ ¼ Eðωq Þ and PðnÞ ðωq Þ ¼ PðnÞðωq Þ according to the definition of the complex field; therefore, Eðωq Þ ¼ Eðωq Þ ¼ Eðωq Þ

and PðnÞ ðωq Þ ¼ PðnÞ ðωq Þ ¼ PðnÞðωq Þ: ð5:12Þ

By taking Fourier transform on (5.2)–(5.4), or by applying (5.9) and (5.10) to (5.2)–(5.4), while using (5.12), we find the following relations in the frequency domain, Pð1Þ ðωq Þ ¼ ϵ 0 χð1Þ ðωq ¼ ωm Þ  Eðωm Þ; X Pð2Þ ðωq Þ ¼ ϵ 0 χð2Þ ðωq ¼ ωm þ ωn Þ :Eðωm ÞEðωn Þ; Pð3Þ ðωq Þ ¼ ϵ 0

X

ð5:13Þ ð5:14Þ

m;n

. χð3Þ ðωq ¼ ωm þ ωn þ ωp Þ .. Eðωm ÞEðωn ÞEðωp Þ;

ð5:15Þ

m;n;p

where the notation χðnÞ ðωq ¼ ω1 þ ω2 þ . . . þ ωn Þ ¼ χðnÞ ðω1 ; ω2 ; . . . ; ωn Þ is used for ω1 þ ω2 þ . . . þ ωn ¼ ωq, and

ð5:16Þ

148

Nonlinear Optical Properties

ð∞ ð∞

ð∞

ðnÞ

χ ðω1 ; ω2 ; . . . ; ωn Þ ¼

... 0

χðnÞ ðt1 ; t2 ; . . . ; tn Þeiω1 t1 þiω2 t2 þ ... þiωn tn dt1 dt2 . . . dtn :

0 0

ð5:17Þ Note that for the nonlinear optical polarizations expressed in (5.14) and (5.15), the summation is performed for a given ωq over all positive and negative values of frequencies that satisfy the constraint ωm þ ωn ¼ ωq in the case of (5.14) and the constraint ωm þ ωn þ ωp ¼ ωq in the case of (5.15), with an electric field that has a negative index for its frequency, thus a negative frequency, interpreted as the complex conjugate field of a positive frequency according to (5.12). For a conducting material such as graphene, the nonlinear conductivity is frequently used in the literature to characterize its nonlinear optical properties contributed by conduction charge carriers. Then, in a manner similar to the perturbation expansion given in (5.1) for the polarization, the total current density is expanded in terms of a perturbation series of linear and nonlinear current densities: Jðr; tÞ ¼ J ð1Þ ðr; tÞ þ J ð2Þ ðr; tÞ þ J ð3Þ ðr; tÞ þ . . . :

ð5:18Þ

In the time domain, the linear and nonlinear current densities in the perturbation series are related to the excitation field through convolution relations: J

ð1Þ

ðt ðr; tÞ ¼

σð1Þ ðt  t0 Þ  Eðr; t0 Þdt0;

ð5:19Þ

∞

J

ð2Þ

ðt ðt ðr; tÞ ¼

σð2Þ ðt  t1; t  t2 Þ:Eðr; t1 ÞEðr; t2 Þdt1 dt2;

ð5:20Þ

∞ ∞

J ð3Þ ðr; tÞ ¼

ðt ðt ðt

. σð3Þ ðt  t1 ; t  t2 ; t  t3 Þ .. Eðr; t1 ÞEðr; t2 ÞEðr; t3 Þdt1 dt2 dt3; ð5:21Þ

∞ ∞ ∞

where σð1Þ is the linear conductivity discussed in the preceding chapters, σð2Þ is the second-order nonlinear conductivity, and σð3Þ is the third-order nonlinear conductivity. By following a procedure similar to that taken in the above for the linear and nonlinear optical polarizations, we find the following relations for the linear and nonlinear current densities in the frequency domain: Jð1Þ ðωq Þ ¼ σð1Þ ðωq ¼ ωm Þ  Eðωm Þ; X σð2Þ ðωq ¼ ωm þ ωn Þ : Eðωm ÞEðωn Þ; Jð2Þ ðωq Þ ¼ m;n

ð5:22Þ ð5:23Þ

5.1 Nonlinear Susceptibility and Nonlinear Conductivity

Jð3Þ ðωq Þ ¼

X

. σð3Þ ðωq ¼ ωm þ ωn þ ωp Þ .. Eðωm ÞEðωn ÞEðωp Þ;

149

ð5:24Þ

m;n;p

where the notation σðnÞ ðωq ¼ ω1 þ ω2 þ . . . þ ωn Þ ¼ σðnÞ ðω1 ; ω2 ; . . . ; ωn Þ

ð5:25Þ

is used for ω1 þ ω2 þ . . . þ ωn ¼ ωq, and ð∞ ð∞

ð∞

ðnÞ

σ ðω1 ; ω2 ; . . . ; ωn Þ ¼

... 0

σðnÞ ðt1 ; t2 ; . . . ; tn Þeiω1 t1 þiω2 t2 þ ... þiωn tn dt1 dt2 . . . dtn :

0 0

ð5:26Þ Again the summation is performed for a given ωq over all positive and negative values of frequencies that satisfy the constraint of ωm þ ωn ¼ ωq in the case of (5.23) and the constraint of ωm þ ωn þ ωp ¼ ωq in the case of (5.24), with an electric field that has a negative index for its frequency, thus a negative frequency, interpreted as the complex conjugate field of a positive frequency according to (5.12). As discussed in Section 3.2, for a graphene sheet located at z ¼ 0, the 3D polarization density Pðx; y; zÞ and current density Jðx; y; zÞ can be expressed in terms of the 2D e yÞ and surface current density e e and surface polarization density Pðx; Jðx; yÞ as P ¼ P=d J¼e J=d for d=2 < z < d=2 in the 3D model, where d is the thickness of the e graphene sheet, or as P ¼ PδðzÞ and J ¼ e JδðzÞ in the 2D model. The linear surface susceptibility e χ and linear surface conductivity e σ of graphene are defined through the relations expressed in (3.50) and (3.49), respectively, in terms of the surface polarization density and surface current density, respectively; they are related to χ and σ as e χ ¼ χd and e σ ¼ σd in the 3D model, or χ ¼ e χ δðzÞ and σ ¼ e σ δðzÞ in the 2D model. Because the electric field E is always a 3D quantity, we find from (5.13)–(5.15) and (5.22)–(5.24) that similar relations can be applied to the polarization densities and current densities of all orders on the graphene sheet of a thickness d located at z ¼ 0: ðnÞ ðnÞ e ðnÞ ¼ PðnÞ d and e P J ¼ J d; for  d=2 < z < d=2;

ð5:27Þ

in the 3D model, or ðnÞ

P

ðnÞ e ðnÞ δðzÞ and JðnÞ ¼ e ¼P J δðzÞ

ð5:28Þ

in the 2D model. Therefore, e ð1Þ ðx; y; ωq Þ ¼ ϵ 0 e χ ð1Þ ðx; y; ωq ¼ ωm Þ  Eðx; y; ωm Þ; P e ð2Þ ðx; y; ωq Þ ¼ ϵ 0 P

ð5:29Þ

X ð2Þ e χ ðx; y; ωq ¼ ωm þ ωn Þ : Eðx; y; ωm ÞEðx; y; ωn Þ; ð5:30Þ m;n

150

Nonlinear Optical Properties

e ð3Þ ðx; y; ωq Þ ¼ ϵ 0 P

X

e χ ð3Þ ðx; y; ωq ¼ ωm þ ωn þ ωp Þ

m;n;p

ð5:31Þ

.. . Eðx; y; ωm ÞEðx; y; ωn ÞEðx; y; ωp Þ; and

ð1Þ e J ðx; y; ωq Þ ¼ e σ ð1Þ ðx; y; ωq ¼ ωm Þ  Eðx; y; ωm Þ; ð5:32Þ X ð2Þ e e J ðx; y; ωq Þ ¼ σ ð2Þ ðx; y; ωq ¼ ωm þ ωn Þ : Eðx; y; ωm ÞEðx; y; ωn Þ; ð5:33Þ m;n ð3Þ e J ðx; y; ωq Þ ¼

X

. e σ ð3Þ ðx; y; ωq ¼ ωm þ ωn þ ωp Þ.. Eðx; y; ωm ÞEðx; y; ωn ÞEðx; y; ωp Þ;

m;n;p

ð5:34Þ where e σ ðnÞ ¼ σðnÞ d χ ðnÞ ¼ χ ðnÞ d and e

ð5:35Þ

χ ðnÞ δðzÞ and σðnÞ ¼ e σ ðnÞ δðzÞ χ ðnÞ ¼ e

ð5:36Þ

in the 3D model, or

in the 2D model. The susceptibility χðnÞ and the conductivity σðnÞ are tensors because PðnÞ , JðnÞ , and E are vectors. The linear susceptibility χð1Þ and the linear conductivity σð1Þ are second-order ð1Þ

ð1Þ

tensors of nine elements, χij and σ ij , respectively, where i and j each can be x, y, or z. The second-order susceptibility χð2Þ and the second-order conductivity σð2Þ are third-order ð2Þ

ð2Þ

tensors of 27 elements, χijk and σ ijk , respectively, and the third-order susceptibility χð3Þ and ð3Þ

ð3Þ

the third-order conductivity σð3Þ are fourth-order tensors of 81 elements, χijkl and σ ijkl , respectively, where i, j, k, and l each can be x, y, or z. Note that here x, y, and z are the Cartesian coordinates defined along the principal axes of the material; these coordinates cannot be arbitrarily defined because the symmetry of the material determines the structures and properties of the susceptibility and conductivity tensors χðnÞ and σðnÞ of the material. The number of nonzero, independent elements of the linear and nonlinear susceptibilities and that of the linear and nonlinear conductivities of a given material are generally reduced by the spatial symmetry of the material. The linear susceptibility depends only on the crystal symmetry, whereas the nonlinear susceptibilities further depend on the point group [1]. Monolayer graphene has hexagonal symmetry, and its space group is P6=mmm, which belongs to the 6=mmm point group [2]. As discussed in Section 3.2, the linear susceptibility tensor of graphene has only two independent elements for the three principal susceptibilities due to its hexagonal symmetry: χx ¼ χy ¼ χ∥ and χz ¼ χ⊥ ; thus ð1Þ ð1Þ χð1Þ xx ¼ χ yy ¼ χ ∥ and χ zz ¼ χ ⊥ . All elements of the second-order nonlinear susceptibility

5.1 Nonlinear Susceptibility and Nonlinear Conductivity

151

tensor χð2Þ contributed by electric dipole interaction vanish for the hexagonal 6=mmm ð2Þ

point group because it is centrosymmetric [1]. Therefore, χijk ¼ 0 for graphene if only electric dipole interaction is considered. Nevertheless, nonvanishing χð2Þ elements can exist due to a number of other effects, including magnetic dipole and electric quadrupole interactions [3]. The nonvanishing χð3Þ elements for the hexagonal 6=mmm point group ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ are χð3Þ xxxx ¼ χ yyyy ¼ χ xxyy þ χ xyyx þ χ xyxy , χ xxyy ¼ χ yyxx , χ xyyx ¼ χ yxxy , χ xyxy ¼ χ yxyx , ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ χð3Þ yyzz ¼ χ xxzz , χ zzyy ¼ χ zzxx , χ zyyz ¼ χ zxxz , χ yzzy ¼ χ xzzx , χ yzyz ¼ χ xzxz , χ zyzy ¼ χ zxzx , and χ zzzz [4,5]. For a 3D material, the nth-order optical conductivity tensor σðnÞ generally has the same structure as the nth-order optical susceptibility tensor χðnÞ. For a graphene sheet,

however, the surface optical conductivity tensor e σ ðnÞ contributed by free charge carriers does not have exactly the same structure as the surface optical susceptibility tensor e χ ðnÞ that accounts also for the contribution from bound charges besides free charge carriers. This difference arises from the fact that the surface current density e J only has components on the graphene surface such that e J

ðnÞ

¼

ðnÞ

ðnÞ e J x ^x

from free carriers ðnÞ

þe J y ^y , whereas

e ðnÞ can have a component perpendicular to the grathe surface polarization density P e ðnÞ^x þ P e ðnÞ ¼ P e ðnÞ^y þ P e ðnÞ^z . For this phene surface from bound charges such that P x y z reason, we find that e χ ð1Þ χz ¼ e χ ⊥ ≠ 0 but e σ ð1Þ σ z ¼ 0 for the linear surface susceptzz ¼ e zz ¼ e ibility and linear surface conductivity of graphene, as discussed in Section 3.2. For a graphene sheet, the only nonvanishing elements of e σ ðnÞ that are contributed by free charge

carriers

are

e σ ð1Þ σ ð1Þ xx ¼ e yy

e σ ð2Þ σ ð2Þ σ ð2Þ σ ð2Þ xyx ¼ e xxy ¼ e yxy ¼ e yyx for e σ ð3Þ σ ð3Þ σ ð3Þ σ ð3Þ xxyy ¼ e xyyx ¼ e xyxy ¼ e yyxx ¼ ð2Þ ments of e σ

for

ð2Þ

e σ ; and

e σ ð1Þ ; e σ ð3Þ xxxx

¼

e σ ð2Þ σ ð2Þ xxx ¼ e yyy , e σ ð3Þ yyyy

¼

e σ ð3Þ xxyy

e σ ð2Þ σ ð2Þ xyy ¼ e yxx , þe σ ð3Þ xyyx

þ

and

e σ ð3Þ xyxy

and

e σ ð3Þ σ ð3Þ σ ð3Þ. For monolayer graphene, all eleyxxy ¼ e yxyx for e

vanish in the electric dipole approximation due to its P6=mmm space

symmetry, as discussed above for χð2Þ. Nonvanishing e σ ð2Þ elements caused by other interactions appear beyond the electric dipole approximation. e By using the relation e J ¼ ∂P=∂t from Maxwell’s equations, we obtain ðnÞ e e ðnÞ ðωÞ: J ðωÞ ¼ iωP

ð5:37Þ

From (5.29)‒(5.34), the effective surface conductivity contributed by bound electrons can be found through the relation e χ ðnÞ ðωÞ: σ ðnÞ ðωÞ ¼ iωϵ 0e

ð5:38Þ

We can also obtain the effective surface susceptibility contributed by free electrons by rearranging (5.38) as e χ ðnÞ ðωÞ ¼ i

e σ ðnÞ ðωÞ ; ϵ0 ω

ð5:39Þ

152

Nonlinear Optical Properties

which was discussed in the paragraph following (4.19). In this chapter, we only consider the optical nonlinearity contributed through the nonlinear surface optical conductivity e σ ðnÞ by free electrons on the Dirac cone. In this case, the nonvanishing elements of e σ ð1Þ , ð2Þ ð3Þ e σ have the symmetry relations discussed in the preceding paragraph. Once σ , and e e χ ðnÞ that is contributed by σ ðnÞ is known, the effective nonlinear surface susceptibility e such free electrons can be obtained using (5.39).

5.2

Semiclassical Approach for Intraband Transitions In this section and the following section, we consider the optical nonlinearities of graphene contributed by intraband transitions of free electrons caused by the perturbation of an external optical field. Such nonlinear optical responses of graphene due to intraband carrier transitions can be treated using a semiclassical approach by employing classical transport theory in combination with the behavior of the electrons on the Dirac cone. Optical nonlinearities contributed by interband carrier transitions on the Dirac cone are treated using a formal quantum mechanical approach in Sections 5.4–5.6. Consider a monochromatic plane field that is polarized in the x direction and propagates in the z direction: Eðr; tÞ ¼ ^x E x ðeikziωt þ eikzþiωt Þ;

ð5:40Þ

where the field amplitude E x is taken to be a real quantity by properly choosing the reference point of the time variable t. This field is normally incident on a graphene sheet located at z ¼ 0 such that there is no spatial field variation on the graphene surface. Thus, the field on the graphene surface is EðtÞ ¼ Eðx; y; 0; tÞ ¼ ^x E x ðeiωt þ eiωt Þ ¼ ^x 2E x cos ωt:

ð5:41Þ

The electrons in this graphene sheet are subject to an electric force in the x direction. Therefore, the electron momentum ℏkx in the x direction is related to the electric field as ℏ

dkx ¼ 2eE x cos ωt; dt

ð5:42Þ

2eE x sin ωt; ω

ð5:43Þ

which gives ℏkx ¼ 

whereas ℏky ¼ 0 because there is no field component in the y direction. As discussed following (2.14) and (3.100), the carrier velocity is given by the relation: vx ¼ vF

kx ¼ vF sgn ð sin ωtÞ: jkj

Then, the surface current density can be written as [6]

ð5:44Þ

5.2 Semiclassical Approach

153

  4 1 1 e J x ðtÞ ¼ ee n v x ¼ ee n v F sgn ðsin ωtÞ ¼ ee n vF sin ωt þ sin 3ωt þ sin 5ωt þ . . . ; π 3 5 ð5:45Þ where e n is the two-dimensional electron density of the graphene sheet. Equation (5.45) contains high-order odd harmonics of frequencies mω, with m ¼ 3; 5; 7. . . . The highorder harmonics appear because of the linear band structure of graphene, resulting in the carrier velocity given by (5.44). For a parabolic band structure, the carrier velocity is simply proportional to the electric field; then, the corresponding current only has one Fourier component of the frequency ω. In the above analysis, the carrier distribution on the Dirac cone is ignored, and all electrons are assumed to contribute equally to the current, as shown in (5.45). However, in reality, the current is actually contributed by the deviation of the carrier distribution away from the thermal equilibrium Fermi–Dirac distribution f0 ðkÞ due to the perturbation of the electric field. Without perturbation, the net electric current is zero even when there is a high density of electrons, as discussed in Chapter 2. To find the perturbed carrier distribution, we use the Boltzmann transport equation by combining (2.10) and (2.12):

Δ

∂f ðk; tÞ þF ∂t

p

f ðk; tÞ ¼ 

f ðk; tÞ  f0 ðkÞ ; τ

ð5:46Þ

where f ðk; tÞ is the perturbed distribution function that gives the probability of an electron having a momentum of ℏk at time t, F is an external force, and the term on the right-hand side of the equation is the rate of change of f ðk; tÞ due to scattering events. Here we consider a homogeneous system under a uniform electric field E so that f ðk; tÞ is independent of the location and F ¼ eE. To find the nonlinear response of graphene, we first ignore the scattering effect by setting τ1 → 0. Then, we obtain ∂f ðk; tÞ ∂f ðk; tÞ  2eE x cos ωt ¼ 0; ∂t ℏ∂kx

ð5:47Þ

where we have used (5.41) for the electric field E. Equation (5.47) can be solved using the method of characteristics. The characteristic curves for (5.47) are given by dkx 2eE x ¼ cos ωt; dt ℏ

ð5:48Þ

or kx ¼ 

2eE x sin ωt þ kc ; ℏω

ð5:49Þ

where kc is a constant. For each value of kc , there is a characteristic curve on the ðkx ; tÞ plane satisfying (5.49), thus satisfying (5.48). The union of these characteristic curves gives the solution to f ðk; tÞ, which remains constant along each of these characteristic curves. When the electric field is turned off, the distribution function f ðk; tÞ of graphene is the thermal equilibrium Fermi–Dirac distribution f0 ðkÞ given in Chapter 1:

154

Nonlinear Optical Properties

f0 ðkÞ ¼ f0 ðkx ; ky Þ ¼

1 : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ðℏvF kx 2 þ ky 2  μÞ=kB T þ 1

½



ð5:50Þ

Therefore, in the limit that E x → 0 we have lim f ðk; tÞ ¼ f0 ðkÞ ¼ f0 ðkc ; ky Þ;

Ex → 0

ð5:51Þ

where the identity kx ¼ kc is used when E x → 0 from (5.49). Because the value of f ðk; tÞ remains constant along each characteristic curve given by (5.49), we obtain the solution of f ðk; tÞ after the electric field is switched on as f ðk; tÞ ¼ f0 ðkc ; ky Þ;

ð5:52Þ

  2eE x sin ωt; ky f ðk; tÞ ¼ f0 kx þ ℏω

ð5:53Þ

or [6]

using (5.49). By comparing (5.50) and (5.53), we find that the x-polarized electric field shifts the distribution of the electrons in the x direction, as expected. Consider the situation when EF ≫ kB T so that the hole current can be ignored and μ ≈ EF . Under this condition, the real surface current induced by the electric field is found by using (2.14) as ðð  g e e J x ðtÞ þ J x ðtÞ ¼ e v x f ðk; tÞdk ð2πÞ2 ð∞ ð∞ 1 kx ¼ e 2 vF f ðk; tÞdkx dky ð5:54Þ π k vF ¼ e 2 π

∞ ∞ 2ðπð ∞

0 0

  2eE x sin ωt; k sin θ kdkdθ; cos θ f0 k cos θ þ ℏω

where e J x ðtÞ is the complex scalar current that is related to the complex vector current as  e JðtÞ ¼ ^x e J x ðtÞ, and e J x ðtÞ þ e J ðtÞ is the current that is related to the real vector

real scalar  x  current as Je ðtÞ ¼ e JðtÞ þ e J ðtÞ ¼ ^x e J x ðtÞ þ e J x ðtÞ . For a weak electric field, (5.54) is approximately given by [6]     3 2 1 2 e J x ðtÞ ≈ 2ee n v F E x 1  E x sin ωt þ E x sin 3ωt ; ð5:55Þ J x ðtÞ þ e 8 8 where e n¼

kF2 π

is the surface electron density given by (1.38) in the limit that EF ≫ kB T, and

ð5:56Þ

5.2 Semiclassical Approach

155

1

0

x

= 0.1

(J

x

+ Jx

)

en

F

0.5

0.5 −0.5 1 x

=5

−1 2

0 t

2

Figure 5.1 Current response of graphene to an optical field for various field strengths. The

dimensionless field strength parameter E x is marked next to the curves. The surface current is obtained from (5.54), and the approximation (5.55) is plotted as dotted symbols for E x ¼ 0:1.

Ex ¼

ev F Ex ωEF

ð5:57Þ

is a dimensionless field strength parameter. The field strength parameter E x is the ratio of the energy ev F E x =ω that an electron obtains from the electric field during one oscillation period to the Fermi energy EF of electrons. The surface current calculated using (5.54) is plotted in Figure 5.1 for various strengths of the electric field. When the electric field is weak such that E x ≪1, the current given by (5.54) is well approximated by (5.55); in this case, the current is  predominately contributed by the linear term e J x ðtÞ þ e J ðtÞ ≈ ee n vF E x sin ωt, which is x

proportional to the amplitude of the electric field. For a strong field such that E x ≫ 1, the energy obtained by an electron from the electric field is much larger than EF ; the carrier distribution is then predominately determined by the electric field and less by EF. In this case, it is sufficient to only consider the kinetic response of electrons to the electric field; then, the current response approaches the behavior described by (5.45), that is,  e J x ðtÞ þ e J ðtÞ ¼ ee n v F sgn ðsin ωtÞ. Clearly, the response becomes increasingly nonlinear x

as the field strength increases. For the electric field given by (5.41), according to (5.18) and (5.32)‒(5.34) the relation between the electric field and the total complex surface current density can be written in the form: iωt iωt 2 e J x ðtÞ ¼ e σ ð1Þ Þþe σ ð3Þ Þ ðE x eiωt Þ xx ðω ¼ ωÞðE x e xxxx ðω ¼ ω þ ω  ωÞðE x e iωt 3 þe σ ð3Þ Þ : xxxx ð3ω ¼ ω þ ω þ ωÞðE x e

ð5:58Þ

156

Nonlinear Optical Properties

As an approximation for weak fields, (5.58) does not consist of any second-order nonlinear term because only odd harmonics are found in (5.55). The subscripts of e σ ð1Þ xx ð3Þ and e σ xxxx indicate that the induced linear and nonlinear currents and the participating electric fields are all in the x direction. The first term e σ ð1Þ xx ðωÞ in (5.58) is the linear term ð3Þ proportional to E x , the second term e σ xxxx ðωÞ is the nonlinear term proportional to E 3x , and the third term e σ ð3Þ xxxx ð3ωÞ is responsible for the third harmonic at 3ω. To obtain the values ð1Þ ð3Þ e of e σ xx ðωÞ, e σ xxxx ðωÞ, and e σ ð3Þ xxxx ð3ωÞ, we can find J x ðtÞ from (5.55) as   3 3 iωt 1 3 i3ωt iωt e J x ðtÞ ≈ iee n vF E x e  Exe þ Exe : ð5:59Þ 8 8 By comparing (5.59) with (5.58), we obtain e σ ð1Þ xx ðωÞ ¼ i e σ ð3Þ xxxx ðωÞ ¼ i e σ ð3Þ xxxx ð3ωÞ ¼ i

e2 EF πℏ2 ω

;

3e4 v F2 8πℏ2 ω3 EF e4 v F2 8πℏ2 ω3 EF

ð5:60Þ ;

ð5:61Þ

:

ð5:62Þ

Equation (5.60) is just the Drude intraband conductivity given by (4.3) in the limit that EF ≫ kB T and τ1 → 0. The nonlinear conductivity for a finite scattering rate τ1 and an electric field that has components in both x and y directions can be derived in a similar manner using the Boltzmann transport equation (5.46) and the method of characteristics. The results are listed in Table 5.1. Note that in Table 5.1, e σ ð3Þ σ ð3Þ xxxx ðωÞ and e xxxx ð3ωÞ in the limit that τω ≫ 1 do not converge to (5.61) and (5.62), respectively, because (5.61) and (5.62) are obtained

Table 5.1 Linear and nonlinear surface optical conductivities of graphene due to intraband transitions [8]. Surface optical conductivity

e2 EF

e σ ð1Þ σ ð1Þ xx ðωÞ ¼ e yy ðωÞ e σ ð3Þ xxyy ðωÞ ¼ ¼e σ ð3Þ yyxx ðωÞ

e σ ð3Þ xyxy ðωÞ ¼ ¼e σ ð3Þ yxyx ðωÞ

e σ ð3Þ xyyx ðωÞ ¼e σ ð3Þ yxxy ðωÞ

e σ ð3Þ σ ð3Þ xxxx ðωÞ ¼ e yyyy ðωÞ ¼

¼

e σ ð3Þ yxyx ð3ωÞ

e σ ð3Þ σ ð3Þ xxxx ð3ωÞ ¼ e yyyy ð3ωÞ

πℏ 

2

ðτ1

i

 iωÞ

4πℏ2 EF

3e4 v2F ðτ2 þ ω2 Þðτ1

 i2ωÞ

3e σ ð3Þ xxyy ðωÞ

e σ ð3Þ σ ð3Þ σ ð3Þ xxyy ð3ωÞ ¼ e xyxy ð3ωÞ ¼ e xyyx ð3ωÞ e σ ð3Þ yyxx ð3ωÞ

τ1 → 0

Finite scattering rate

¼

e σ ð3Þ yxxy ð3ωÞ



e4 v2F 2

4πℏ EF

ðτ1

3e σ ð3Þ xxyy ð3ωÞ

 iωÞðτ1  i2ωÞðτ1  i3ωÞ

e2 EF

πℏ2 ω 3e4 v2F i 8πℏ2 EF ω3 9e4 v2F i 8πℏ2 EF ω3 e4 v2F i 24πℏ2 EF ω3 e4 v2F i 8πℏ2 EF ω3

5.3 Bistability

157

in the collisionless limit τ1 → 0, whereas the results in Table 5.1 are derived by assuming a finite value of τ1 , and the condition τω ≫ 1 represents a limit when the oscillation period is much smaller than the relaxation time [7].

5.3

Bistability Optical bistability is a characteristic feature of a system that has two stable output states for a given input. The phenomenon of optical bistability for graphene is due to the optical nonlinearity of graphene as shown in (5.59), where the presence of the nonlinear terms of E x signifies possible multiple solutions of E x for a given induced current e J x. Consider normal incidence of an optical wave on a graphene sheet that is located at z ¼ 0 and is surrounded by free space for n1 ¼ n2 ¼ 1, as shown in Figure 3.8(a). As in (3.117) and (3.118), we define the incident field as Ei ¼ ^x Ei eiωt ¼ ^x E i eikziωt , the reflected wave as Er ¼ ^x Er eiωt ¼ ^x E r eikziωt , and the transmitted field as Et ¼ ^x Et eiωt ¼ ^x E t eikziωt . By applying the boundary conditions that E1 jz¼0 ¼ E2 jz¼0 and ^z  ðH2  H1 Þj ¼ e J dictated by Maxwell’s equations, we obtain z¼0

2ðE i  E t Þeiωt ¼

e Jx : cϵ 0

ð5:63Þ

By substituting e J x of (5.58) in (5.63) with E x ¼ E t as E t is the amplitude of the electric field polarized in the x direction, we obtain [7]  i 1 h ð1Þ ð3Þ 2 Ei ¼ Et 1 þ e σ ðωÞ þ e σ xxxx ðωÞE t ð5:64Þ 2cϵ 0 xx after keeping only the eiωt terms for e J x. Here we take E t to be a real quantity and E i to be a complex quantity so that the information of the phase difference between E i and E t is expressed in E i . By using the definition of the dimensionless field parameter given in (5.57) and the values of e σ ðnÞ ðωÞ given by (5.60) and (5.61), (5.64) can be expressed in a concise form:    3 2 E i ¼ E t 1 þ iα 1  E t ; ð5:65Þ 8 where α¼

e2 E F 2cϵ 0 πℏ2 ω

ð5:66Þ

is a dimensionless parameter [7]; E i and E t are the dimensionless field parameters for E i and E t , respectively, as defined in (5.57). The bistability curves given by (5.65) are drawn for different values of α as shown in   Figure 5.2. As can be seen, for a given amplitude E i  of the incident field, there are one,

158

Nonlinear Optical Properties

2 B

U

C 1.5

t

=4

1

A

=2

=8

0.5 L 0

D 0

2

4

6

i

Figure 5.2 Bistability curves given by (5.65) for different values of α. For the case of α ¼ 8, the thick and thin curves represent stable and unstable states, respectively.

pffiffiffi two, or three possible values for the transmitted field amplitude jE t j. For α > 3, bistability can be experimentally observed by measuring the transmitted field amplitude as a function of the amplitude of the incident field. For EF ¼ 300 meV, the frequency pffiffiffi has to be smaller than 0.6 THz so that α > 3. For the case of α ¼ 8, by slowly increasing the incident field amplitude, the transmitted field amplitude jE t j increases along the lower thick curve to point “A,” as shown in Figure 5.2. Further increase of jE i j leads to a sudden jump in jE t j from point “A” to point “B,” represented by an arrow pointing upward. If jE i j is reduced from this point, jE t j follows the upper thick curve from point “B” to point “C” rather than going from “B” to “A.” Further decrease of jE i j leads to a sudden drop in jE t j from point “C” to point “D,” represented by an arrow pointing downward in Figure 5.2. Thus the middle curve, shown as a thin solid curve in Figure 5.2, is never visited and represents unstable states, whereas all points on the thick curves are stable states. For any input E i that has a value such that pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 16ðα α2  3Þð2αþ α2  3Þ2 =81α > jE i j2 > 16ðα þ α2  3Þ ð2α α2 3Þ2 =81α, two stable states exist, such as the states that are respectively marked by “U” and “L” on the upper and lower branches of the hysteresis curves. Whether the system is in state “U” or “L” is determined by the past values of E i , signifying the hysteresis nature of the bistability system.

5.4

Quantum Mechanical Approach for Interband Transitions In the preceding two sections, the nonlinear conductivity is calculated by considering only intraband carrier transitions, whereas interband carrier transitions are ignored. This

5.4 Quantum Mechanical Approach

159

is justified if the photon energy ℏω is much smaller than the chemical potential jμj. In this section and in the following two sections, we consider the nonlinear conductivity contributed by interband transitions in the opposite situation, where the energy of a single photon or a combination of photons involved in the nonlinear interaction is comparable to or larger than jμj so that the contribution to the nonlinear conductivity from intraband transitions is negligible. Similar to the calculation of the optical conductivity in Chapter 3, the nonlinear conductivity due to interband transitions can also be calculated by using the density matrix combined with the perturbation theory. As discussed in Chapter 3, the perturbation Hamiltonian accounting for the interaction between the electrons on the Dirac cone of graphene and an electromagnetic field is expressed in terms of the vector potential given by (3.63) as [9–11] H^ 0 ¼ v F eσ  A ¼vF eσ  ðA þ AÞ;

ð5:67Þ

where σ is the Pauli vector. Consider the mixing of multiple frequencies. The real electric field that consists of multiple discrete frequency components has the form of (5.9): X X Eðr; tÞ ¼ Eðr; tÞ þ Eðr; tÞ ¼ Eðωq Þeiωq t þ Eðωq Þeiωq t ¼ Eðωq Þeiωq t ; q>0

½



q

ð5:68Þ where Eðωq Þ ¼ Eðωq Þ ¼ Eðωq Þ according to (5.12). Using (3.61), we find the vector potential: A ¼ A þ A ¼

X 1 X 1 Eðωq Þeiωq t  Eðωq Þeiωq t ¼ Eðωq Þeiωq t : ð5:69Þ iω q iωq q>0

½



q

Using (5.67) and (5.69), we obtain X X  H^ 0 ðtÞ ¼ iEF σ  Eðωq Þeiωq t  E ðωq Þeiωq t ¼ iEF σ  Eðωq Þeiωq t ; ð5:70Þ q>0

½



q

where Eðωq Þ is defined in a manner similar to (5.57): Eðωq Þ ¼

ev F Eðωq Þ; ωq EF

ð5:71Þ

and, according to (5.11) and (5.12), Eðωq Þ ¼ 

ev F  E ðωq Þ: ωq EF

ð5:72Þ

In the following, to reduce mathematical complexity, we assume that the electric field is polarized along the x direction such that Eðωq Þ ¼ ^x Eðωq Þ and Eðωq Þ ¼ ^x Eðωq Þ, where

160

Nonlinear Optical Properties

Eðωq Þ ¼

ev F ev F  Eðωq Þ and Eðωq Þ ¼  E ðωq Þ: ωq EF ωq EF

ð5:73Þ

Then, the matrix elements of (5.70) in the interaction picture can be expressed as X X 〈k0 jH^ 0 ðtÞjk〉 ¼ V k0 k Eðωq Þeiðωk0 ωk ωq Þt ¼ V k0 k Eðωq Þeiðωk0 k ωq Þt ; ð5:74Þ q

q

where V k0 k ¼ iEF 〈k0 jσ x jk〉;

ð5:75Þ

ωk ¼ ωnk by dropping the band index n to simplify the expression, and ωk0 k ¼ ωk0  ωk ;

ð5:76Þ

ωnk is given by ωnk ¼

Enk ¼ nv F jkj; ℏ

ð5:77Þ

and we define ωn0 k0 ;nk as ωn0 k0 ;nk ¼ ωn0 k0  ωnk ¼ ωk0 k ¼ ωk0  ωk :

ð5:78Þ

As done in Chapter 3, we drop the band indices n and n0 in the notations of jk〉 and jk0 〉, which respectively represent jnk〉 and jn0 k0 〉, and in the subscripts of V k0 k to simplify the mathematical expressions. Equation (5.74) is essentially the same as (3.89) in Chapter 3 for the calculation of the linear conductivity. The only difference is that only a finite number of plane wave modes of discrete frequencies are considered in (5.74), whereas in (3.89) infinite modes of a continuous frequency spectrum are considered. For the consideration of the linear optical conductivity of graphene in Chapter 3, the density matrix is expressed in perturbation expansion only to the first order as given in (3.93): ^ρ ðtÞ ¼ ^ρ ð0Þ þ ^ρ ð1Þ ðtÞ, where ^ρ ð0Þ represents the unperturbed thermal equilibrium distribution f0 that does not contribute to the conductivity, while the first-order perturbation term ^ρ ð1Þ ðtÞ is found to be proportional to the electric field E and is responsible for the linear conductivity. To find the nonlinear optical conductivity of graphene, it is necessary to include high-order terms in the perturbation expansion of ^ρ ðtÞ: ^ρ ðtÞ ¼ ^ρ ð0Þ þ ^ρ ð1Þ ðtÞ þ ^ρ ð2Þ ðtÞ þ ^ρ ð3Þ ðtÞ þ . . . ;

ð5:79Þ

where we require that ^ρ ðnÞ ∝ En . Then, by using the fact that H^ 0 ðtÞ ∝ A ∝ E and the Liouville–von Neumann equation given in (3.92), ∂ i ^ρ ðtÞ ¼  H^0 ðtÞ; ^ρ ðtÞ ; ∂t ℏ

½



ð5:80Þ

for the time evolution of the density matrix in the interaction picture, we find the relation

5.4 Quantum Mechanical Approach

161

i ∂ ðnÞ ih ^ρ ðtÞ ¼  H^ 0 ðtÞ; ^ρ ðn1Þ ðtÞ ∂t ℏ

ð5:81Þ

by collecting terms that have the same orders of E in (5.80). For the zeroth-order term, we have d^ρ ð0Þ =dt ¼ 0 because ^ρ ð0Þ does not vary with time. The time evolution of the matrix elements of ^ρ ðnÞ ðtÞ in the interaction picture is found from (5.81) as ∂ 〈k2 j ^ρ ðnÞ ðtÞjk1 〉 ∂t  ð5:82Þ  iX ðn1Þ ðn1Þ 0 0 ^ ^ ðtÞjk1 〉  〈k2 j ^ρ ðtÞjk3 〉〈k3 jH ðtÞjk1 〉 ; ¼ 〈k2 jH ðtÞjk3 〉〈k3 j^ρ ℏ jk 〉 3

where jki 〉 represents a state with a band index ni . The summation on the right-hand side of (5.82) runs through all the possible states jk3 〉 of different wave vectors for k3 over the two possible bands of n3 ¼ 1 and n3 ¼ 1. Using (5.82) and (5.74), the matrix elements of the first-order ^ρ ð1Þ ðtÞ are obtained as given in (3.97), which are given here again as 〈k2 j^ρ ð1Þ ðtÞjk1 〉 ¼ 

X Eðωm Þ V k2 k1 ð fk 1  fk 2 Þ eiðωk2 k1 ωm Þt ; ω ℏ  ω k k m 2 1 m

ð5:83Þ

where fki ¼ f0 ðEki Þ is the thermal equilibrium Fermi–Dirac distribution. Once ^ρ ðn1Þ is obtained, ^ρ ðnÞ can be successively obtained. For example, by inserting (5.83) back into (5.82), we obtain the matrix elements of ^ρ ð2Þ ðtÞ as 〈k3 j^ρ ð2Þ ðtÞjk1 〉  XX V k k V k k Eðωm ÞEðωn Þeiðωk3 k1 ωm ωn Þt  fk  fk fk2  fk3 3 2 2 1 1 2 ¼  : ωk3 k1  ωm  ωn ωk2 k1  ωm ωk3 k2  ωm ℏ2 jk 〉 m;n 2

ð5:84Þ Accordingly, the matrix elements of ^ρ ð3Þ ðtÞ are obtained by inserting (5.84) back into (5.82) again: 〈k4 j^ρ ð3Þ ðtÞjk1 〉 ¼

XX X V k jk2 〉 jk3 〉 m;n;p

4 k3

V k3 k2 V k2 k1 Eðωm ÞEðωn ÞEðωp Þeiðωk4 k1 ωm ωn ωp Þt ωk4 k1  ωm  ωn  ωp ℏ3



  1 fk1 fk2 fk2 fk3   ωk3 k1 ωm ωn ωk2 k1 ωm ωk3 k2 ωm   1 fk 2  fk 3 fk3  fk4   ; ωk4 k2  ωm  ωn ωk3 k2  ωm ωk4 k3  ωm

ð5:85Þ

g

where the dummy index k2 has been swapped with k3 in the last term so that the expression can be made more compact.

162

Nonlinear Optical Properties

In the above, we have ignored for simplicity the relaxation rate γk0 k ¼ γn0 k0 ;nk of the off-diagonal elements 〈k0 j ^ρ ðnÞ ðtÞjk〉, which has n0 ≠ n or k0 ≠ k so that jk0 〉 ≠ jk〉, of the density matrix [4,5]. This relaxation rate characterizes the phase relaxation of the coherence between the two different states jk0 〉 and jk〉 [1]. The effect is to broaden the linewidth. This relaxation rate can be restored in (5.83)–(5.85) by simply replacing ωk0 k with ωk0 k  iγk0 k for k0 ¼ k2, k3 , or k4, and k ¼ k1 , k2 , or k3. Once the density matrix is known, the nth-order nonlinear surface current density in the x direction is obtained using the relation i ðnÞ ðnÞ e h e J x ðtÞ þ e J x ðtÞ ¼  Tr ^ρ ðnÞ ðtÞ^v x ðtÞ ; A

ð5:86Þ

ðnÞ

where e J x is the complex nth-order nonlinear surface current density, the ^v x ðtÞ operator is given by (3.100), and ^ρ ðnÞ ðtÞ is defined in (5.79). As seen in Chapter 3, ^ρ ð0Þ does not contribute to the current, and ^ρ ð1Þ contributes to a current that is linearly proportional to the electric field; this linear response is characterized by the linear optical conductivity e σ ðωÞ discussed in Chapter 3, or e σ ð1Þ xx ðωÞ using the notation of this chapter. As we shall see in the following sections, ^ρ ð2Þ contributes to the second-order nonlinear optical conductivity e σ ð2Þ ρ ð3Þ contributes to the third-order nonlinear optical conductivity xxx, and ^ ð3Þ e σ xxxx . Equations (5.83)–(5.86) are generally applicable to any material. The characteristics of graphene come into play through the velocity operator ^v x in (5.86) and the matrix elements V k0 k given in (5.75), which characterize the interaction of the electrons on the Dirac cone with an electromagnetic field through the matrix elements given in (5.74) for the interaction Hamiltonian H^ 0 given in (5.67). The matrix elements of the velocity operator ^v x ¼ vF σ x for the electrons on the Dirac cone are discussed in Chapter 3. By using (1.28) and (5.75), we obtain V k0 k as V k0 k ¼ iEF 〈k0 jσ x jk〉 ¼ EF

neiφ þ n0 eiφ δk0 ;k ; 2i

ð5:87Þ

where φ ¼ tan1 ðky =kx Þ ¼ tan1 ðk 0y =k 0x Þ because k ¼ k0 due to the factor δk0 ;k. 0 Alternatively, 〈k0 jσ x jk〉 in (5.87) can be replaced by v1 v x jk〉 using the identity F 〈k j^ ^v x ¼ v F σ x .

5.5

Second-Order Optical Nonlinearity Because graphene is centrosymmetric, as discussed in Section 5.1, there is no second-order optical nonlinearity contributed by electric dipole interaction. This fact can be checked by taking ^ρ ðnÞ ðtÞ in (5.86) to be the second-order ^ρ ð2Þ ðtÞ to ð2Þ ð2Þ J x ðtÞ expressed in the form of (5.10) in terms of its frequency compofind e J x ðtÞ þ e nents as

5.5 Second-Order Optical Nonlinearity

ð2Þ ð2Þ e J x ðtÞ þ e J x ðtÞ ¼

X

163

ð2Þ e J x ðωq Þeiωq t

q

¼

e XXXX Eðωm ÞEðωn Þeiðωm þωn Þt Aℏ2 m;n jk3 〉 jk2 〉 jk1 〉

ð5:88Þ

  〈k1 j^v x ðtÞjk3 〉V k3 k2 V k2 k1 fk 1  fk 2 fk 2  fk 3   : ωk3 k1  ωm  ωn ωk2 k1  ωm ωk3 k2  ωm As discussed above, for simplicity the phase relaxation rate γk0 k for the relaxation of the coherence between states jk0 〉 and jk〉 is ignored in (5.88); it can be restored by simply replacing ωk0 k with ωk0 k  iγk0 k in (5.88) for k0 ¼ k2 or k3, and k ¼ k1 or k2. Consider the simple case in which the optical field is a monochromatic plane wave that consists of only one frequency component at ω. In this case, ωm and ωn can each only take the value of either ω or ω when the summation over the indices m and n in (5.88) runs through all positive and negative values for m and n. Consequently, the second-order nonlinear current density in (5.88) contains frequency components of ω  ω and 2ω. The ω  ω component results in a DC current due to optical rectification, whereas the 2ω harmonics result in a second-harmonic current. For the second-harmonic current at the frequency ωq ¼ 2ω ¼ ω þ ω, we focus on the term expði2ωtÞ, which requires that ωm ¼ ωn ¼ ω. Then, by using (5.88) and (5.33), we find that e σ ð2Þ x ð2ωÞ ¼

   ievF X X V n3 k3 ;n2 k2 V n3 k3 ;n2 k2 V n2 k2 ;n1 k1 fn1 k1  fn2 k2 fn2 k2  fn3 k3  ; ωn3 k3 ;n1 k1  2ω ωn2 k2 ;n1 k1  ω ωn3 k3 ;n2 k2  ω ℏ2 AEF k ;k ;k n3 ;n2 ;n1 3

2

1

ð5:89Þ where (5.87) and the relation v^ x ¼ v F σ x are used to replace 〈k1 j^v x ðtÞjk3 〉, and the summation over jki 〉 is written out explicitly as the summations over the wave vector ki and the band index ni . In (5.89), the index ni is restored for the subscripts where ki is present. Knowing that k3 ¼ k2 ¼ k1 due to the term δk0 ;k in the matrix elements V n0 k0 ;nk of the form given in (5.87), the summations in (5.89) are simplified. By using (5.87) and by following the procedure used to calculate e σ 0 ðωÞ in (3.78), we obtain from (5.89) that e σ ð2Þ x ð2ωÞ

2 2evF X V 1k;1k V 1k;1k ð f1k  f1k Þ ¼ i 2 ℏ AEF k ðω1k;1k  ωÞðω1k;1k þ ωÞ

ð∞2ðπ ∝ 0 0

cos φ sin 2 φ ½ f ðEÞ  f ðEÞdφkdk 4k 2 ð2E  ℏωÞð2E þ ℏωÞ

ð5:90Þ

¼ 0; where we have set k3 ¼ k2 ¼ k1 ¼ k because of the delta function in the expression of the matrix element given by (5.87). Because of the term cos φ sin2 φ, the integral in (5.90)

164

Nonlinear Optical Properties

ð2Þ

has a zero value. The same result can be obtained for other harmonics in e J x ðtÞ as well. Therefore, graphene does not have second-order optical nonlinearity. The above discussion is valid only when the inversion symmetry of the graphene crystal is conserved and electric dipole interaction is considered. If the symmetry is broken or other effects are considered beyond the electric dipole approximation, nonvanishing second-order optical nonlinearity is possible for graphene [3,12]. The symmetry can be destroyed if the surface fluctuation of graphene is considered, or if graphene is on a substrate so that the interface is asymmetric. The second-order optical nonlinearity also arises when magnetic dipole or electric quadrupole interaction beyond the electric dipole interaction is considered. The calculation of these effects is usually carried out by taking into account the inplane propagation of electromagnetic waves. This spatial variation of electric field is accounted for in Chapter 4 by expressing the electric field as a function of both wave vector and frequency, Eðq; ωÞ; to consider the effect of nonlocal response in real space. It is found that by considering the spatial dispersion in the momentum space, contributions to the conductivity analogous to those due to magnetic dipole and electric quadrupole interactions arise, resulting in nonvanishing second-order optical nonlinearity [3,12].

5.6

Third-Order Optical Nonlinearity Similar to the second-order optical nonlinearity, the third-order optical nonlinearity is obtained by taking ^ρ ðnÞ ðtÞ in (5.86) to be the third-order ^ρ ð3Þ ðtÞ. The third-order ð3Þ ð3Þ J x ðtÞ expressed in a form in terms of its nonlinear surface current density e J x ðtÞ þ e frequency components similar to that of the nonlinear polarization expressed in (5.10) is given as ð3Þ ð3Þ e J x ðtÞ þ e J x ðtÞ X ð3Þ e ¼ J x ðωq Þeiωq t q

¼i

evF X XXXX V k1 k4 V k4 k3 V k3 k2 V k2 k1 Eðωm ÞEðωn ÞEðωp Þeiðωm þωn þωp Þt ωk4 k1  ωm  ωn  ωp ℏ AEF m;n;p jk4 〉 jk3 〉 jk2 〉 jk1 〉 3

 

  1 fk 1  fk 2 fk 2  fk 3  ωk3 k1 ωm ωn ωk2 k1  ωm ωk3 k2  ωm   1 fk2  fk3 fk3  fk4   ; ωk4 k2  ωm  ωn ωk3 k2  ωm ωk4 k3  ωm

ð5:91Þ

v x ðtÞjk4 〉 is replaced by V k1 k4 using (5.87); the summations where again as in (5.89), 〈k1 j^ run over all possible states jki 〉 of different values of ki on different bands of ni ¼ 1. Then, from (5.34), we find that

5.6 Third-Order Optical Nonlinearity

e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ evF XXXX V k1 k4 V k4 k3 V k3 k2 V k2 k1 ¼i 3 ℏ AEF jk4 〉 jk3 〉 jk2 〉 jk1 〉 ωk4 k1  ωm  ωn  ωp    1 fk 1  fk 2 fk2  fk3   ωk3 k1  ωm  ωn ωk2 k1  ωm ωk3 k2  ωm   1 fk2  fk3 fk 3  fk 4   : ωk4 k2  ωm  ωn ωk3 k2  ωm ωk4 k3  ωm

165

ð5:92Þ

As discussed above, for simplicity the phase relaxation rate γk0 k for the relaxation of the coherence between states jk0 〉 and jk〉 is ignored in (5.91) and (5.92); it can be restored by simply replacing ωk0 k with ωk0 k  iγk0 k in (5.91) and (5.92) for k0 ¼ k2, k3 , or k4, and k ¼ k1 , k2 , or k3. The effect of this phase relaxation rate is to broaden the spectral linewidth of the nonlinear conductivity e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ near each resonance frequency, which is further discussed later. Because of the term δk0 ;k in the matrix elements V k0 k , all ki vectors in (5.92) have to be the same so that momentum is conserved. Therefore, we can set k4 ¼ k3 ¼ k2 ¼ k1 ¼ k and only consider different values of ni in the summation of (5.92). After lengthy but straightforward algebra, we obtain, from (5.92), e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ ¼i

4evF X 3

Aℏ EF (



k

2

ðf1k  f1k ÞV 1k;1k

ω1k;1k ωq

 2 V 1k;1k ωq ωm  ωq  ω1k;1k ωm  ω1k;1k ðωq þ ωm Þðωm þ ωn Þðωn þ ωp Þ 2

þV 1k;1k

2

V 1k;1k

ðω2q

ωm ðωm þ ωn Þ þ ωq ð2ωm þ ωn þ ωq Þ h i  ω2m Þ ω2q  ðωm þ ωn Þ2 ðωq  ω1k;1k Þ

ωq ðωm þ ωn Þ þ ωm ð2ωm þ ωn þ ωq Þ h i 2 ðωq  ω2m Þ ω2m  ðωm þ ωn Þ2 ðωm  ω1k;1k Þ

2

V 1k;1k h

ω2m

ð5:93Þ

ωq ωm þ ðωm þ ωn Þð2ωm þ ωn þ ωq Þ ih i  ðωm þ ωn Þ2 ω2q  ðωm þ ωn Þ2 ðωm þ ωn  ω1k;1k Þ

g

þðω1k;1k → ω1k;1k Þ ; where ωq ¼ ωm þ ωn þ ωp is the sum frequency, and we have restored the dependence on ni in the subscripts that contain ki . The last term ðω1k;1k → ω1k;1k Þ in the curly brackets of (5.93) represents the same terms as the preceding terms in the curly brackets with ω1k;1k replaced by ω1k;1k ¼ ω1k;1k. Except for this term, the other terms in the curly brackets are divergent if ωm > 0, ωq > 0, or ωm þ ωn > 0, respectively, because

166

Nonlinear Optical Properties

the denominator can be zero for a certain value of ω1k;1k > 0. In the reverse situation when ωm < 0, ωq < 0, or ωm þ ωn < 0, the last term is divergent. The divergence is removed if we restore the relaxation rate by replacing ω1k;1k with ω1k;1k  iγ1k;1k , as discussed above, which is equivalent to considering an infinitesimal scattering rate by writing ωm as ωm þ iγ, as we did in Chapter 3. In replacing ω1k;1k with ω1k;1k  iγ1k;1k to restore the relaxation rate, ω1k;1k in the last term ðω1k;1k → ω1k;1k Þ is replaced with ω1k;1k  iγ1k;1k ¼ ω1k;1k  iγ1k;1k because γ1k;1k ¼ γ1k;1k though ω1k;1k ¼ ω1k;1k . By using (5.87) and by following the procedure used to calculate e σ 0 ðωÞ in (3.78), we obtain the real part of e σ ð3Þ xxxx ðωq Þ from (5.93) as e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ 2ðπð ∞ 4eEF3 1 ¼ 4 FðEÞE2 sin2 φ πℏ vF ℏωq 0

0 0

  

     ℏωq sin2 φ ℏωm ωm δ E   ωq δ E  ðωq þ ωm Þðωm þ ωn Þðωn þ ωp Þ 2 2   ωm ðωm þ ωn Þ þ ωq ð2ωm þ ωn þ ωq Þ ℏωq h i þ cos2 φ δ E 2 2 2 2 2 ðωq  ωm Þ ωq  ðωm þ ωn Þ ð5:94Þ   ω ðω þ ω Þ þ ω ð2ω þ ω þ ω Þ ℏω q m n m m n q m h i δ E  cos2 φ 2 ðω2  ω2 Þ ω2  ðωm þ ωn Þ2 q

 cos2 φ h

m

m

g

ωq ωm þ ðωm þ ωn Þð2ωm þ ωn þ ωq Þ  ω þ ωn ih iδ E  ℏ m dEdφ; 2 ω2m  ðωm þ ωn Þ2 ω2q  ðωm þ ωn Þ2

where the integration over k in (5.93) is converted to an integration over energy E, and FðEÞ ¼ f0 ðEÞ  f0 ðEÞ:

ð5:95Þ

Finally, we obtain, from (5.94), e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ "   e4 v 2F 1 ℏωm þ ℏωn ðωm þ ωn Þ2 ¼ 3 F 2 ωm ωn ωq 4ℏ ωm ωn ωp 0

  ω2m ℏωm 2ω2m þ 4ωn ðωn þ ωp Þ þ ωm ð6ωn þ ωp Þ F þ ðωm þ ωn Þðωn þ ωp Þðωm þ ωq Þ ωn ωq 2 #   2 ωq ℏωq 2ωm þ ωm ð3ωn  2ωp Þ þ ωn ðωn þ ωp Þ þ F : ðωm þ ωn Þðωn þ ωp Þðωm þ ωq Þ ωp 2 ð5:96Þ

5.6 Third-Order Optical Nonlinearity

167

The intrinsic permutation symmetry [1,5], which is purely a result of the convention used for the representation of frequency-dependent nonlinear susceptibilities, requires that ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

e χ ijkl ðωq ¼ ωm þ ωn þ ωp Þ ¼ e χ ijlk ðωq ¼ ωm þ ωp þ ωn Þ χ ikjl ðωq ¼ ωn þ ωm þ ωp Þ ¼e χ iklj ðωq ¼ ωn þ ωp þ ωm Þ ¼ e

ð5:97Þ

χ ilkj ðωq ¼ ωp þ ωn þ ωm Þ; ¼e χ iljk ðωq ¼ ωp þ ωm þ ωn Þ ¼ e where the frequencies ωm , ωn , and ωp , which represent the frequencies of the excitation fields, can be freely permutated on the right-hand side of the equals sign without changing the value of the nonlinear susceptibility if the corresponding Cartesian coordinate indices j, k, and l, which represent the polarization directions of the excitation fields, are simultaneously permuted. Therefore, the intrinsic permutation symmetry also requires that ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

e σ ijkl ðωq ¼ ωm þ ωn þ ωp Þ ¼ e σ ijlk ðωq ¼ ωm þ ωp þ ωn Þ σ ikjl ðωq ¼ ωn þ ωm þ ωp Þ ¼e σ iklj ðωq ¼ ωn þ ωp þ ωm Þ ¼ e

ð5:98Þ

¼e σ iljk ðωq ¼ ωp þ ωm þ ωn Þ ¼ e σ ilkj ðωq ¼ ωp þ ωn þ ωm Þ: This relation implies that the value of e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ, given by (5.92)

and (5.93), and that of e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ, given by (5.94) and (5.96), are not changed by interchanging the three field frequencies ωm , ωn , and ωp in any manner. 0

While this is true, it is not apparent in the expressions of e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ seen in (5.92) and (5.93), nor in those of e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ seen in (5.94) and (5.96), because all of these expressions are not symmetric in form for the three frequen0

cies ωm , ωn , and ωp . The expression of e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ can be made formally symmetric for the three frequencies ωm , ωn , and ωp without changing its ^ i to it: value by applying an intrinsic permutation operator P e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ ¼i

evF ^ XXXX V k1 k4 V k4 k3 V k3 k2 V k2 k1 Pi ℏ AEF jk4 〉 jk3 〉 jk2 〉 jk1 〉 ωk4 k1  ωm  ωn  ωp 3

 

ωk3 k1 

  1 fk 1  fk 2 fk 2  fk 3   ωm  ωn ωk2 k1  ωm ωk3 k2  ωm

ωk4 k2

ð5:99Þ

 # 1 fk 2  fk 3 fk 3  fk 4  ;  ωm  ωn ωk3 k2  ωm ωk4 k3  ωm

^ i averages everything to its right-hand side by summing over all where the operator P possible permutations of the input frequencies ωm , ωn , and ωp and then dividing the sum by the number of possible intrinsic permutations among these frequencies [5]. In other ^ i can be expressed as words, P

168

Nonlinear Optical Properties

^ i f ðωm ; ωn ; ωp Þ ¼ P

1 Number of permutations

X ωm ; ωn ; ωp ωm þ ωn þ ωp ¼ ωq

f ðωm ; ωn ; ωp Þ: ð5:100Þ

For example, consider the mixing of two fields of different frequencies at ω1 and ω2 through e σ ð3Þ xxxx . Note that with fields at ω1 and ω2 , both ω1 and ω2 are available for frequency mixing. For ωq ¼ 2ω1 þ ω2 , there are three intrinsic permutations in total: ωq ¼ ω1 þ ω1 þ ω2 ¼ ω1 þ ω2 þ ω1 ¼ ω2 þ ω1 þ ω1 . From (5.98), it is clear that e σ ð3Þ σ ð3Þ σ ð3Þ xxxx ðωq ¼ ω1 þω1 þω2 Þ ¼ e xxxx ðωq ¼ ω1 þω2 þω1 Þ ¼ e xxxx ðωq ¼ ω2 þω1 þω1 Þ. ð3Þ Therefore, e σ xxxx ð2ω1 þ ω2 Þ is the sum of the expressions of these three per^ i is to ensure that both the form mutations divided by three. The purpose of P ð3Þ and the value of e σ xxxx ðωq ¼ ωm þ ωn þ ωp Þ are not changed under intrinsic permutation. When the optical field is monochromatic at a frequency of ω, the frequencies ωm , ωn , and ωp in (5.96) can each only take the value of either ω or ω. Therefore, the only possible frequency combinations for ωq to have a positive frequency are ωq ¼ 3ω, with σ ð3Þ e σ ð3Þ xxxx ð3ωÞ ¼ e xxxx ð3ω ¼ ω þ ω þ ωÞ for third-harmonic generation, and ωq ¼ ω, with σ ð3Þ σ ð3Þ σ ð3Þ e σ ð3Þ xxxx ðωÞ ¼ e xxxx ðω ¼ ω þ ω  ωÞ ¼ e xxxx ðω ¼ ω  ω þ ωÞ ¼ e xxxx ðω ¼ ω þ ω þ ωÞ for optical Kerr effect or optical saturation. By contrast, if the optical field contains two different frequencies at ω1 and ω2 , there are many possible frequencies that can result from four-wave mixing through the third-order optical nonlinearity, including ω1 , ω2 , ð2ω1  ω2 Þ, ð2ω2  ω1 Þ, and other high harmonics. Many more frequencies generated through four-wave mixing are possible when the optical field consists of three distinct frequencies at ω1 , ω2 , and ω3 . In the following, we use (5.96) to calculate the nonlinear conductivities that are responsible for third-harmonic generation, optical Kerr effect, and four-wave mixing.

5.6.1

Third-Harmonic Generation Consider a monochromatic optical field at a frequency of ω. The third-order nonlinear surface conductivity can generate a third-harmonic nonlinear surface current that has a temporal variation of expði3ωtÞ. This nonlinear surface current can be found using (5.34) to be 3 e J ð3Þ σ ð3Þ xxxx ð3ωÞEx ðωÞ; x ð3ωÞ ¼ e

ð5:101Þ

where e σ ð3Þ σ ð3Þ σ ð3Þ σ ð3Þ xxxx ð3ωÞ ¼ e xxxx ð3ω ¼ ω þ ω þ ωÞ. The real part e xxxx ð3ωÞ ¼ e xxxx ð3ω ¼ ω þ ω þ ωÞ of the third-harmonic surface conductivity is found by requiring ωm ¼ ωn ¼ ωp ¼ ω in (5.96) to obtain [11] 0

0

e σ ð3Þ xxxx ð3ωÞ ¼ 

e4 v2F 192ℏ3 ω4

 45F

0

    3ℏω ℏω  64FðℏωÞ þ 17F : ð5:102Þ 2 2

5.6 Third-Order Optical Nonlinearity

(a)

(b)

20 (3)

)

xxxx

(3 )

0

(3) xxxx

10

(3 )

−0.2

0

(3)

xxxx

4

(

μ 10 20 e3 meV A m 2

169

−10

−0.4 (3) xxxx

−20

(3)

(3 )

xxxx

1

0

2

3

1.99

(3 )

2

μ

2.01

μ (d)

(c)

μ

μ

ð3Þ

Figure 5.3 (a) Nonlinear surface optical conductivity e σ xxxx ð3ω ¼ ω þ ω þ ωÞ as a function of

frequency in the limit that jμj ≫ kB T. 00The gray region is enlarged and shown in (b). The real 0 part e σ ð3Þ σ ð3Þ xxxx ð3ωÞ and imaginary part e xxxx ð3ωÞ are shown in solid and dashed curves, respectively. The curves are calculated by taking the temperature to be T ¼ 1 K for jμj ≈ 103 kB T. The interband transitions responsible for the resonances in (a) and (b) denoted by ①, ②, and ③ are shown in (c) and (d) marked by the corresponding numbers. The solid and dashed lines in (c) and (d) represent the eigenstates of graphene and the virtual states, respectively.

00

The real part e σ ð3Þ σ ð3Þ xxxx ð3ωÞ, given in (5.102), and the imaginary part e xxxx ð3ωÞ, obtained 0

σ ð3Þ from e σ ð3Þ xxxx ð3ωÞ using (3.79), of the nonlinear surface conductivity e xxxx ð3ωÞ are both 0

4 plotted in Figures 5.3(a) and (b). The value of e σ ð3Þ xxxx jμj is plotted against ℏω=jμj so that the figure is independent of jμj. Unlike the linear conductivity shown in Figure 3.3 where

divergence only occurs at ℏω ¼ 2jμj, divergence seen in Figure 5.3(a) for e σ ð3Þ xxxx ð3ωÞ

170

Nonlinear Optical Properties

appears at 3ℏω ¼ 2jμj, marked by ①, at 2ℏω ¼ 2jμj, marked by ②, and at ℏω ¼ 2jμj, marked by ③, due to the first, second, and third terms in the brackets of (5.102), respectively. These resonances correspond to the interband transitions shown in Figures 5.3(c) and (d) marked by the corresponding numbers in Figures 5.3(a) and (b). As can be seen, the process of converting the frequency from ω to 3ω is accomplished by converting three photons of energy ℏω to a photon of energy 3ℏω. The process of promoting electrons to states of higher energies can be expressed as a series of multiplication of ðΔE1  ℏωm Þ1 ðΔE2  ℏωm  ℏωn Þ1 ðΔE3  ℏωm  ℏωn  ℏωp Þ1 in (5.91), where ΔEi ¼ ℏωki0 ki is the energy difference between two energy levels jki0 〉 and jki 〉, and ℏðωm þ . . . Þ is the photon energy involved in the process. Because the resonances for interband transitions on the Dirac cone can only happen when ΔE ¼ 2jμj, it is only possible for one of the three terms in the multiplication to satisfy one of the resonance conditions while the other two terms in the multiplication are nonresonant and represent transitions through virtual states. A virtual state is not an eigenstate of graphene but is a state of the system consisting of graphene and the radiation field of one or more photons. These virtual states are represented by dashed horizontal lines in Figures 5.3(c) and (d) in contrast to the solid horizontal lines that represent real eigenstates of graphene. A resonant transition results in the divergence of the nonlinear surface conductivity at the resonance frequency, whereas nonresonant transitions through virtual states do not cause divergence in the nonlinear surface conductivity. The reason for the divergence of the nonlinear surface conductivity at a resonance frequency is that the phase relaxation rate γk0 k is ignored in the calculation of e σ ð3Þ xxxx ð3ωÞ. However, the resonance peaks seen in Figure 5.3 are not infinitely sharp because the 00

σ ð3Þ curves plotted in Figure 5.3 for e σ ð3Þ xxxx ð3ωÞ and e xxxx ð3ωÞ are calculated by taking a nonzero temperature of T ¼ 1 K, which results in finite widths for the resonance peaks due to temperature-dependent broadening of the Fermi distribution. This effect is evident for the resonance at ℏω ¼ 2jμj seen in Figure 5.3(b). Temperature-dependent spectral broadening of the linear surface conductivity is discussed in Section 3.5 and demonstrated in Figure 3.4. The spectral linewidths of nonlinear surface conductivities broaden with increasing temperature as well. Clearly, both zero relaxation rate and divergence of the nonlinear surface conductivity at resonance are unphysical. As discussed above, the phase relaxation rate can be restored by properly replacing ωk0 k with ωk0 k  iγk0 k in the general expression 0

for e σ ð3Þ xxxx ðωq ¼ ωm þ ωn þ ωp Þ given in (5.91), which changes the expressions for e σ ð3Þ xxxx ð3ω ¼ ω þ ω þ ωÞ. Restoration of the phase relaxation rate removes the divergence at a resonance frequency, but it leads to a broadening of the spectral linewidth of e σ ð3Þ xxxx ð3ωÞ at the resonance frequency. Spectral broadening caused by finite relaxation time, thus a nonzero relaxation rate, for the linear surface conductivity is discussed in Section 3.5 and demonstrated in Figure 3.4. Nonlinear surface conductivity also has similar spectral broadening caused by a nonzero relaxation rate.

5.6 Third-Order Optical Nonlinearity

5.6.2

171

Parametric Frequency Conversion For a radiation field that has two frequency components at ω1 and ω2 , many frequency combinations are possible for frequency mixing through the third-order optical nonlinearity to result in many different frequencies: ω1 , ω2 , ð2ω1  ω2 Þ, ð2ω2  ω1 Þ, ð2ω1 þ ω2 Þ, ðω1 þ 2ω2 Þ, 3ω1 , and 3ω2 . Here we only consider the parametric frequency of 2ω1  ω2 as an example. The possible combinations of ðωm ; ωn ; ωp Þ that satisfy the requirement that ωq ¼ ωm þ ωn þ ωp ¼ 2ω1  ω2 are ðωm ; ωn ; ωp Þ ¼ ðω1 ; ω1 ; ω2 Þ, ðω1 ; ω2 ; ω1 Þ, ð3Þ and ðω2 ; ω1 ; ω1 Þ. Therefore, the nonlinear current e J x ð2ω1  ω2 Þ that has a temporal variation of exp½ið2ω1  ω2 Þt in (5.91) is found using (5.34) to be ð3Þ 2  e σ ð3Þ J x ð2ω1  ω2 Þ ¼ 3e xxxx ð2ω1  ω2 ÞEx ðω1 ÞEx ðω2 Þ;

ð5:103Þ

where the factor 3 accounts for the threefold intrinsic permutation symmetry for 0 e σ ð3Þ σ ð3Þ xxxx ð2ω1  ω2 Þ according to (5.98). The real part e xxxx ð2ω1  ω2 Þ of the nonlinear surface conductivity e σ ð3Þ xxxx ð2ω1  ω2 Þ for the parametric frequency of 2ω1  ω2 , assuming 2ω1 > ω2 , is found by requiring ωq ¼ ωm þ ωn þ ωp ¼ 2ω1  ω2 in (5.96) as 0

0

0

ð2ω1  ω2 Þ ¼ e σ ð3Þ σ ð3Þ σ ð3Þ e σ ð3Þ xxxx ðω1 ; ω1 ; ω2 Þ ¼ e xxxx ðω1 ; ω2 ; ω1 Þ ¼ e xxxx ðω2 ; ω1 ; ω1 Þ xxxx 0

¼

e4 v2F 48ℏ ω41 ω22 ðω1  ω2 Þ2 ð2ω1  ω2 Þ    ℏω1  ℏω2  16Fðℏω1 Þω31 ðω1  ω2 Þ2 þ 4F ðω1  ω2 Þ5 2    ℏω1 3  2 2F ω1 2ω1  10ω1 ω2 þ 5ω22 2     2ℏω1  ℏω2 F ð2ω1  ω2 Þ3 2ω21  ω1 ω2 þ 2ω22 2    ℏω2 3  2 F ω2 8ω1  7ω1 ω2 þ 2ω22 : 2 3

ð5:104Þ

g

The real part e σ ð3Þ xxxx ð2ω1  ω2 Þ, given in (5.104), and the imaginary part 0

00

e σ ð3Þ σ ð3Þ xxxx ð2ω1  ω2 Þ, obtained from e xxxx ð2ω1  ω2 Þ using (3.79), of the nonlinear surface 0

conductivity e σ ð3Þ xxxx ð2ω1  ω2 Þ are both plotted in Figure 5.4(a) as a function of ω1 , assuming ℏω2 ¼ jμj=2 in the limit that jμj ≫ kB T. Four resonances are observed, which are given by the first four terms in the curly brackets in (5.104). These resonances are located at 2ℏω1 ¼ 2jμj, ℏω1  ℏω2 ¼ 2jμj, ℏω1 ¼ 2jμj, and 2ℏω1  ℏω2 ¼ 2jμj, marked by ①, ②, ③, and ④, respectively, in Figure 5.4. Examples of the responsible interband transitions are showed in Figures 5.4(b) and (e), marked by the corresponding numbers. The radiation at ωq ¼ 2ω1  ω2 can be generated through the third-order

Nonlinear Optical Properties

μ 10 20 e3 meV A m 2

(b)

15

)

(a)

10

(3) xxxx

(2

1

2

)

5 (c)

2

0

(3) xxxx

−15 0

(2

1 1

(c)

1

1

μ

2

2

1

)

2

3 (e)

(d)

0.5

(3)

)

xxxx

(2

1

2

) (3)

0.3

xxxx

(2

1

2

)

2

μ

2

0.1 1

1

(3)

4

(

μ 10 20 e3 meV A m 2

μ

−5

xxxx

(3)

4

(

(d)

−10

xxxx

172

−0.1

(3) xxxx

(2

1

2

)

(3) xxxx

(2

1

2

)

−0.3 1.99

2 1

2.01

μ

2.4

2.5 1

2.6

μ

Figure 5.4 (a) Nonlinear optical conductivity e σ ð3Þ xxxx ð2ω1  ω0 2 Þ as a function of frequency assuming ℏω200 ¼ jμj=2 in the limit that jμj ≫ kB T. The real part e σ ð3Þ xxxx ð2ω1  ω2 Þ and imaginary part ð3Þ e σ xxxx ð2ω1  ω2 Þ are shown in solid and dashed curves, respectively. The curves are calculated by taking the temperature to be T ¼ 1 K for jμj ≈ 103 kB T. The interband transitions responsible for the resonances in (a) denoted by ① and ④ are shown in (b) marked by the corresponding numbers. The interband transitions in the shaded regions are enlarged in (c) and (d). They correspond to the resonances denoted by ③ and ②, respectively; the corresponding interband transitions are shown in (e) marked by the corresponding numbers. The solid and dashed horizontal lines in (b) and (e) represent the eigenstates of graphene and the virtual states, respectively.

optical nonlinearity by converting two photons of energy ℏω1 to a photon of energy ℏωq ¼ 2ℏω1  ℏω2 together with a photon of energy ℏω2. Divergence at each resonance frequency is seen in the spectrum of e σ ð3Þ xxxx ð2ω1  ω2 Þ shown in Figure 5.4, as is seen in the spectrum of e σ ð3Þ xxxx ð3ωÞ shown in Figure 5.3.

5.6 Third-Order Optical Nonlinearity

173

00

Because the curves plotted in Figure 5.4 for e σ ð3Þ σ ð3Þ xxxx ð2ω1  ω2 Þ and e xxxx ð2ω1  ω2 Þ are calculated by taking a nonzero temperature of T ¼ 1 K, the resonance peaks seen are not infinitely sharp due to temperature-dependent broadening of the Fermi distribution. As discussed above, spectral divergence at resonance is caused by the unphysical 0

neglect of the phase relaxation rate in the calculation of e σ ð3Þ xxxx ð2ω1  ω2 Þ that is plotted in Figure 5.4. Restoration of the phase relaxation rate leads to a broadening of the spectral linewidth of e σ ð3Þ xxxx ð2ω1  ω2 Þ at each resonance frequency while removing the divergence at each resonance. Increasing temperature also broadens the spectral linewidths.

5.6.3

Optical Kerr Effect The optical Kerr effect is another important phenomenon of the third-order optical nonlinearity. Consider a monochromatic wave of a frequency ω traveling through a graphene sheet. The optical Kerr effect causes the phase difference between the incident light and the transmitted light to depend on the light intensity. The nonlinear process is optical-field-induced birefringence, which makes the refractive index of graphene a function of light intensity. Optical-field-induced-birefringence is a third-order parametric process similar to the one discussed in the preceding subsection but with the emitted light generated through the parametric process having the same frequency as the incident light. In the fully degenerate case discussed here, only one frequency at ω is involved in the process, such that ωq ¼ ω þ ω  ω ¼ ω  ω þ ω ¼ ω þ ω þ ω ¼ ω. Therefore, the nonlinear ð3Þ

current e J x ðωÞ that has a temporal variation of expðiωtÞ in (5.91) is found using (5.34) to be ð3Þ 2  e σ ð3Þ J x ðωÞ ¼ 3e xxxx ðωÞEx ðω1 ÞEx ðω1 Þ;

ð5:105Þ

where the factor 3 accounts for the threefold intrinsic permutation symmetry for e σ ð3Þ xxxx ðωÞ according to (5.98). This nonlinear process responsible for optical-field-induced birefringence is a degenerate frequency-mixing process that is similar to the case of parametric frequency conversion discussed above for ωq ¼ ω1 þ ω1  ω2 ¼ ω1  ω2 þ ω1 ¼ ω2 þ ω1 þ ω1 ¼ 2ω1  ω2 , but with ω1 ¼ ω2 ¼ ω. Therefore, the optical nonlinear conductivity e σ ð3Þ xxxx ðωÞ for the optical Kerr effect can be found by setting ω2 → ω1 ¼ ω in (5.104). However, directly replacing ω2 with ω1 in (5.104) is problematic because both the denominator and the numerator are zero if ω2 ¼ ω1 . This issue is resolved by setting ω1 ¼ ω and ω2 ¼ ω þ δ in (5.104) and expanding the equation as a Taylor series in terms of δ. In the limit that δ → 0, only the zeroth-order term for δ0 needs to be considered; thus, we find that

174

Nonlinear Optical Properties

e σ ð3Þ σ ð3Þ σ ð3Þ σ ð3Þ xxxx ðωÞ ¼ e xxxx ðω; ω; ωÞ ¼ e xxxx ðω; ω; ωÞ ¼ e xxxx ðω; ω; ωÞ        e4 v 2F 7 ℏω 3 3 2 2 00 ℏω 0 ℏω ¼ 3 FðℏωÞ  F  ℏωF  ℏ ω F ; 8 2 8 2 64 2 3ℏ ω4 0

0

0

0

ð5:106Þ where F 0 and F 00 are the first and second derivatives of the function F given in 0 00 σ ð3Þ (5.95). The real part e σ ð3Þ xxxx ðωÞ, given in (5.106), and the imaginary part e xxxx ðωÞ, 0 obtained from e σ ð3Þ σ ð3Þ xxxx ðωÞ using (3.79), of the nonlinear surface conductivity e xxxx ðωÞ are both plotted in Figures 5.5(a) and (b), over two different frequency ranges, as a function of ℏω=jμj. Two resonances located at ℏω ¼ jμj and ℏω ¼ 2jμj are observed in Figures 5.5(a) and (b), respectively; their corresponding interband transitions are shown in Figure 5.5(c). It can be seen from Figure 5.5(b) that the resonance of e σ ð3Þ xxxx ðωÞ at ℏω ¼ 2jμj is two to three orders of magnitude larger than the resonances of e σ ð3Þ xxxx ð3ωÞ ð3Þ and e σ xxxx ð2ω2  ω1 Þ shown in Figures 5.3 and 5.4, respectively. A large imaginary part of e σ ð3Þ χ ð3Þ xxxx ðωÞ leads to a large real part of e xxxx ðωÞ, which in turn gives a large nonlinear refractive index of graphene and a strong optical Kerr effect, as we shall see in the following section. Again, divergence caused by the unphysical neglect of the phase relaxation rate is seen at each of the two resonance frequencies in the spectrum of e σ ð3Þ xxxx ðωÞ shown in Figure 5.5. Nonetheless the resonance peaks seen in the figure are not infinitely sharp due to temperature-dependent broadening of the Fermi distribution at the σ ð3Þ nonzero temperature of T ¼ 1 K taken to calculate e σ ð3Þ xxxx ðωÞ and e xxxx ðωÞ. Restoration of the phase relaxation rate leads to a broadening of the spectral linewidth 0

00

of e σ ð3Þ xxxx ðωÞ at each resonance frequency while removing the divergence at each resonance. Increasing temperature causes further broadening of the spectral linewidths.

5.7

Experiments on Nonlinear Optical Properties From (5.39), the effective third-order nonlinear surface susceptibility of graphene is given by e χ ð3Þ xxxx ðωÞ ¼ i

e σ ð3Þ xxxx ðωÞ ; ϵ0 ω

ð5:107Þ

which can be regarded as a 2D nonlinear surface susceptibility of graphene contributed by free electrons on the Dirac cone. To compare with the nonlinear susceptibility of a 3D material, the 3D effective third-order nonlinear susceptibility of graphene is defined by (5.35): χð3Þ xxxx ðωÞ ¼

e χ ð3Þ xxxx ðωÞ ; d

ð5:108Þ

5.7 Experiments on Nonlinear Optical Properties

(10

0

( )

(3)

4

xxxx

( )

2 0

(3)

−2 (3)

−4 −6 0.96

xxxx

( )

xxxx

4

(

2

4

μ xxxx

(3)

6

)

(3) xxxx

μ 10 17 e3 meV A m 2

e meV A m 2

4

20 3

(b)

6

)

(a)

175

−2 (3)

−4

xxxx

−6 1

1.97

1.04

μ

2

( ) 2.03

μ

(c)

μ

ð3Þ

Figure 5.5 (a) Nonlinear optical conductivity e σ xxxx ðωÞ as a function of frequency in the limit that 0 e σ ð3Þ xxxx ðωÞ

jμj ≫ kB T. The real part and imaginary part e σ ð3Þ xxxx ðωÞ are shown in solid and dashed curves, respectively. The curves are calculated by taking the temperature to be T ¼ 1 K for jμj ≈ 103 kB T. The interband transitions responsible for the resonances in (a) and (b) marked by ① and ②, respectively, are shown in (c) with the corresponding numbers. The solid and dashed horizontal lines in (c) represent the eigenstates of graphene and the virtual states, respectively. 00

where d ¼ 0:335 nm is the thickness of a monolayer graphene sheet. Then, the effective third-order nonlinear susceptibility of graphene is of the order of χð3Þ xxxx ðωÞ ≈

e4 v2F ϵ 0 ℏ3 ω5 d

;

which is obtained by using (5.96), (5.107), and (5.108).

ð5:109Þ

176

Nonlinear Optical Properties

In an experiment on the optical Kerr effect, the intensity of the optical beam is usually measured to obtain χð3Þ xxxx ðωÞ. The relation between the intensity I of the optical beam and the refractive index n of graphene is given by the relation [1] n ¼ n0 þ n2 I;

ð5:110Þ

where n0 is the intensity-independent linear refractive index and n2 is the nonlinear refractive index given by [1,13], ð3Þ0

n2 ¼

3χxxxx ðωÞ : 4cϵ 0 n20

ð5:111Þ

17 2 2 Taking λ ¼ 1 μm and n0 to be of the order of unity, we obtain χð3Þ m V xxxx ðωÞ ≈ 10 15 1 2 from (5.109) and n2 ≈ 10 m W from (5.111). Hendry et al. [14] and Wu et al. [15] respectively measured the effective nonlinear susceptibility of graphene in four-wave mixing (ω ¼ ω1 þ ω1  ω2 ) and self-phase modulation (optical Kerr effect with 7 ω ¼ ω þ ω  ω) experiments; they found a value of χð3Þ e:s:u:, or xxxx ðωÞ ≈ 10 15 ð3Þ 2 2 χxxxx ðωÞ ≈ 10 m V , which is two orders of magnitude larger than the theoretical value. Gu et al. [16] observed the optical Kerr effect by using a graphene–silicon hybrid optoelectronic device consisting of graphene and a 2D photonic crystal; they obtained a value of n2 ≈ 4:8  1017 m2 W1 in the experiment. However, a very large value of n2 ≈ 1011 m2 W1 was obtained by Zhang et al. using the Z-scan technique [17]. The discrepancy might be caused by the large variance in the value of the nonlinear susceptibility around the resonance frequency at ℏω ¼ 2jμj, as shown in Figure 5.5; depending on the energy difference between the photon energy ℏω and 2jμj, the measured n2 can be very large around the resonance point at 2jμj, or very small away from resonance. More experimental data are required to quantify the nonlinear susceptibility and the nonlinear refractive index of graphene near and away from the resonance points that are seen in Figures 5.3, 5.4, and 5.5. The dependence of the nonlinear optical properties of graphene on temperature and chemical potential also has to be experimentally established and compared with the theoretical prediction.

References 1. J. M. Liu, Photonic Devices (Cambridge University Press, 2005). 2. L. M. Malard, M. H. D. Guimarães, D. L. Mafra, M. S. C. Mazzoni, and A. Jorio, “Grouptheory analysis of electrons and phonons in N-layer graphene systems,” Physical Review B, Vol. 79, 125426 (2009). 3. J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Second order optical nonlinearity of graphene due to electric quadrupole and magnetic dipole effects,” Scientific Reports, Vol. 7, 43843 (2017). 4. Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, 1984). 5. R. W. Boyd, Nonlinear Optics (Elsevier Science, 2013).

References

177

6. S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhysics Letters, Vol. 79, 27002 (2007). 7. N. M. R. Peres, Y. V. Bludov, J. E. Santos, A. P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Physical Review B, Vol. 90, 125425 (2014). 8. R. Li, X. Lin, and H. Chen, “Graphene induced bifurcation of energy levels at low input power,” arXiv:1505.02957 (2015). 9. A. R. Wright, X. G. Xu, J. C. Cao, and C. Zhang, “Strong nonlinear optical response of graphene in the terahertz regime” Applied Physics Letters, Vol. 95, 072101 (2009). 10. X. G. Xu, S. Sultan, C. Zhang, and J. C. Cao, “Nonlinear optical conductance in a graphene pn junction in the terahertz regime,” Applied Physics Letters, Vol. 97, 011907 (2010). 11. J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New Journal of Physics, Vol. 16, 053014 (2014). 12. Y. Wang, M. Tokman, and A. Belyanin, “Second-order nonlinear optical response of graphene,” Physical Review B, Vol. 94, 195442 (2016). 13. R. del Coso and J. Solis, “Relation between nonlinear refractive index and third-order susceptibility in absorbing media,” Journal of the Optical Society of America B, Vol. 21, pp. 640–644 (2004). 14. E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Physical Review Letters, Vol. 105, 097401 (2010). 15. R. Wu, Y. Zhang, S. Yan, et al, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Letters, Vol. 11, pp. 5159–5164 (2011). 16. T. Gu, N. Petrone, J. F. McMillan, et al., “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photonics, Vol. 6, pp. 554–559 (2012). 17. H. Zhang, S. Virally, Q. Bao, et al., “Z-scan measurement of the nonlinear refractive index of graphene,” Optics Letters, Vol. 37, pp. 1856–1858 (2012).

6

Plasmonics

6.1

Plasmons, Surface Plasmons, and Surface Plasmon Polaritons Graphene has unique plasmonic properties because of its unusual electromagnetic properties and its 2D geometric structure. Graphene supports plasma oscillations when it has a nonzero chemical potential. Different from a metal, which has a fixed carrier density, thus a fixed plasma frequency, graphene is similar to a semiconductor in that its carrier density, thus its plasma frequency, can be varied by varying its chemical potential through, for example, impurity doping, electrical modulation, or optical illumination. Nevertheless, graphene is very different from an ordinary semiconductor because of its unique band structure and its 2D geometry. As a 2D material, graphene naturally supports true surface plasmons and, in a certain spectral region depending on its chemical potential, surface plasmon polaritons. As discussed in preceding chapters, the electromagnetic properties of graphene approach those of a conductor at low frequencies and those of a dielectric at optical frequencies. Between the two limits, particularly in the terahertz frequency region, graphene has sophisticated electromagnetic properties that can be tuned through varying its chemical potential. For this reason, graphene has very interesting plasmonic properties that lead to many useful applications in the terahertz region.

6.1.1

Plasmons Before discussing graphene plasmons, we first review the plasmonic properties of threedimensional materials. Plasmons are the quantized collective oscillations of free charge carriers that exist on the surface or in the bulk of a three-dimensional conducting material. These free carriers are electrons in a metal, or electrons and holes in a semiconductor. By using the Drude model for the equation of motion of a free charge carrier in an electric field, it can be shown that for a conducting material, the permittivity is given by [1,2] " # ω2p τ2 ϵðωÞ ¼ ϵ ∞ 1  ; ð6:1Þ ωτðωτ þ iÞ where ϵ ∞ is the permittivity of the material in the limit that ω → ∞, which is contributed by bound charges; τ is the average momentum relaxation time of the free charge carriers;

6.1 Plasmons, Surface Plasmons, Polaritons

179

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and ωp ¼ ne2 =ϵ ∞ m is the plasma frequency, where n is the electron density and m is the effective mass of the free charge carriers. In the high-frequency limit that ωτ ≫ 1, the permittivity given in (6.1) can be approximated as   ωp 2 ϵðωÞ ¼ ϵ ∞ 1  2 ; ω

ð6:2Þ

which is generally valid for the discussions in this chapter. For a perfect conductor, ϵ ∞ ¼ ϵ 0 , which is a good approximation for highly conducting metals such as Ag, Au, and Cu. Therefore, we set ϵ ∞ ¼ ϵ 0 for (6.1) and (6.2) in the following. The characteristic dispersion of a monochromatic propagating mode at a frequency of ω in a medium can be found from Maxwell’s equations,  E ¼ μ0

∂H ∂t

ð6:3Þ

 H ¼ ϵðωÞ

∂E : ∂t

ð6:4Þ

Δ and

Δ E¼

ΔΔ

Δ

Δ

Using the identity  from (6.3) and (6.4) as

ð  EÞ  ∇2 E; the general wave equation is found

ΔΔ

∇2 E  ð  EÞ  μ0 ϵðωÞ

∂2 E ¼ 0: ∂t2

ð6:5Þ

There are two types of modes that satisfy this wave equation: a longitudinal mode that is polarized in the direction of wave propagation and a transverse mode that is polarized in a direction perpendicular to the direction of wave propagation. In the following, we take the direction of wave propagation to be the x direction and the propagation constant of a mode to be β, as shown in Figure 6.1. A longitudinal mode polarized in the x direction, E ¼ ^x Ex ¼ ^x E x expðiβx  iωtÞ, is shown in Figure 6.1(a). For this mode, the first and second terms in (6.5) cancel so that (6.5) can be satisfied only when ϵðωÞ ¼ 0 or ω ¼ ωp :

ð6:6Þ

This longitudinal mode is known as a volume plasmon mode, or simply a volume plasmon, which is characterized by collective longitudinal oscillation of free electrons in the direction of the wave propagation; that is, the longitudinal displacement of the electron gas. Note that (6.6) indicates that the frequency ω of a volume plasmon mode does not depend on its propagation constant β but has a constant value of the plasma frequency ωp. The displaced electron gas against the positive ionic background creates a polarization density P and a restoring force from the electric field E ¼ P=ϵ 0 ; thus, D ¼ ϵðωÞE ¼ ϵ 0 E þ P ¼ 0 or ϵðωÞ ¼ 0. The self-sustaining longitudinal oscillation frequency ωp in (6.6) can be found by using the classical equation of motion, which

180

Plasmonics

(a) Volume plasmon −1

−0.5

Re Ex 0

0.5

1

z x

( )

Ex

(b) Volume plasmon polariton −1

−0.5

Re Ez 0

0.5

1

z x

( )

Ez

(c) Surface plasmon polariton −1

−0.5

Re Ex 0

0.5

1

z x

Ez

E Ex

d

( )

Figure 6.1 Various plasmonic oscillations and their characteristic dispersion curves. (a) Volume plasmon. (b) Volume plasmon polariton. (c) Surface plasmon polariton. The Re Ex field is also plotted for each excitation. The absolute values of Re Ez and Re Ex are normalized to have a unit peak value of 1.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gives ωp ¼ ne2 =ϵ 0 m [1]. As discussed in Chapter 4, the response of a material can be spatially nonlocal. If spatial nonlocality is considered, (6.1) is modified and (6.6) is no qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi longer dispersionless but depends on the wave number β as ω ¼ ωp 2 þ 0:6ðvF βÞ2 [3],

6.1 Plasmons, Surface Plasmons, Polaritons

1

p

Volume plasmon

sp

mon polariton ce plas a f r Su

0.5 0

0.5

n

Ph oto n

pol ari to

mon polariton

n oto Ph

e p la s

1

1.5

r fa c

Vo

e l um

ton

p

Su

1.5

n mo s a pl

i lar po

c ,ck

2 p

sp

2

Volu me pla sm on

(b) 2.5

Volume plasmon

(a)

181

0 0

0.5

1 c ,ck

1.5

2

0

0.5

1

1.5

2

2.5

p

Figure 6.2 Characteristic dispersion curves as a function of (a) wave number, and (b) frequency.

where vF is the Fermi velocity of the electrons [1]. Nevertheless, in the limit that β → 0, we obtain ω ¼ ωp as given by (6.6). A transverse mode can also be found from (6.5). A transverse mode is polarized in the z direction, E ¼ ^z Ez ¼ ^z E z expðiβx  iωtÞ, as shown in Figure 6.1(b). For this mode, the second term of (6.5) vanishes so that (6.5) gives the relation β2 ¼ ω2 μ0 ϵðωÞ, which yields the dispersion relation: ω¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωp 2 þ c2 β2 :

ð6:7Þ

This transverse mode is called a volume plasmon polariton mode, or simply a volume plasmon polariton. A plasmon polariton is a coupled state of a photon and a plasmon. The coupling is strongest near the intersection of the dispersion curve of the photon, given by ω ¼ ck, and that of the volume plasmon, given by (6.6). Away from the intersection, the volume plasmon polariton behaves more photon-like for a large β and more plasmon-like for a small β, as can be seen by respectively taking the limits cβ ≫ ωp and cβ ≪ ωp for (6.7). In the limit that β → 0, the field is no longer propagating and there is no variation of the electric field in the propagation direction x; then, the definitions of longitudinal and transverse modes are no longer applicable as the wave is not propagating. Consequently, the volume plasmon polariton mode is degenerate with the volume plasmon mode in the limit that β → 0. The dispersion curves for photons, volume plasmons, and volume plasmon polaritons are drawn in Figure 6.2.

6.1.2

Surface Plasmon Polaritons We now consider the situation in which half of the space (z < 0) is occupied by a semiinfinite conducting medium that has a frequency-dependent permittivity ϵðωÞ characterized

182

Plasmonics

by (6.2) with the rest of the space (z ≥ 0) filled with a dielectric of a frequencyindependent permittivity ϵ d, as shown in Figure 6.1(c). A surface mode that is confined at the interface can be found. This mode is supported by the coupling between the surface plasmon of the conducting medium and a transverse magnetic (TM) optical field; it is known as a surface plasmon polariton (SPP) mode, or simply a surface plasmon polariton. The electric field of this TM surface mode has the form ( E2 ¼ ^x E2x þ ^z E2z ¼ ð^x E 2x þ ^z E 2z Þeiβxγ2 ziωt ; z ≥ 0; E¼ ð6:8Þ E1 ¼ ^x E1x þ ^z E1z ¼ ð^x E 1x þ ^z E 1z Þeiβxþγ1 ziωt ; z < 0:

Δ

By using (6.3), (6.4), and  D ¼ 0 because there is no external charge, we obtain E 1z ¼ iβE 1x =γ1 , E 2z ¼ iβE 2x =γ2 , β2  γ1 2 ¼ ω2 μ0 ϵðωÞ;

β2  γ2 2 ¼ ω2 μ0 ϵ d ;

ð6:9Þ

and

H¼

8 ωϵ d iβxγ2 ziωt > ^ ; >

ωϵðωÞ > :H1 ¼ ^y H1y ¼ ^y i E 1x eiβxþγ1 ziωt ; γ1

z < 0:

ð6:10Þ

Unlike a volume plasmon mode or a volume plasmon polariton mode, the SPP mode is confined at the interface between the conducting material and the dielectric, with its fields decaying away from the interface boundary located at z ¼ 0, as shown in Figure 6.1(c). By applying the continuity boundary conditions at the interface to the tangential components of the electric and magnetic fields, such that E 2x ¼ E 1x and H2y ¼ H1y , we obtain ϵðωÞ ϵ d þ ¼ 0; γ1 γ2

ð6:11Þ

which yields the relation for the dispersion characteristics of the SPP mode: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ d þ ϵðωÞ ; ω ¼ βc ϵ 0 ϵ d ϵðωÞ

or

ω β¼ c

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ d ϵðωÞ : ϵ 0 ϵ d þ ϵ 0 ϵðωÞ

ð6:12Þ

This characteristic dispersion curve of the SPP mode is plotted in Figure 6.2. For a small value of β, the dispersion curve of SPP approaches that of photons and the field is weakly confined at the interface. For a large value of β, the field is highly confined at the interface.

6.1.3

Surface Plasmons In the limit that β → ∞, the SPP mode is no longer coupled to the free electromagnetic field while the oscillation frequency of the SPP mode asymptotically approaches the value of

6.1 Plasmons, Surface Plasmons, Polaritons

ωsp ¼ ωp

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ0 ; ϵd þ ϵ0

183

ð6:13Þ

which is known as the surface plasma frequency. This characteristic is seen in Figure 6.2. This region, where the SPP frequency ω ≈ ωsp in the limit that ω=β ≪ c, is usually referred to as the nonretarded region in contrast to the retarded region where ω=β is comparable to or larger than c. Because the SPP mode in the nonretarded region is essentially decoupled from the photon, this SPP mode is sometimes referred to as a surface plasmon (SP) mode, or simply a surface plasmon, by dropping the term “polariton” from SPP. For an SP mode, the propagation of its field is instantaneous in the physical region under consideration because the speed c of light is much larger than the phase velocity ω=β of the field so that the effect of time retardation can be ignored [4]. Therefore, the SP mode is a quasistatic approximation of the SPP mode in the nonretarded region where ω=β ≪ c so that β ≫ ω=c.

6.1.4

Surface Current Model The semi-infinite conducting medium shown in Figure 6.1(c) has been assumed to be a homogeneous material with a frequency-dependent permittivity ϵðωÞ given by (6.2). As shown in (3.37), we can express the permittivity of an isotropic conductor in terms of its conductivity as ϵðωÞ ¼ ϵ ∞ þ i

σðωÞ ω

ð6:14Þ

to distinguish the contribution iσðωÞ=ω of free electrons to the permittivity from the contribution ϵ ∞ of bound electrons. Therefore, we can express the current density in the x direction as J ¼ σðωÞE1  ^x , where σðωÞ is the conductivity of the conductor and E1 is given by (6.8). The magnitude of the current density exponentially diminishes away from the interface into the conductor, following the trend of the electric field as shown in Figure 6.3(a). The total surface current density in the x direction is obtained by integrating the volume current density from z ¼ ∞ to z ¼ 0: e J ðxÞ ¼

ð0

ð0 Jðx; zÞdz ¼

∞

σðωÞE 1x eiβxþγ1 ziωt dz ¼ ∞

σðωÞ E 1x eiβxiωt : γ1

ð6:15Þ

For a good conductor such that σðωÞ ≫ ωϵ ∞ , this current can be considered as a surface current because it is confined within a very thin layer at the surface of the conductor. Therefore, we can model the conductor as a combination of a dielectric that has a permittivity of ϵ ∞ and a surface conductivity of e σ ðωÞ ¼ σðωÞ=γ1 so that the surface current density is given by e J ¼e σ ðωÞE1  ^x at z ¼ 0, which is consistent with (6.15). This is shown in Figure 6.3(b). Because the surface current can only flow on the plane of the interface, it only has an x component in this situation; therefore, e J¼e J ^x ¼ e σ ðωÞE1x^x

184

Plasmonics

(a) z x

d

( )

(b) z x

d

Figure 6.3 Physical configurations for (a) the metal SPP mode, (b) the metal SPP mode in an

equivalent model. In (b), the metal is modeled as a dielectric of a permittivity ϵ ∞ with a conducting surface layer of a surface conductivity e σ.

with E1x ¼ E 1x eiβxiωt for z ¼ 0. Because of the presence of the surface current, the continuity of the magnetic field at the interface z ¼ 0 as used above in the derivation of (6.11) is no longer valid, though the continuity of the electric field that is parallel to the interface is still valid. In this case, the boundary conditions at the interface are given by ^z  ðH2  H1 Þ ¼ e J

ð6:16Þ

for the magnetic field and the continuity of the parallel electric field components. Using these boundary conditions and (6.10), together with ϵðωÞ ¼ ϵ ∞ in the bulk, we obtain ϵ∞ ϵd e σ ðωÞ : þ ¼ i ω γ1 γ2

ð6:17Þ

Equation (6.17) is obtained by considering the current that is confined in a thin layer near the interface as a surface current that is supported by a surface conductivity e σ ðωÞ. In this model, there is no current in the bulk of the conductor because the conductor is modeled as a nonconducting dielectric with a conducting surface layer, which has a permittivity of ϵ ∞ , given by (6.14) with σðωÞ → 0, for its volume and a surface conductivity of e σ ðωÞ for its surface, as shown in Figure 6.3(b). The dispersion characteristics of the SPP mode should be the same regardless of whether the permittivity of

6.2 Graphene Surface Excitations

2x

e

i x

2z

185

z x

2 1 1x

ei

x+ 1z

Figure 6.4 Physical configuration for the graphene SPP mode.

the conductor is expressed as (6.2) or (6.14). Furthermore, these characteristics remain unchanged when the conductivity of the conductor is replaced by an equivalent surface conductivity that exists only at the interface, as described above. It can be shown that (6.17) is identical to (6.11) by replacing e σ ðωÞ with σðωÞ=γ1 and (6.14) with (6.2). Therefore, the two models shown in Figures 6.3(a) and (b) for the SPP mode at the interface between a metal and a dielectric are equivalent and yield the same dispersion characteristics for the SPP mode.

6.2

Graphene Surface Excitations

6.2.1

Transverse Magnetic Mode Consider the geometry shown in Figure 6.4, where a sheet of monolayer graphene is sandwiched between two dielectrics characterized by permittivities ϵ 1 and ϵ 2 . This situation is identical to that shown in Figure 6.3(b), with ϵ ∞ and ϵ d replaced by ϵ 1 and ϵ 2 , respectively, and the equivalent conducting surface layer of the metal replaced by the graphene sheet; the current is confined within the graphene sheet and is absent in the dielectrics. Therefore, the graphene sheet supports a TM SPP mode in a manner similar to a metallic surface that is modeled with an equivalent surface conductivity of e σ ðωÞ. From (6.17), we obtain the eigenvalue equation for the dispersion characteristics of the graphene SPP mode as ϵ1 ϵ2 e σ ðωÞ ; þ ¼ i ω γ1 γ2

ð6:18Þ

where e σ ðωÞ is now the 2D surface conductivity of graphene. To be precise, e σ ðωÞ should also account for the bound electrons, as given by the effective surface conductivity e σ ∥ ;eff ðωÞ in (3.123). Here, for simplicity, we ignore the contribution of bound electrons σ ðωÞ and write the effective surface conductivity e σ ∥ ;eff ðωÞ as the surface conductivity e solely contributed by free electrons. Although the dispersion characteristics of the SPP modes for a graphene sheet and a metallic surface are solved using similar eigenvalue equations, the SPP dispersion curves of the two are very different due to very different values of e σ ðωÞ and very different charge carrier concentrations between the two types of

186

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materials. For most metals, the SPP resonance is observed in the ultraviolet spectral region, whereas the SPP resonance for graphene appears in the terahertz spectral region. As in the case of the metal SP mode, the characteristics of the graphene SP mode can be obtained from (6.18) by setting c → ∞ so that c ≫ ω=β and thus β ¼ γ1 ¼ γ2 from (6.9); we then obtain β ¼ iω

ϵ1 þ ϵ2 : e σ ðωÞ

ð6:19Þ

The dispersion curves of the graphene SPP mode and the graphene SP mode are plotted in Figures 6.5(a) and (b), together with the two light lines for the two dielectric media of ϵ 1 and ϵ 2 on the two sides of the graphene sheet. The Drude model for the surface conductivity e σ ðωÞ given by (4.2) in the limit that the Fermi energy EF ≫ kB T is used: e σ ðωÞ ¼

e2 EF πℏ2 ðτ1  iωÞ

:

ð6:20Þ

For simplicity, we first ignore the scattering rate τ1 so that β has a real value and β ¼ β0. The effect of the scattering rate will be discussed shortly. As can be seen in Figures 6.5(a) and (b), the SPP mode is a coupled state of photon and the SP mode. The coupling is the strongest near the intersection of the light line and the dispersion curve of the SP mode. The location of this intersection depends on the values of the permittivities ϵ 1 and ϵ 2 of the two dielectrics and the Fermi energy EF of the graphene sheet. The SPP mode exists only in a small window centered around this intersection. Outside of this spectral window away from the intersection, the SPP mode essentially becomes either the photon in the dielectric that has a higher permittivity, which is assumed to be the substrate with ϵ 1 > ϵ 2 , or the SP mode. As shown in Figures 6.5(a) and (b), for ϵ 1 ¼ 3:9ϵ 0 and ϵ 2 ¼ ϵ 0 , the SPP mode only exists in a small spectral window near ℏω ¼ 102 EF for graphene on a flat substrate in the configuration shown in Figure 6.4. This window can be tuned by adjusting the Fermi energy of graphene. For example, for a Fermi energy of EF ¼ 100 meV, the SPP mode is only meaningful in the spectral range of approximately 0:1 to 0:5 THz, whereas for a Fermi energy of EF ¼ 400 meV, the resonance frequency of the SPP mode can reach 2 THz. In some literature in which a more complicated structure is considered, the derivation of the plasmon mode is sometimes carried out without making the assumption c ≫ ω=β so that the result is applicable to the whole spectrum; then, the term “SPP” is used instead of “SP,” even though for most parts of the spectrum the coupling is extremely weak between the SP mode and the optical wave. The field distributions at the locations marked by (c), (d), and (e) in Figures 6.5(a) and (b) are plotted in Figures 6.5(c), (d), and (e), respectively. For small values of the propagation constant β0, the SPP mode behaves like a photon in the substrate as the SPP dispersion curve approaches the light line of the substrate. As a result, the SPP field is weakly confined on the graphene sheet and spreads far into the substrate, as shown in Figure 6.5(c). When β0 has a value near the intersection of the SP mode and the photon

(a)

(b) 1

10

(e)

3

SP

1

P SP

SP P 10

4

(d)

(d) (c)

(c)

10

4

10

3

10

(c) Photon-like SPP = 0.9746c1 = p

2

10

1

EF

kF , k kF

Graphene

2

SP

10

2

1

n oto h P n oto Ph

(e)

kF , k kF

EF

Ph o to n Ph o to n

2

10

−1

−0.5

Re Ex 0

0.5

1

−1

−0.5

Re Ex 0

0.5

1

−1

−0.5

Re Ex 0

0.5

1

2

1

(d) SPP = p

= 0.7344c1

Graphene

2

1

(e) SP-like SPP = = 0.2778c1 p

Graphene

2

1

Figure 6.5 (a,b) Dispersion curves of the graphene SPP mode, the graphene SP mode, and the light lines of the dielectrics of ϵ1 ¼ 3:9ϵ0 and ϵ2 ¼ ϵ0 for the structure shown in Figure 6.4. The Fermi energy of the graphene sheet is taken to be EF ¼ 100 meV. The Drude model ispused for ffiffiffiffiffiffiffiffiffiffi ffi the surface conductivity pffiffiffiffiffiffiffiffiffiffiffi e σ of graphene. The two straight light lines are given by ω ¼ ck= ϵ1 =ϵ0 and ω ¼ ck= ϵ2 =ϵ0 , respectively. The dispersion curve of the SPP mode is given by (6.18), and that of the SP mode is

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Plasmonics

line, the strongly coupled SPP mode field is well confined on the graphene sheet as the coupling between the photon and the SP mode is strong, as shown in Figure 6.5(d). For an even larger value of β0, the SPP mode behaves like an SP mode, and its field is highly confined on the graphene sheet, as shown in Figure 6.5(e). The phase velocity of this SPlike SPP mode is very small compared to the speed of light in the substrate. In this case, the quasistatic approximation can be adopted by assuming that ω=β0 ≪ c; the resulting approximated field distribution shows little difference compared to that shown in Figure 6.5(e). Eventually, for an extremely large value of β0, the SPP mode is essentially an SP mode; the field distribution and the dispersion curve in this limit can be obtained by setting c → ∞. Now we consider the effect of a nonzero scattering rate τ 1 ≠ 0. By using (6.20) from the Drude model in the limit that EF ≫ kB T, we find the dispersion characteristics of the graphene SP mode from (6.19): β¼

πℏ2 ω ðω þ iτ1 Þðϵ 1 þ ϵ 2 Þ: e2 E F

ð6:21Þ

Clearly, β is a complex number because of the nonzero scattering rate τ1 ; it can be expressed as β ¼ β0 þ iβ00 , where β0 and β00 have real values. The relation between β0 and ω defines the dispersion characteristics of the graphene SP mode, as shown in Figure 6.6 (a). For this plot, the graphene sheet is taken to have EF ¼ 100 meV and τ1 ¼ 5  1012 s1 , and the permittivities of the surrounding media are assumed to be ϵ 1 ¼ 3:9ϵ 0 andp ϵ 2ffiffiffiffi¼ ϵ 0 . As can be seen from Figures 6.6(a) and (b) and (6.21), ω is proportional to β0 , a characteristic that is different from a metal SPP mode for which ω approaches the constant surface plasma frequency of the metal in the limit that β0 → ∞. The propagation distance of the graphene SP mode is determined by β00 because the imaginary part of β leads to an exponentially decaying field in the form of expðβ00 xÞ given by (6.8) or (6.10). Therefore, β00 is the attenuation constant of the graphene SP, which travels a distance of 1=β00 before its field amplitude reduces to 1=e of its original value. A figure of merit can be defined as β0=2πβ00 , i.e., the propagation distance 1=β00 divided by the plasmonic wavelength 2π=β0. From (6.21), we find that pffiffiffiffi β0=2πβ00 ¼ ωτ=2π ∝ β0 ; clearly, a large value of the scattering rate constant τ1 leads to a small value of β0=2πβ00 . As shown in Figure 6.6(c), β0=2πβ00 increases with β0. Because β0 increases with frequency, β0=2πβ00 also increases as the frequency increases, as shown in Figure 6.6(d), indicating that a graphene SP propagates a distance covering a larger number of the plasmonic wavelength. As discussed in Chapter 4, spatial nonlocality is not accounted for in the Drude conductivity. The losses due to interband absorption are not considered in the Drude Caption for Figure 6.5 (cont.)

given by (6.19). Plotted in (c), (d), and (e), respectively, are the spatial distributions of the on itsffi longitudinal field component Ex of the SPP mode at the locations marked by (c), (d), and (e) pffiffiffiffiffiffiffiffiffiffi dispersion curve. The phase velocity vp is expressed in terms of the light velocity c1 ¼ c= ϵ1 =ϵ0 in the substrate that has a permittivity of ϵ1 , where c is the speed of light in free space.

6.2 Graphene Surface Excitations

(a)

189

(b) 2

1.5

1.5 kF

EF

2

1

1

0.5

0.5

0

0

0.5

1

1.5 kF

(c)

0

2

0

0.5

1

(d)

10

10

8

8

6

6

2

1.5

2

2

2

1.5 EF

4

4

2

2

0

0

0.5

1

1.5

2

0

0

kF

0.5

1 EF

Figure 6.6 (a,b) Dispersion and (c,d) attenuation characteristics of the SP mode. The Fermi energy and the scattering rate of graphene are taken to be 100 meV and 5  1012 s1 , respectively. The Drude model is used for the conductivity e σ of graphene.

model either. Therefore, the dispersion and attenuation of a graphene SP mode characterized by the complex β given in (6.21) are only a good approximation for small wave numbers and low frequencies so that the response of graphene to the external optical field is approximately local and the losses due to interband absorption are negligible. To account for the effects of nonlocal response and interband absorption that are ignored in the Drude model, we use the RPA–RT surface conductivity, e σ ðβ; ωÞ ¼

iωe2 e Π γ ðβ; ω; γÞ; β2

ð6:22Þ

which is given by (4.26), to replace the Drude surface conductivity e σ ðωÞ in (6.18) for the eigenvalue equation of the graphene SPP mode:

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Plasmonics

ϵ1 ϵ2 e σ ðβ; ωÞ ; þ ¼ i ω γ1 γ2

ð6:23Þ

and to replace the Drude surface conductivity e σ ðωÞ in (6.19) for the dispersion relation of the graphene SP mode: β ¼ iω

ϵ1 þ ϵ2 : e σ ð β; ωÞ

ð6:24Þ

e γ ð β; ω; γÞ on the scattering rate γ is explicitly expressed. The dependence of Π As discussed in Chapter 4, the scattering rate itself is a function of frequency. Here, for simplicity, we assume that the scattering rate has a fairly constant value within the frequency region under consideration for the graphene SPP and SP modes, and we use the phenomenological parameter γ ¼ τ1 for the total scattering rate. Note that in (6.22) the wave number β replaces the variable q that is used for the wave number of the electric field in (4.26) because β is a symbol usually reserved for a propagating eigenmode such as the graphene SPP mode. In (6.22), the RPA–RT surface polarizability function e γ ðβ; ω; γÞ is given by (4.63) as Π e γ ðβ; ω; γÞ ¼ Π

ω þ iγ

e ω; γÞ þ iγ=Πðβ; e 0; 0Þ ω=Πðβ;

:

ð6:25Þ

By using e σ ðβ; ωÞ given in (6.22) for (6.24), we obtain β¼

e γ ðβ; ω; γÞ e2 Π : ϵ1 þ ϵ2

ð6:26Þ

Unlike (6.21), where β is explicitly expressed, (6.26) is an implicit relation for β because e ω; γÞ, which is obtained both its right- and left-hand sides contain β. Furthermore, Πðβ; by replacing q in (4.62) with β, is a complex-valued function of β. Therefore, proper approximations have to be taken in order to solve for β. e ω; γÞ given by (4.62) with Upon closer observation of the mathematical form of Πðβ; q replaced by β, we find that ω and iγ always appear together in the form of ω þ iγ. e ω; γÞ ¼ Πðβ e 0 þ iβ00 ; ω þ iγÞ and regard Πðβ; e ω; γÞ as Therefore, we can write Πðβ;

a function of the complex wave number β ¼ β0 þ iβ00 and the complex frequency ω þ iγ. Then, we assume that the scattering rate constant has a small value such that γ ≪ ω, which also implies that β0 is much larger than β00 according to (6.21). We can later check if the assumption that β0 ≫ β00 is justified by comparing their relative magnitudes once β0 and β00 are found. Because β0 ≫ β00 and ω ≫ γ, we can expand e 0 þ iβ00 ; ω þ iγÞ as Πðβ e 0 e 0 e 0 þ iβ00 ; ω þ iγÞ ¼ Πðβ e 0; ωÞ þ iβ00 ∂Πðβ ; ωÞ þ iγ ∂Πðβ ; ωÞ þ . . . ; Πðβ 0 ∂ω ∂β e 0; ωÞ is a function of β0 and ω, which are both real. where Πðβ

ð6:27Þ

6.2 Graphene Surface Excitations

191

e 0; ωÞ is a complex-valued function as shown in Figure 4.7 in Chapter 4, Although Πðβ e 00 ðβ0; ωÞ is small compared to its real part Π e 0ðβ0; ωÞ for ðβ0; ωÞ values its imaginary part Π on the dispersion curve calculated using the Drude model, as shown in Figure 6.6(c). e 00 ðβ0; ωÞ is identically zero, as Outside the interband and intraband scattering regions, Π

e 0ðβ0; ωÞ ≫ Π e 00 ðβ0; ωÞ. Note shown in Figure 4.7. Therefore, it is justified to assume that Π e 0; ωÞ=∂β0 or that this assumption does not lead to the assumption that the real part of ∂Πðβ e 0; ωÞ=∂ω is much larger than their respective imaginary parts. Then, we can that of ∂Πðβ rewrite (6.27) as e 0 þ iβ00 ; ω þ iγÞ ¼ Π e 0ðβ0; ωÞ þ δΠ e 0 þ iδΠ e 00 þ higher-order terms; Πðβ

ð6:28Þ

where 0

e ¼ β00 δΠ

e 00 ðβ0; ωÞ e 00 ðβ0; ωÞ ∂Π ∂Π ;  γ ∂ω ∂β0

e0 0 e0 0 e 00 ¼ Π e 00 ðβ0; ωÞ þ β00 ∂Π ðβ ; ωÞ þ γ ∂Π ðβ ; ωÞ : δΠ ∂ω ∂β0

ð6:29Þ ð6:30Þ

e 0 and δΠ e 00 is much smaller than Π e 0ðβ0; ωÞ because the imaginary parts γ, Every term in δΠ e 00 ðβ0; ωÞ of the complex variables ω þ iγ, β, and Πðβ e 0; ωÞ, respectively, are each β00 , and Π e 0ðβ0; ωÞ. an order of magnitude smaller than the respective real parts ω, β0, and Π e 0 þ iβ00 ; ω þ iγÞ given in (6.28), Π e 0ðβ0; ωÞ is Therefore, in the series expansion of Πðβ 0

00

e and δΠ e are both first-order terms; higher-order the only zeroth-order term, whereas δΠ e 0ðβ0; ωÞ is the terms beyond the first order are not explicitly shown. The zeroth-order Π

e 0 þ iβ00 ; ω þ iγÞ, whereas the three terms of the firstleading term for the real part of Πðβ e 00 given in (6.30) are the leading terms for the imaginary part of order δΠ e 0 þ iβ00 ; ω þ iγÞ. Πðβ

We now plug (6.25) and (6.28) into (6.26). The real part of (6.26) gives the dispersion characteristics of the graphene SPP mode. By keeping only the zerothe γ ðβ; ω; γÞ, i.e., by dropping the first-order δΠ e 0 and iδΠ e 00 terms and order term for Π e 0 þ iβ00 ; ω þ iγÞ to find Π e γ ðβ; ω; γÞ through (6.25), we all high-order terms of Πðβ obtain [5] β0 ¼

e2 e 0 0 Π ðβ ; ωÞ: ϵ1 þ ϵ2

ð6:31Þ

This relation shows that in the lowest order, the dispersion relation is independent of the scattering rate because the equation is free of β00 and contains only the unknown β0 for a given frequency ω. Nevertheless, it is an implicit relation for β0, which has to be numerically solved to find β0. By using the relation

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Plasmonics

e 0; ωÞ Πðβ 2kF ¼ þ πℏvF

"    # 2 β0 2vF kF þ ω 2v F kF  ω qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G  sgnð2vF kF  ωÞG ; 2 v F β0 v F β0 4πℏ ω2  v2F β0 ð6:32Þ

which is obtained from (4.89) by replacing q and ω þ iγ with β0 and ω, respectively. With the function G given by (4.72), we can solve (6.31) to obtain the characteristic dispersion curve of the graphene SP mode, as shown in Figure 6.7(a) in the limit that the temperature T → 0. Because the RPA surface polarizability function is used in (6.31), we call the relation between β0 and ω found from (6.31) the RPA dispersion curve in contrast to the semiclassical dispersion curve found using the Drude model. By comparing the RPA and semiclassical results, both shown in Figure 6.7(a), we find that they differ the most in the interband scattering region because the Drude model fails to account for the effect of interband scattering. The imaginary part β00 of β found from (6.26) gives the attenuation of the graphene SP mode. As discussed above, the leading terms of the imaginary part are of the first order. By keeping only the leading first-order terms of the imaginary part of e 0 þ iβ00 ; ω þ iγÞ to find Π e γ ðβ; ω; γÞ through (6.25), followed by finding β00 through Πðβ (6.26), we obtain [5]

(a)

(b) 2

10 Interband

8

1.5

6 2

EF

D

del mo e rud

1 Intraband 0.5

0

Interband

4

R PA

2

0

0.5

1 kF

1.5

2

0

0

0.5

1

model 1.5

2

kF

Figure 6.7 (a) Dispersion and (b) attenuation characteristics of the graphene SP mode in the zerotemperature limit that T → 0. The Fermi energy and the scattering rate of graphene are taken to be 100 meV and 5  1012 s1 , respectively, for the curves plotted in this figure. The Drude model (solid curves) and the RPA model (dashed curves) are respectively used for the surface conductivity e σ of graphene.

6.2 Graphene Surface Excitations

β00 ¼

e0 0 e0 0 e0 0 e 00 ðβ0; ωÞ þ γ ∂Π ðβ ; ωÞ þ γΠ ðβ ; ωÞ 1  Π ðβ ; ωÞ Π ∂ω ω e 0ðβ0; 0Þ Π e 0ðβ0; ωÞ ∂Π e 0ðβ0; ωÞ Π  β0 ∂β0

193

! :

ð6:33Þ

In contrast to (6.31), which gives the real part β0 in an implicit relation, (6.33) gives the imaginary part β00 as an explicit function of β0, ω, and γ. As can be seen from (6.33), the losses of the graphene SP mode are contributed not only by the scattering rate γ, but also e 00 ðβ0; ωÞ, which manifests interband scattering. The figure of merit β0=2πβ00 by the term Π calculated using (6.31) and (6.33) as the RPA result is plotted in Figure 6.7(b) along with the semiclassical result found using the Drude model. Inside the interband scattering region, β0=2πβ00 drops significantly as plasmons decay into electron–hole pairs through interband scattering. Note that the RPA results shown in Figure 6.7 can only be regarded as approximations, especially in the low-frequency region, where the assumption ω ≫ γ is not valid, and in the high-frequency region, where the assumption β0 ≫ β00 is not true anymore.

Transverse Electric Mode Now we examine whether a transverse electric (TE) mode can be supported by an interface between a conductor and a dielectric or by a sheet of monolayer graphene. Because a TE mode does not have a longitudinal electric field component, it cannot couple to the longitudinal plasma oscillations of graphene. Therefore, if a TE mode supported by a graphene sheet exists, it is not an SP or SPP mode but is only a guided electromagnetic mode. By contrast, a TM mode supported by a graphene sheet can couple to the longitudinal plasma oscillations because it has a longitudinal electric field component; therefore, it is an SPP mode, as discussed in the preceding subsection. The electric field of a TE mode that propagates in the x direction but is confined by an interface or by a graphene sheet, located at z ¼ 0, has the form: ( E2 ¼ ^y E2y ¼ ^y E 2y eiβxγ2 ziωt ; z ≥ 0; E¼ ð6:34Þ E1 ¼ ^y E1y ¼ ^y E 1y eiβxþγ1 ziωt ; z < 0: For an interface between an ordinary conductor, such as a metal, and a dielectric, the continuity of the tangential electric field across the boundary at z ¼ 0 requires that E 1y ¼ E 2y ¼ E y . By using Faraday’s law,  E ¼ μ∂H=∂t, the corresponding magnetic field is obtained: 8 iγ2^x þ β^z > > E y eiβxγ2 ziωt ; z ≥ 0; >H2 ¼ ^x H2x þ ^z H2z ¼ < ωμ2 ð6:35Þ H¼ > iγ1^x þ β^z > iβxþγ1 ziωt > Eye ; z < 0; :H1 ¼ ^x H1x þ ^z H1z ¼ ωμ1

Δ

6.2.2

where μ1 and μ2 are respectively the magnetic permeabilities of the media below and above the boundary at z ¼ 0. Then, we obtain the condition γ1 =μ1 þ γ2 =μ2 ¼ 0 for the TE mode to exist by applying to (6.35) the continuity of the tangential magnetic field

194

Plasmonics

component, H ^x , across the boundary at z ¼ 0. This condition can only be satisfied if one of the two media has a negative permeability because both γ1 and γ2 are real and positive. Because the permeability of an ordinary conductor or dielectric is generally positive, which actually has the value of μ0 at an optical frequency, a TE mode is usually not supported by an interface between an ordinary conductor and a dielectric, such as a metallic surface. By contrast, a monolayer graphene sheet can support a TE mode if certain conditions are met. By applying the boundary condition given in (6.16) to the magnetic fields given in (6.35), with e J¼e σ ðωÞEy^y ¼ ^y e σ ðωÞE y eiβxiωt at z ¼ 0 for the electric field given in (6.34), we obtain the eigenvalue equation for the TE mode: σ ðωÞ; γ1 þ γ2 ¼ iωμ0 e

ð6:36Þ

where the permeabilities of the two media surrounding the graphene sheet are taken to be qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ1 ¼ μ2 ¼ μ0 as discussed above, and γi ¼ β2  ω2 μ0 ϵ i as given in (6.9). This eigenvalue equation has the form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β2  k12 þ β2  k22 ¼ a;

ð6:37Þ

pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi where k1 ¼ ω μ0 ϵ 1 and k2 ¼ ω μ0 ϵ 2 are the propagation constants in the two dielecσ ðωÞ, and β is the unknown that we intend to solve. Because k1 and tric media, a ¼ iωμ0 e k2 have positive real values, (6.37) has a solution for a positive real value of β only if three conditions are simultaneously satisfied: (1) β > k1 and β > k2 ; (2) a has a positive real value; and (3) β2 < k12 þ a2 and β2 < k22 þ a2 . Condition (1) is imposed by the requirement that γ1 and γ2 have positive real values so that the fields can be confined on graphene. Condition (1) leads to condition (2) because the left-hand side of the equation is positive and real. Therefore, from (6.36) and (6.37), we find that to have a positive real solution of β that β ¼ β0 > 0, the surface optical conductivity e σ ðωÞ has to be negative purely imaginary such that e σ ðωÞ ¼ ie σ 00 ðωÞ and e σ 00 ðωÞ < 0. Then, we find that a ¼ ωμ0 e σ 00 ðωÞ ¼ ωμ0 je σ 00 ðωÞj > 0, which satisfies condition (2). In the limits of low temperature, T → 0, and vanishing scattering rate, γ → 0, the surface conductivity e σ ðωÞ of monolayer graphene is given by (4.5), which is plotted in Figure 6.8(a). As is highlighted by the shaded area in Figure 6.8(a), there is a spectral region 1:67EF < ℏω < 2EF where e σ ðωÞ is purely imaginary and its imaginary part is negative qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [6]. Condition (3) is not that obvious; one can move, say, β2  k12 to the right-hand side qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the equation, and then we have the condition a  β2  k12 > 0, which is tantamount to the relation β2 < k12 þ a2 . By combining condition (3) with condition (1), we find that k12 < β2 < k12 þ a2 and k22 < β2 < k22 þ a2 ; the solution of β exists within the overlapping region of these two regions. By examining the values of k1 , k2 , and a, it is found that a2 ≪ k12 and a2 ≪ k22 . Therefore, condition (3) states that for a positive real solution of β to exist for (6.37), it is necessary that k1 ≈ k2 , namely ϵ 1 ≈ ϵ 2 so that γ1 ≈ γ2 for (6.36).

6.2 Graphene Surface Excitations

(a)

195

(b) 4

1.0008

( )

k

0

2 1.0004 0

−2 1 0

1

3

2

Light line 1.7

EF

k 1.8

1.9

2

EF

Re E y

(c)

1 Graphene

0

z x

−1

Figure 6.8 (a) The real (solid curve) and imaginary (dashed curve) parts of the optical conductivity of

monolayer graphene in the limit of low temperature T → 0 and vanishing scattering rate γ → 0, calculated using (4.4). The shaded region indicates the spectral region where e σ ðωÞ has a negative purely imaginary value. The dispersion curve solved by using (6.36) and e σ ðωÞ given by (a) is shown in (b). The light line of a propagation constant k for the surrounding medium of a permittivity ϵ is also plotted. (c) The distribution of the field Ey of the TE mode. The graphene sheet is at the middle of the figure.

In Figure 6.8(b), the propagation constant β and hence the dispersion characteristics of the TE mode are found by solving (6.36). In order to have a sustainable environment for a TE mode, the graphene sheet is assumed to be surrounded by the same dielectric medium of the permittivity ϵ 1 ¼ ϵ 2 ¼ ϵ. Then the eigenvalue equation of the TE mode given in (6.36) reduces to the form: 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β2  k 2 ¼ iωμ0 e σ ðωÞ;

ð6:38Þ

where k 2 ¼ ω2 μ0 ϵ. In the limit that the scattering rate γ → 0 and under the condition ℏω < 2EF so that interband scattering is not possible, e σ 0ðωÞ ¼ 0 because of the absence of the scattering mechanism. A lossless TE mode, for which β00 is zero and β ¼ β0 has a positive real value, is found in the spectral region 1:67EF < ℏω < 2EF , where e σ 0ðωÞ ¼ 0 and e σ 00 ðωÞ < 0, as shown in Figure 6.8(a). As discussed above, in order to support a TE mode, the condition k 2 < β2 < k 2 þ a2 has to be satisfied, where a2 ¼ ω2 μ20 e σ 002 ðωÞ is contributed by the e σ ðωÞ term in (6.36). Because a2 =k 2 ¼ μ0 e σ 002 ðωÞ=ϵ < μ0 e σ 002 ðωÞ=ϵ 0 and je σ 00 ðωÞj is of the order of

196

Plasmonics

e σ 0 ¼ e2 =4ℏ ≈ 6:1  105 S, we find that a2 =k 2 is of the order of 5:2  104 . Because a2 =k 2 ≪ 1, we find from (6.38) that ! 2 μ0 e σ 00 ðωÞ 0 β ¼ β ≈k 1 þ ð6:39Þ 8ϵ σ 00 ðωÞ < 0. It can be seen from the above discussion that in the case when e σ 0ðωÞ ¼ 0 and e the propagation constant β of the TE mode is very close to the propagation constant k of the free electromagnetic wave in the surrounding medium, as shown in Figure 6.8(b). By comparing the difference between β and k from (6.39) and from Figure 6.8(b), we qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi find that γ1 ¼ γ2 ¼ β2  k 2 is very small, and hence the confinement of the TE electromagnetic mode is very poor compared to that of the TM SPP mode. The field distribution is very similar to that of a free-propagating wave in the surrounding medium of a permittivity ϵ, as shown in Figure 6.8(c). For ℏω > 2EF, we find from Figure 6.8(a) that e σ 0ðωÞ ¼ e σ 0 > 0 and e σ 00 ðωÞ < 0 in the σ 02 ðωÞ=k 2 ≪ 1 and limit that the scattering rate γ → 0. In this case, we find that ω2 μ20 e ω2 μ20 e σ 002 ðωÞ=k 2 ≪ 1 because both are of the order of 5:2  104 . Then, the eigenvalue equation given in (6.38) for the TE mode has the solution: ! 2 2 μ0 e σ 00 ðωÞ  μ0 e σ 0 ðωÞ 0 β ≈k 1þ ; 8ϵ ð6:40Þ 00 00 2 2 0 2 2 0 ω μ0 e σ ðωÞe σ ðωÞ ω μ0 e σ ðωÞe σ ðωÞ : β00 ¼  ≈ 4k 4β0 Because e σ 0ðωÞ > 0 and e σ 00 ðωÞ < 0 in this spectral region, we find that β00 > 0 so that this TE mode is attenuated though β00 has a small value of the order of 104 k. For the TE mode to be guided, however, it is necessary that β0 > k; otherwise, the value of γ1 ¼ γ2 becomes purely imaginary so that the TE mode becomes a radiation mode and is not guided. From (4.5), we find that je σ 00 ðωÞj < e σ 0ðωÞ ¼ e σ 0 for ℏω > 2:024EF, as can be seen in Figure 6.8(a). Then, we find from (6.40) that for ℏω > 2:024EF, β0 < k and the TE mode is not guided but becomes a radiation mode. Therefore, in the limit that the scattering rate γ → 0 and that the temperature T → 0, the guided lossy TE mode that has a propagation constant β0 > k and a positive attenuation constant β00 > 0 exists only in a very small spectral region that 2EF < ℏω < 2:024EF . For the case of a finite temperature and a nonvanishing scattering rate, the optical conductivity e σ ðωÞ is plotted in Figure 4.1 in Chapter 4. The spectral region where e σ ðωÞ is purely imaginary becomes very small or even disappears. The nonlocality effect can also shrink this region as interband scattering becomes possible for ℏω < 2EF, as shown in Figure 6.7(a) where below ℏω ¼ 2EF the interband scattering region occupies a triangular area within a frequency range indicated in light gray. Therefore, in a realistic situation in which the temperature is finite and the graphene sheet is imperfect, the TE mode is most likely unsupported or hardly observable in an experiment.

6.2 Graphene Surface Excitations

197

z 3

z =0

d2

2

x

ds

s

1

Figure 6.9 Four-layer system setup for the calculation of coupled SP modes. The graphene sheet shown as the dotted line located at z ¼ 0 has a surface conductivity of e σ.

6.2.3

Graphene Surface Plasmons Coupled with Phonons Besides coupling with photons to form the SPP mode, the graphene SP mode can also couple with other quasiparticles if the coupling strength is sufficiently strong. Consider a sheet of graphene that is located at z ¼ 0 at a distance of d2 above a substrate that has a thickness of ds , as shown in Figure 6.9. The substrate is characterized by a permittivity of ϵ s , and the space between the graphene sheet and the substrate is filled with a medium of a permittivity ϵ 2. Above the graphene sheet and below the substrate are media of permittivities ϵ 3 and ϵ 1 , respectively. Here we study the coupling of the SP mode of graphene with the surface optical (SO) phonons on the substrate. We can characterize the permittivity ϵ s of the substrate by extending (2.46) as ϵ s ¼ ϵ high þ ðϵ high  ϵ int Þ

ω2TO2 ω2TO1 þ ðϵ  ϵ Þ ; int low 2 ω2  ωTO2 ω2  ω2TO1

ð6:41Þ

where ωTO1 and ωTO2 are the frequencies of the two transverse optical (TO) phonon modes of the substrate, and ϵ low , ϵ int , and ϵ high are the approximate permittivities of the substrate at low, intermediate, and high frequencies, respectively. Clearly from (6.33), ϵ s ¼ ϵ low in the limit when ω → 0, and ϵ s ¼ ϵ high when ω → ∞, whereas ϵ int is the permittivity at a specific intermediate frequency ωint such that ωTO1 < ωint < ωTO2 . These parameters are given in Table 2.1. The electric field distribution of the phonon-coupled SP mode that propagates in the x direction in the structure shown in Figure 6.9 has the form: 8 iβxγ z 3 ^ Ae x þ Beiβxγ3 z^z ; > > > > > ðC1 cosh γ2 z þ C2 sinh γ2 zÞeiβx^x > > > > < þ ðD1 cosh γ2 z þ D2 sinh γ2 zÞeiβx^z ; E¼ > >ðC3 cosh γs z þ C4 sinh γs zÞeiβx^x > > > > > þ ðD3 cosh γs z þ D4 sinh γs zÞeiβx^z ; > > : iβxþγ ðzþd þd Þ 2 s 1 ^x þ Feiβxþγ1 ðzþd2 þds Þ^z ; Ee

z ≥ 0; 0 > z ≥  d2 ;

ð6:42Þ

d2 > z ≥  d2  ds ; z ω as justified by Figure 6.11, we find that Π 0 that β →0 is   2kF r 0 0 0 e e Πðβ ; ωÞ ¼ Πðβ ; β rv F Þ ≈  1  pffiffiffiffiffiffiffiffiffiffiffiffiffi : ð6:53Þ πℏvF r2  1 Accordingly, by plugging (6.53) in (6.22), we find that the RPA surface conductivity is given as e σ ðβ0; ωÞ ¼

  iωe2 e 0 2e2 kF r ffiffiffiffiffiffiffiffiffiffiffiffi ffi p Πðβ ; ωÞ ≈  i r 1  : 2 πℏβ0 r2  1 β0

ð6:54Þ

As can be seen from (6.51) and (6.54), both models show that the surface conductivity is inversely proportional to the wave number β0. Now with the surface conductivity obtained, we can take the limit β0 → 0 for (6.45) with the surface conductivity given by either (6.51) or (6.54). By keeping the lowest order of β0, it can be shown that (6.45) reduces to the form: ϵ2 þ i

e σ d2 0 β ¼ 0: rvF

ð6:55Þ

Note that both terms on the left-hand side of (6.55) are of the order of β00 because e σ is proportional to β 01 . By using e σ of the Drude model given by (6.51), we can obtain r from (6.55), which gives the dispersion relation of the graphene SP mode as

204

Plasmonics

0

ω ¼ rvF β ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 E F d2 2

πℏ ϵ 2

β0 ðDrude modelÞ:

ð6:56Þ

Alternatively, by using e σ of the RPA model given by (6.54), we find from (6.55) the dispersion relation of the graphene SP mode as 1þΛ ω ¼ rvF β0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vF β0 ðRPA modelÞ; 1 þ 2Λ

ð6:57Þ

where Λ is given by Λ¼

2e2 EF d2 πℏ2 v 2F ϵ 2

:

ð6:58Þ

As can be seen by comparing (6.56) and (6.57), the two models give different results, signifying that the effect of nonlocality is still important in the limit that β0→0, although the difference in ω given by the two models is small for small values of β0. The linear dispersion relations given by (6.56) and (6.57) are plotted in Figure 6.11(b) as the gray dashed and solid curves, respectively. As can be seen, the analytical results are in agreement with the numerical results when β0 is small. However, as β0 increases, the discrepancy increases because the limit β0→0 is no longer valid.

6.3

Surface Plasmons of Two Graphene Sheets A graphene sheet can also be placed near another sheet of graphene to form a coupled mode of two plasmons on the two graphene sheets. Hence we consider in this section a system of two monolayer graphene sheets that are separated at a distance of d2 by a space that is filled with a dielectric medium of a permittivity ϵ 2, as shown in Figure 6.12 (a). This system can be considered to have the form of the structure shown in Figure 6.9 if we regard the lower graphene sheet in Figure 6.12(a) as the substrate. Thus, we can

Caption for Figure 6.12

(a) System of two monolayer graphene sheets. Dispersion curves of graphene SP modes in the limit that γ ¼ τ1 → 0 for (b) a symmetric two-sheet graphene structure with ϵ3 ¼ ϵ1 and (c) an asymmetric two-sheet graphene structure with ϵ3 ¼ 2ϵ1 . The dispersion curves are calculated for ϵ1 ¼ ϵ0 and ϵ2 ¼ 3:9ϵ0 using (6.64) in (b), and (6.63) in (c). For the calculation, the surface conductivity of graphene given by the Drude model in (6.20) in the limit that τ1 → 0 is used. The distance d2 is assumed to be 10 nm for the coupled modes shown in dashed curves, and d2 → ∞ for decoupled modes shown in dotted curves. Other parameters are the same as those used for Figure 6.10. The real parts of the transverse Hy and longitudinal Ex field distributions of the symmetric and antisymmetric modes marked by (d) and (e) in (a) are shown in (d) and (e), respectively. The locations of the graphene sheets are marked by the dotted lines.

(a)

Figure 6.12

3 upper

z x d2

2

1

(b)

lower

(c) 2

0.5 Interband

Interband 0.4 EF

EF

1.5

1

(e)

0.2

Intraband 0.5

0

0

1 kF

0

2

0

1

Re Ex

40

20 z (nm)

20

0 −20

0 −20

0

−40

4

2

0

(e)

x

Re H y

40

Re Ex

40 20 z (nm)

20 z (nm)

4

2

x

0 −20 −40

2 kF

Re H y

40

z (nm)

Intraband

0.1

(d)

−40

0.3

(d)

0 −20

0

2

4 x

−40

0

4

2 x

206

Plasmonics

reuse (6.43) without deriving the characteristic equation again. As before, we consider the nonretarded region, where γ3 ≈ γ2 ≈ γs ≈ γ1 ≈ β. In this region, (6.43) reduces to the form: ϵs

ϵ 1 þ ϵ s tanh βds ϵ s þ ϵ 1 tanh βds þ ϵ2 ¼ 0; Γupper þ tanh βd2 1 þ Γupper tanh βd2

ð6:59Þ

where Γupper ¼

βe σ upper ϵ3 þi ; ϵ2 ωϵ 2

ð6:60Þ

and e σ upper is the surface optical conductivity of the upper graphene sheet in a 2D model. For consistency, we shall also use the 2D model for the lower graphene sheet as we do for the upper graphene sheet. For the 2D model, we take the limit that the thickness of the lower graphene sheet approaches zero, ds → 0. In (6.59), ϵ s represents the permittivity of the lower graphene sheet because here we regard the lower graphene sheet as the substrate layer. However, ϵ s is a 3D quantity. To relate the 2D conductivity model with ϵ s , we use (3.41) to find ϵ s ¼ ie σ lower =ωds , where e σ lower is the surface optical conductivity of the lower graphene sheet. As done for e σ upper of the upper graphene sheet, we have ignored the contribution from the bound electrons to e σ lower of the lower graphene sheet. By plugging the relation ϵ s ¼ ie σ lower =ωds in (6.59) and taking the limit that ds → 0, we obtain Γupper þ Γlower þ Γupper Γlower tanh βd2 þ tanh βd2 ¼ 0;

ð6:61Þ

where Γlower ¼

ϵ1 βe σ lower þi : ϵ2 ωϵ 2

ð6:62Þ

It is seen that the form of (6.61) is not changed by exchanging Γupper and Γlower , i.e., by exchanging the upper and lower graphene sheets, as expected. A few observations discussed in the following can be made for (6.61). In the limit that d2 → ∞, the two graphene sheets are decoupled because of the infinite distance between them. Then, (6.61) reduces to ðΓupper þ 1ÞðΓlower þ 1Þ ¼ 0:

ð6:63Þ

This is just the two decoupled graphene SP modes given by (6.19); Γupper þ 1 ¼ 0 gives the SP mode of the upper graphene sheet, and Γlower þ 1 ¼ 0 gives the SP mode of the lower graphene sheet. If the structure is symmetric with ϵ 3 ¼ ϵ 1 ¼ ϵ and e σ upper ¼ e σ lower ¼ e σ , then Γupper ¼ Γlower ¼ Γ. For this symmetric structure, (6.61) becomes

6.3 Surface Plasmons of Two Graphene Sheets

 Γ þ coth

βd2 2

 Γ þ tanh

βd2 2

207

 ¼ 0:

ð6:64Þ

The two solutions of this eigenvalue equation represent symmetric and antisymmetric SP modes of a symmetric two-sheet graphene structure: Γ þ coth ðβd2 =2Þ ¼ 0 for the symmetric mode, which has a lower energy, and Γ þ tanh ðβd2 =2Þ ¼ 0 for the antisymmetric mode, which has a higher energy. Because the structure is symmetric, the two graphene sheets have the same properties and identical settings: Both have the same surface conductivity e σ, and both are sandwiched between dielectrics of permittivities ϵ and ϵ 2 . The two SP modes become degenerate in the limit that d2 → ∞ because the eigenvalue equations found from (6.64) for the two modes reduce to the same form Γ þ 1 ¼ 0 in this limit. The common dispersion curve of the two decoupled degenerate modes in the limit that d2 → ∞ when the two graphene sheets are far apart is plotted as the dotted curve in Figure 6.12(b). This degeneracy is lifted when the two graphene sheets are placed in close proximity at a sufficiently small distance d2 so that the two modes are coupled to result in the symmetric and antisymmetric modes that have different dispersion characteristics shown as dashed curves in Figure 6.12(b). The lower dashed curve represents the dispersion characteristics of the low-energy symmetric mode, whereas the upper dashed curve represents those of the high-energy antisymmetric mode. The difference between the nondegenerate low-energy symmetric mode and high-energy antisymmetric mode increases with decreasing distance between the two graphene sheets. The Drude model given by (6.20) is used for the dispersion curves plotted in Figure 6.12(b), but the scattering rate is neglected by taking τ1 → 0 so that β ¼ β0 has a real value. The parameters used for the calculation are ϵ 3 ¼ ϵ 1 ¼ ϵ ¼ ϵ 0 , ϵ 2 ¼ 3:9ϵ 0 , EF ¼ 100 meV, and d2 ¼ 10 nm. Note that the interband and intraband scattering regions are also plotted in Figure 6.12(b) for later comparison with the characteristics calculated using the RPA model. The Drude model does not account for these intrinsic scattering mechanisms; therefore, the scattering mechanisms in these regions do not have any effect on the dispersion curves shown in Figure 6.12(b). For an asymmetric structure, such as a structure of different permittivities ϵ 3 ≠ ϵ 1 for the two dielectrics or one of different Fermi energies for the two graphene sheets, there are always two nondegenerate modes, as shown in Figure 6.12(c) for ϵ 3 ¼ 2ϵ 1, with other parameters being the same as those for Figure 6.12(b). As in the case of a symmetric structure, the energy difference between the optical and acoustic modes increases with decreasing d2 . For a large value of βd2 , the two modes are decoupled, as can be seen in the large-β region of Figure 6.12(c), where the dashed curves, for which d2 ¼ 10 nm, approach the dotted curves, for which d2 → ∞. The field distributions of the symmetric and antisymmetric modes can be found by using (6.42). If we follow the procedure taken to obtain (6.43) from (6.42), an identity D1 ¼ ΓD2 can be found. Then, by plugging D1 ¼ ΓD2 with Γ ¼ coth ðβd2 =2Þ and Γ ¼ tanh ðβd2 =2Þ for the two SP modes into (6.42), it can be shown that the spatial patterns of the transverse field components, Ez and Hy , are symmetric (in-phase) for the low-energy mode, and antisymmetric (out-of-phase) for the high-energy mode, as shown in Figures 6.12(d) and (e), respectively. Therefore, the low-energy mode is

Plasmonics

2 Interband 1.5 EF

208

1 Intraband 0.5

0

0

1 kF

2

Figure 6.13 Dispersion curves of graphene SP modes, calculated using (6.64) in the limit that

γ ¼ τ1 → 0, for a symmetric two-sheet graphene structure with ϵ3 ¼ ϵ1 . The parameters for the calculation are the same as those used for the curves shown in Figure 6.12(b). The Drude model is used for the dashed curves assuming d2 ¼ 10 nm, which are also shown as the dashed curves in Figure 6.12(b). The RPA model is used for the solid and dotted curves assuming d2 ¼ 10 nm and d2 → ∞, respectively.

considered to be a symmetric mode, whereas the high-energy mode is an antisymmetric mode. However, the spatial pattern of the longitudinal component Ex is symmetric for the high-energy mode and antisymmetric for the low-energy mode. These characteristics are consistent with the characteristics of the guided modes of a planar optical waveguide: The spatial distributions of the transverse field components are symmetric with respect to the waveguide structure for the fundamental mode, which has the highest propagation constant for a given frequency among all guided modes or the lowest frequency for a given propagation constant, and they are antisymmetric for the first-order mode [8]. In the small-β limit, analytical solutions can be found from (6.64). Using the Drude qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi model, we obtain ω ¼ β e2 EF d2 =2πℏ2 ϵ 2 and ω ¼ e2 EF β=πℏ2 ϵ 3 for the low-energy and high-energy modes, respectively. Because the low-energy mode has a linear dispersion relationship ω ∝ β in the small-β limit, it is also called the acoustic SP mode. pffiffiffi The high-energy mode has a dispersion relationship ω ∝ β and is sometimes referred to as the optical SP mode of the two-sheet graphene structure. We find that the energy of the acoustic mode decreases with decreasing d2 , whereas the optical mode does not depend on d2 . This feature can also be seen in Figure 6.12(b), where in the limit β → 0 the dispersion curve of the optical mode, for which d2 ¼ 10 nm, approaches that of the decoupled mode, for which d2 → ∞, because in this limit neither is a function of d2 . As discussed in the preceding section, the analytical solution of (6.64) in the small-β limit can also be found by using the RPA model. Although the resultant acoustic mode is given in a different mathematical form because of the consideration of nonlocality, the conclusion is unchanged: There is one acoustic mode and one optical mode, both of which have the aforementioned dispersion relationships [9].

6.4 Excitation and Detection of SPPs and SPs

209

The dispersion curves of the coupled and decoupled modes found by using the RPA surface conductivity model for a symmetric two-sheet graphene structure are plotted in Figure 6.13. Also shown for comparison are those of the coupled modes found by using the Drude surface conductivity model and plotted in Figure 6.12(b). The dashed curves are still the dispersion curves of the nondegenerate optical and acoustic modes calculated using the Drude model, whereas the solid and dotted curves are those calculated using the RPA model given by (6.22). As can be seen, when nonlocality is considered, the dispersion curves do not enter the intraband scattering region as opposed to the dispersion curves calculated using the Drude model that ignores the effect of nonlocality. In the large-β0 region where the plasmonic wavelength is small and the fields are highly confined on the graphene sheets, the fields on the two graphene sheets are essentially decoupled. This phenomenon is shown by comparing in the large-β0 region the solid curves for the coupled modes, which are calculated using a distance of d2 ¼ 10 nm between the two graphene sheets, with the dotted curve for the decoupled degenerate modes, calculated using d2 → ∞. Eventually, the solid curves merge with the dotted curve in the limit that β0 → ∞, and the two modes become decoupled SP modes that are degenerate.

6.4

Excitation and Detection of Graphene SPPs and SPs For a structure with a graphene sheet sandwiched between two dielectric media of permittivities ϵ 1 and ϵ 2 as shown in Figure 6.4, in the entire spectral region the dispersion curve of a graphene SPP mode is below the light line of an optical wave that propagates freely in the dielectric medium on either side of the graphene sheet, as is seen in Figure 6.5. At a given frequency, the propagation constant of the SPP mode is always larger than that of a free-propagating optical wave in the pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi dielectric medium on either side: β > k1 ¼ ω ϵ 1 =ϵ 0 =c and β > k2 ¼ ω ϵ 2 =ϵ 0 =c. When an optical wave is incident from either side at any incident angle θ, the SPP mode can never be excited by the incident wave because of phase mismatch. It is clear that to excite a graphene SPP mode, a proper arrangement that provides the necessary phase matching to couple an incident optical wave and a graphene SP mode is required.

6.4.1

Coupling Through Prism Consider an optical wave that is incident from the ϵ 2 side. Phase matching requires that β ¼ k2 sin θ, which is not possible without assistance due to the fact that β > k2 . One possible way to solve this problem is to use a dielectric prism coupler that has a permittivity of ϵ p to facilitate phase matching. The phase-matching condition for coupling an optical wave through a dielectric prism to a graphene SPP mode is that the parallel component of the incident wave vector in the prism is equal to the propagation constant of the SPP mode:

210

Plasmonics

(a)

1

2

(b)

1

2


ϵ2 . Bottom figures show the dispersionp offfiffiffiffiffiffiffiffiffiffi the SPP of theffi incident wave pffiffiffiffiffiffiffiffiffiffi ffi mode, as a solid curve, and the parallel component k∥ ¼ k2 sin θ ¼ ω ϵ 2 =ϵ0 sin θ=c, as a dashed line, and k∥ ¼ k3 sin θ ¼ ω ϵ3 =ϵ0 sin θ=c as a dotted line. The SPP mode cannot be excited using the Kretschmann configuration for any incident angle θ because β > k2 ≥ k2 sinθ at all frequencies.

ω k∥ ¼ kp sin θ ¼ c

rffiffiffiffiffi ϵp sin θ ¼ β; ϵ0

ð6:65Þ

where θ is the incident angle, as shown in Figure 6.14. Two configurations that are used for the excitation of metal SPPs are shown in Figures 6.14(a) and (b), known as the Kretschmann configuration and the Otto configuration, respectively. In the following, the possibility of using either arrangement for the excitation of a graphene SPP mode is examined. In the Kretschmann configuration shown in Figure 6.14(a), the prism has direct contact with the graphene sheet without any other dielectric medium between them; thus, effectively, the prism becomes the dielectric medium on the upper side of the graphene sheet such that ϵ p ¼ ϵ 2 . Because k∥ ¼ kp sin θ ¼ k2 sin θ ≤ k2 < β in this case, the phase-matching condition expressed in (6.65) can never be satisfied, as

6.4 Excitation and Detection of SPPs and SPs

211

illustrated in Figure 6.14(a). Therefore, the Kretschmann configuration cannot be used to excite the graphene SPP mode. Note that increasing ϵ 2 does not solve the problem because doing so only scales both dispersion curves in Figure 6.14(a) without changing their relationship. In the case when the graphene sheet is replaced by a metal slab, the Kretschmann configuration is used to excite the SPP mode on the opposite side of the metal slab at its interface with a rarer medium of a permittivity ϵ 1 smaller than ϵ 2 . In the Otto configuration, the prism is located at a small distance above the graphene sheet. The phase-matching condition can then be possibly satisfied at a certain incident angle θ if the following conditions are satisfied: (1) the permittivity, ϵ p ¼ ϵ 3 , of the prism is larger than the permittivity, ϵ 2 , of the dielectric spacer between the graphene sheet and the prism; and (2) the incident angle is larger than the critical angle, i.e., pffiffiffiffiffiffiffiffiffiffiffi θ > θc ¼ sin1 ϵ 2 =ϵ 3 , so that the incident wave does not freely propagates in the spacer but only its evanescent field exists in the spacer. Therefore, it is theoretically possible to use the Otto configuration to excite the graphene SPP mode if the permittivity of the prism is sufficiently large such that an incident angle θ can be found to satisfy the condition expressed in (6.65). However, as seen in Figure 6.6(c), for this condition to be satisfied, the slope of the light line has to be of the order of the group velocity of the graphene SPP mode, which is approximately EF =ℏkF ¼ v F as shown in Figure 6.6. This pffiffiffiffiffiffiffiffiffiffiffi requirement means that c= ϵ 3 =ϵ 0 ≈ v F , or the refractive index of the prism is as large as pffiffiffiffiffiffiffiffiffiffiffi ϵ 3 =ϵ 0 ≈ c=vF ≈ 102 , which is practically impossible. Therefore, the Otto configuration cannot be used to excite a graphene SPP mode, either. Other techniques are needed to excite graphene plasmons.

6.4.2

Coupling Through Near-Field Scanning Optical Microscopy One way to excite and detect the large-β SP mode is to use near-field scanning optical microscopy (SNOM). The technique of SNOM [10] was first demonstrated by Pohl et al. using a configuration similar to the one shown in Figure 6.15(a) [11]. SNOM is now widely used not only for the detection but also for local excitation of surface modes. For the excitation of a graphene SP mode using SNOM as shown in Figure 6.15(a), a fiber is used to deliver light onto the surface of graphene. The end of the fiber is tapered so that the incident light can be confined in a tiny spot at a small distance above graphene. The diameter of the spot has to be much smaller than the wavelength, 2π=β,

(a)

Fiber

(b) AFM Graphene

Figure 6.15 (a) Aperture SNOM. (b) Apertureless SNOM (a-SNOM), also known as scattering-

type SNOM (s-SNOM), which can be incorporated with atomic-force microscopy (AFM).

212

Plasmonics

of the SP mode to be excited. By doing so, the highly confined light field at the tip of the fiber has a component of a finite amplitude that has a propagation constant of a value matching the propagation constant β of the SP mode in the direction parallel to the graphene sheet. For a graphene sheet that has a Fermi energy EF of a few hundred meV and an SP mode of a propagation constant β of the order of kF , the needed aperture of the fiber tip is a few tens of nanometers. Because of the small aperture of the tip, the light in the highly focused spot defined by the tip covers a large range of propagation constants in the k space. Therefore, by tuning the wavelength of the light carried by the fiber, the SP mode can be excited at different frequencies as phase matching is fulfilled. SNOM is also used to detect the evanescent field of the SP mode. The spatial resolution is also determined by the size of the aperture. To achieve the smallest possible aperture, the tapered fiber is usually coated with metal. When the size of the aperture reaches the order of the skin depth of the metallic coating, the effective size of the aperture is determined by the skin depth and can no longer be reduced by physically shrinking the aperture. Therefore, the resolution of the aperture SNOM schematic shown in Figure 6.15(a) is limited within 10 to 100 nm, depending on the light frequency. The small aperture also leads to a small light throughput, which cannot be remedied by further increasing the power of the light source because of potential thermal damage of the tip. To address the issues mentioned above for the aperture SNOM and to achieve a resolution below 10 nm, apertureless SNOM (a-SNOM), also named scattering-type SNOM (s-SNOM), was developed [12]. The schematic of s-SNOM is shown in Figure 6.15(b). Unlike the aperture SNOM, which needs a fiber to deliver the excitation wave, there is no fiber, thus no aperture, for s-SNOM; the metallic tip of s-SNOM is illuminated by a free-space excitation beam. The tip is polarized as a response to the external excitation, creating a strong coupling between the metallic tip and the surface of the sample. Such an interaction gives rise to an amplified and strongly localized field near the apex of the tip, resulting in a sub-10 nm resolution. The near-field information is then obtained by collecting the reflected light. An s-SNOM setup is usually incorporated with atomic-force microscopy (AFM) by using an AFM metallic tip as the scattering center for s-SNOM. The topography of the sample is detected by measuring the forces, such as the van der Waals force and the electrostatic force, between the tip and the surface. Therefore, by using s-SNOM together with AFM, the topography of the graphene sample, such as defects and grain boundaries of graphene, can be concurrently measured while detecting the graphene SP mode [13‒15].

6.4.3

Coupling Through Structural Patterns The SNOM configurations are impractical for a graphene-based plasmonic device because of the large footprint of the typical device. Instead, a free-space wave is coupled to the SP mode via a pattern on the graphene sheet or via a miniature structure in a material that is in proximity of the graphene sheet. This pattern or structure provides the necessary additional momentum to facilitate phase matching between an incident wave and the excited SP mode.

6.4 Excitation and Detection of SPPs and SPs

(a)

213

(b) 3 2

2

1

s

w

w

Frequency,

SP

Frequency,

SP

Wave number,

K

Wave number,

weff Figure 6.16 Schematics of (a) graphene ribbons, and (b) graphene on a grating structure for the

excitation of the SP mode. For normal incidence, the light line overlaps the ω axis.

One example of patterned graphene is graphene ribbons, as shown in Figure 6.16(a). Although the graphene SP mode can be supported by a single ribbon [16], periodic ribbons of the same width w and the same spacing are usually used in an experiment to enhance the response signal arising from the excitation of the SP mode. The propagation direction of the SP mode is perpendicular to the parallel edges of the ribbons. The plasmonic fields are highly confined on the ribbons; outside the ribbons the fields are comparatively weak so that the coupling of the fields between neighboring ribbons is negligible [17]. Therefore, the resonance frequency is not a function of the distance between neighboring ribbons. The resonance frequency, however, highly depends on the ribbon width. The resonance frequency required for the excitation of the SP modes can be found by assuming that the SP fields vanish at the two edges of each ribbon. Then the relation between the plasmonic wavelength λ and the ribbon width w is given by w ¼ mλ=2 or β¼m

π ; w

ð6:66Þ

where m is a positive integer. The resonance frequency is then found by plugging (6.66) in (6.19) for a graphene sheet that is sandwiched between two dielectrics of permittivities ϵ 1 and ϵ 2 [18]. In an experiment, however, it was found that the edges of a ribbon are

214

Plasmonics

often electrically inactive due to defects and irregularities on the edges [19]. Therefore, an effective width weff that is smaller than w has to be used in (6.66) instead of w. Then, 1=2 from (6.66) and (6.19), we find the relation ω ∝ m1=2 weff ; for the fundamental mode, 1=2 m ¼ 1 and ω ∝ weff . Note that this relation is not valid for a ribbon that has a width smaller than 10 nm because of strong quantum effects [20]. Graphene SP modes can also be excited using a grating structure shown in Figure 6.16(b). Because the incident light provides negligible momentum in the propagation direction of a graphene SP mode, the parallel wave vector component required by phase matching for the excitation of the SP mode is predominantly provided by the grating structure. However, if the incident light is not normally incident, two SP modes of different wave numbers βþ and β that propagate in opposite directions can be excited, with one wave number larger than the other one because of the extra momentum provided by the incident light. In the following, we consider only the case of normal incidence. According to the fundamental grating theory [8], the wave number β of an SP mode that can be phase matched to the normally incident wave through the grating is β ¼ mK ¼ m

2π ; Λ

ð6:67Þ

where m is a positive integer, Λ is the period shown in Figure 6.16(b), and K ¼ 2π=Λ is the wave number of the grating. The corresponding resonance frequency is then found by plugging (6.67) into (6.45) for a three-layer system with the substrate thickness ds → ∞ as shown in Figure 6.9 or Figure 6.16(b) without the gaps in the grating structure. In the experiment, the excitation of an SP mode is observed by comparing the transmittance or reflectance spectrum with and without the graphene sheet [18,21]. When the size of the gaps in the grating structure becomes comparable to the plasmonic wavelength, 2π=β, the reflection of a propagating SP mode at the edges of the gaps becomes strong. In this situation, the propagating SP mode becomes essentially localized, similar to the case of graphene ribbons shown in Figure 6.16(a). Then, (6.66) has to be used instead of (6.67) for the determination of β as the coupling of the field between neighboring periods becomes very weak.

References 1. 2. 3. 4.

C. Kittel, Introduction to Solid State Physics (Wiley, 2004). J. M. Liu, Principles of Photonics (Cambridge University Press, 2016). F. Wooten, Optical Properties of Solids (Elsevier Science, 2013). J. Larsson, “Electromagnetics from a quasistatic perspective,” American Journal of Physics, Vol. 75, pp. 230–239 (2007). 5. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Physical Review B, Vol. 80, 245435 (2009). 6. S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Physical Review Letters, Vol. 99, 016803 (2007).

References

215

7. I. T. Lin and J. M. Liu, “Coupled surface plasmon modes of graphene in close proximity to a plasma layer,” Applied Physics Letters, Vol. 103, 201104 (2013). 8. J. M. Liu, Photonic Devices (Cambridge University Press, 2005). 9. R. E. V. Profumo, R. Asgari, M. Polini, and A. H. MacDonald, “Double-layer graphene and topological insulator thin-film plasmons,” Physical Review B, Vol. 85, 085443 (2012). 10. V. Z. Anatoly and I. S. Igor, “Near-field photonics: Surface plasmon polaritons and localized surface plasmons,” Journal of Optics A: Pure and Applied Optics, Vol. 5, p. S16 (2003). 11. D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: Image recording with resolution λ/20,” Applied Physics Letters, Vol. 44, pp. 651–653 (1984). 12. F. Zenhausern, M. P. O’Boyle, and H. K. Wickramasinghe, “Apertureless near-field optical microscope,” Applied Physics Letters, Vol. 65, pp. 1623–1625 (1994). 13. J. Chen, M. Badioli, P. Alonso-Gonzalez, et al., “Optical nano-imaging of gate-tunable graphene plasmons,” Nature, Vol. 487, pp. 77–81 (2012). 14. Z. Fei, A. S. Rodin, G. O. Andreev, et al., “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature, Vol. 487, pp. 82–85 (2012). 15. Z. Fei, A. S. Rodin, W. Gannett, et al., “Electronic and plasmonic phenomena at graphene grain boundaries,” Nature Nanotechnology, Vol. 8, pp. 821–825 (2013). 16. A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Physical Review B, Vol. 84, 161407 (2011). 17. Z. Fei, M. D. Goldflam, J. S. Wu, et al., “Edge and surface plasmons in graphene nanoribbons,” Nano Letters, Vol. 15, pp. 8271–8276 (2015). 18. J. H. Strait, P. Nene, W. M. Chan, et al., “Confined plasmons in graphene microstructures: Experiments and theory,” Physical Review B, Vol. 87, 241410 (2013). 19. H. Yan, T. Low, W. Zhu, et al., “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nature Photonics, Vol. 7, pp. 394–399 (2013). 20. S. Thongrattanasiri, A. Manjavacas, and F. J. García de Abajo, “Quantum finite-size effects in graphene plasmons,” ACS Nano, Vol. 6, pp. 1766–1775 (2012). 21. I. T. Lin, J. M. Liu, H. C. Tsai, et al., “Family of graphene-assisted resonant surface optical excitations for terahertz devices,” Scientific Reports, Vol. 6, 35467 (2016).

7

Photonic Devices

7.1

Plasmonic Waveguides As discussed in Chapter 6, a graphene sheet can support an SPP mode. The field of the SPP mode is highly confined on the graphene sheet; it exponentially decays away from the graphene sheet into the surrounding dielectric media on both sides. This characteristic field distribution is desirable for a waveguide as the footprint of the waveguide in the vertical direction perpendicular to the graphene sheet can be very small. Because the SPP resonance of graphene is in the terahertz spectral region, SPP-based graphene waveguides are terahertz waveguides. The simplest graphene waveguide is shown in Figure 7.1(a); this planar waveguide consists of a graphene sheet on a flat dielectric substrate. However, such a waveguide fails to confine the field in the lateral direction parallel to the graphene sheet. In order to laterally confine the plasmonic field, several schemes have been considered in the literature [1–5]. In Figure 7.1(b), a graphene nanoribbon waveguide is shown. The plasmonic field is confined within the nanoribbon because there is no free electron outside the ribbon. The surface plasmon mode supported by a graphene nanoribbon has a characteristic dispersion relation given by (6.66). Note that when the width of the nanoribbon is below 10 nm, (6.66) is no longer applicable for the description of the plasmonic dispersion due to strong quantum effects [6]. For undoped graphene, it is found that plasmons can also be supported by the edge states that are not present on a homogeneous graphene sheet [3]. The disadvantage of the nanoribbon structure is the difficulty in fabrication; furthermore, imperfect edges can lead to extra edge scatterings that broaden the spectral linewidth of the plasmon resonance. Instead of using nanoribbons, the spatial inhomogeneity needed for the lateral confinement of the plasmonic field can be created by incorporating a structure in the substrate to form a graphene waveguide. The waveguide based on the spatial inhomogeneity of the surface conductivity of graphene is shown in Figure 7.1(c) [1]. The structure consists of a monolayer graphene sheet and an uneven substrate of a permittivity ϵ 1. A spacer of a permittivity ϵ 2 fills the space between the graphene sheet and the substrate. By applying a voltage across the spacer, the chemical potential on the graphene sheet varies with the distance between the graphene sheet and the substrate. Because of the ridge structure of the substrate, a ribbon-like region on the graphene sheet

7.1 Plasmonic Waveguides

(a)

w

(b)

1

(c)

2

1

217

(d) V

2

2

w 1

w

1

(e)

Core

Cladding

1

Cladding

2

w Figure 7.1 Graphene-based plasmonic waveguides. (a) Planar waveguide. (b) Nanoribbon waveguide. (c‒e) Graphene waveguides defined by structured substrates.

above the ridge is formed with a surface conductivity different from that of the surrounding regions. Then, a plasmon mode can be guided with its field effectively confined in this ribbon-like region if the plasmon resonance is supported by the conductivity e σ 2 of the ribbon-like region, but not by the conductivity e σ1 in the surrounding areas. The applied voltage in the structure shown in Figure 7.1(c) allows the plasmon frequency to be tuned by varying the voltage, thus varying the electrostatic doping of graphene. Nevertheless, even without the electrostatic doping through an applied voltage, the plasmonic field can still be confined laterally if the ridge of the substrate is very close to the graphene sheet, as shown in Figure 7.1(d) [5,7]. Such a structure can be considered as consisting of two distinctive regions of different thicknesses of d2 in the geometry shown in Figure 6.9; the thickness d2 for the ribbon-like region above the ridge of the substrate is smaller than that for the rest of the structure. As discussed in Chapter 6 and seen in (6.43), the plasmon resonance is a function of d2 if the plasmonic wavelength is comparable to d2 . Therefore, similar to the case of the structure shown in Figure 7.1 (c), we can find a plasmon resonance that is supported by the ribbon-like region of graphene directly above the ridge, with the plasmonic field laterally decaying away from this region. However, different from the structure shown in Figure 7.1(c), for which the plasmon frequency can be tuned by varying the applied voltage, the plasmonic characteristics of the structure shown in Figure 7.1(d) are not tunable after the structure is fabricated.

218

Photonic Devices

A structure similar to that shown in Figure 7.1(d), but without a spacer between the graphene sheet and the structure of the substrate, is shown in Figure 7.1(e) [2]. For the structures shown in Figures 7.1(d) and (e), the plasmonic waveguide modes can be found by using the effective-index method [8]. For example, for the structure shown in Figure 7.1 (e), the surface plasmon modes of the cladding and core regions are solved individually as two different planar waveguides of the structure shown in Figure 7.1(a) with ϵ ¼ ϵ 1 and ϵ ¼ ϵ 2 , respectively. The wave number β thus obtained for each region gives the effective permittivity of the corresponding region given by ϵ eff ¼ ϵ 0 β2 c2 =ω2. With the different values of ϵ eff found for both core and cladding regions, the plasmonic waveguide modes are numerically solved in the same manner as the modes of a dielectric slab waveguide that has a core width of w [2]. The typical plasmonic dispersion characteristics for the structure shown in Figure 7.1(e) are plotted in Figure 7.2 as thin solid curves. The fundamental mode is marked by the number 1, with the successive high-order modes marked by successive numbers. The thick curves represent the plasmonic dispersion without the presence of the core region or the cladding region, respectively, with ϵ 2 > ϵ 1 . That is, the thick curve marked with the word “Core” assumes that the structure consists of a graphene sheet on a spatially homogeneous substrate of a permittivity ϵ 2, and that marked with the word “Cladding” assumes a spatially homogeneous substrate of a permittivity ϵ 1. As can be seen from Figure 7.2, the dispersion curves are very similar to those of optical waveguide modes; there are multiple modes, and only the fundamental mode has no cutoff frequency because of the lateral symmetry of the structure. The thin curves are always in the region between the thick curves if extrinsic scattering is not considered [2]. The field of a plasmon mode becomes weakly confined when the dispersion curve of the mode approaches the thick curve marked “Cladding,” and it becomes highly confined when the dispersion curve of the mode approaches the thick curve marked “Core.” It is found that the propagation distance is reduced as a trade-off for a better confinement in the lateral direction [2]. It is also possible to construct a plasmonic waveguide using multiple layers of graphene. In Figure 7.3(a), a plasmonic waveguide constructed using two graphene sheets separated by a finite distance is shown [9]. It is found that when the distance between the two graphene sheets is comparable to or smaller than the wavelength of an excited surface plasmon mode, the plasmon fields supported by the two graphene sheets are coupled, resulting in two nondegenerate plasmon modes: the acoustic and optical modes. The typical dispersion curves of the acoustic and optical modes are plotted in Figure 7.3(b), and the real part of the transverse magnetic field of the two modes along the propagation direction are plotted in Figures 7.3(c) and (d). For simplicity, we assume that ϵ 1 ¼ ϵ 2 ; therefore, the dispersion curves shown in Figure 7.3(b) are those for a planar waveguide of the structure shown in Figure 7.1(a), which has no lateral confinement. Because the field of a surface plasmon that is supported by a graphene sheet is confined on the graphene sheet and exponentially decays away from the graphene sheet in the surrounding dielectric medium on either side with a decaying constant of the order of its propagation constant β, it can be sensed away from the graphene sheet

7.1 Plasmonic Waveguides

219

1 Core 0.8

Intraband

4

kF

0.6 3 0.4 Cladding

2 0.2 0

1

0

0.25

0.5

0.75 EF

1

1.25

Figure 7.2 Typical plasmonic dispersion curves of the structure shown in Figure 7.1(e) with ϵ2 > ϵ1 . The first four waveguide modes are plotted as thin solid curves and marked with numbers. The thick curve marked with “Core” is the dispersion curve of a graphene sheet on a spatially homogeneous substrate as the structure shown in Figure 7.1(a) but with a permittivity ϵ ¼ ϵ2, and that marked with “Cladding” is the dispersion curve of a graphene sheet on a spatially homogeneous substrate of a permittivity ϵ ¼ ϵ1. The intraband and interband scattering regions are shaded in light gray and dark gray, respectively. Reprinted from I. T. Lin and J. M. Liu, “Enhanced graphene plasmon waveguiding in a layered graphene−metal structure,” Applied Physics Letters, Vol. 105, 011604 (2014), with the permission of AIP Publishing.

only at a distance of the order of β1 , i.e., of the order of the plasmon wavelength. If the two graphene sheets are sufficiently far apart so that the plasmon field supported by one graphene sheet barely reaches the other graphene sheet, the two fields that are separately supported by the two graphene sheets are essentially decoupled. Then, the acoustic and optical modes become degenerate and have the same dispersion curve, which is shown in Figure 7.3(b) as the gray curve. By contrast, if the separation of the two graphene sheets is sufficiently small so that the plasmon field supported by a graphene sheet reaches the other graphene sheet, the two fields that are respectively supported by the two graphene sheets become strongly coupled, resulting in nondegenerate acoustic and optical modes. The difference between the dispersion curves of these two modes increases as the coupling increases with decreasing distance between the two graphene sheets. Initially, this difference also increases with increasing propagation constant β and correspondingly increasing frequency ω, as can be seen in Figure 7.3(b). However, after reaching a maximum value, the difference between the dispersion curves of the two modes monotonically decreases with increasing propagation constant β, and increasing frequency ω, as the plasmon wavelength becomes smaller and smaller. Eventually, for a sufficiently large value of β, the fields on the two graphene sheets become decoupled again as the plasmon wavelength becomes much smaller than the distance between the two graphene sheets. Then, the two modes becomes degenerate again while their

220

Photonic Devices

(a)

(b) 1 Intraband

kF

0.8 0.6 0.4 1

2

Acoustic

0.2

w

0

0

Optical

0.5

1 EF

(c)

(d) Acoustic mode

40

20 z (nm)

z (nm)

20 0

0 −20

−20 −40

Optical mode

40

−40 0

4

2 x

0

4

2 x

Figure 7.3 Waveguide based on two layers of graphene. (a) Schematic of the structure. (b) Typical plasmonic dispersion characteristics of the acoustic and optical modes with ϵ1 ¼ ϵ2 and EF ¼ 100 meV for both graphene sheets. The gray curve is obtained by setting the distance between the two graphene infinitely large so that the fields on the two graphene sheets are decoupled and the two modes become degenerate. (c,d) The real part of the transverse magnetic field distributions of the acoustic and optical modes along the propagation direction.

dispersion curves merge into the same curve, as can be seen in Figure 7.3(b) for large values of β and correspondingly high frequencies.

7.2

Photodetectors A photodetector detects photons by converting photon energy to electric energy. In the following, we discuss graphene-based photodetectors based on various photodetection mechanisms. Some of the physical quantities and figures of merit

7.2 Photodetectors

221

Table 7.1 Commonly used physical quantities and symbols for photodetectors [8,10]. Physical quantity

Symbol

Unit

Explanation

Photocurrent

Iph

A

Incident power Photon energy Incident photon flux Absorbed photon flux

Pin Eph ¼ ℏω ϕin ¼ Pin =Eph ϕabs ¼ Aabs ϕin

W eV s1 s1

Responsivity

Rph ¼ Iph =Pin

AW1

External quantum efficiency (EQE) Internal quantum efficiency (IQE)

ηe ¼ ðIph =qÞ=ϕin

1

Electric current generated by incident photons Optical power incident on the photodetector Energy of one incident photon of frequency ω Number of incident photons per second Number of absorbed photons per second Aabs ≤ 1 is the absorbed fraction Output photocurrent signal for a given input optical power Number of generated carriers per incident photon Number of generated carriers per absorbed photon

ηi ¼ ðIph =qÞ=ϕabs 1

for photodetectors are summarized in Table 7.1 for the convenience of discussion in this section.

7.2.1

Photovoltaic Effect When a photon is incident on a piece of graphene, an electron–hole pair is created if the photon energy is high enough, as discussed in Chapter 3. Normally the photogenerated electron–hole pairs in graphene recombine shortly after they are generated within tens of picoseconds. However, if a built-in potential is present to pull the electrons and holes apart, the electrons and holes can be separated and may exit the photogeneration region before they recombine, as shown in Figure 7.4(a), creating a net electric current. This mechanism for generating the photocurrent without an externally applied voltage is called the photovoltaic (PV) effect because it results in a photovoltage due to the separation of the electrons and holes. The internally built-in electric field can be produced by a mismatch in the work functions of the graphene sheet and the nearby metal contacts, or be produced by a junction between two regions of graphene of different doping, such as a p–n junction [11]. The simplest device is shown in Figure 7.4(b), which consists of a source, a drain, and a gate electrode. Because of the mismatched work functions of graphene and metal electrodes, a built-in electric field is created near the boundaries of the drain and graphene without an external bias between the two electrodes. When the photons are incident near the boundaries, electron–hole pairs are excited; the electrons and holes are separated by the built-in electric field, and a photocurrent is created. If the time it takes for the carriers to travel out of the photogeneration region is τtr , the transit-time-limited bandwidth of the photodetector based on the structure shown in Figure 7.4(b) is in the range [11,12]:

222

Photonic Devices

(a)

Built-in electric field

(b)

(c) Drain

Drain Source

Source

Vg

Vg Drain

(d)

Si

(e)

Source PV effect + TMDC

Figure 7.4 (a) Photovoltaic effect. The locations of the conduction band, the valence band, and the Dirac point are shown in the presence of an electric field. (b) A simple graphene-based photodetector. Other more advanced photodetectors for high responsivity consist of (c) interdigitated electrodes, (d) a waveguide, or (e) a transition metal dichalcogenide (TMDC) material sandwiched between two graphene sheets.

ftr ¼

2:8 3:5 to : 2πτtr 2πτtr

ð7:1Þ

The actual value of ftr depends on the absorption rate of graphene, which is a function of wavelength. For a current generation region of 200 nm length and a saturation velocity of 5:5  105 m s1 , the transit time τtr is about 0:36 ps, and thus ftr is around 1:5 THz, determined by the upper limit of (7.1) [11]. The bandwidth of a photodetector is also affected by the RC time constant τRC of the device:

7.2 Photodetectors

fRC ¼

1 : 2πτRC

223

ð7:2Þ

A graphene photodetector that has the structure shown in Figure 7.4(b) is experimentally measured and its RC-limited bandwidth fRC is about 40 GHz [11]; therefore, the bandwidth of this graphene photodetector is limited by the RC time constant as the transit-time-limited bandwidth is much larger. The photodetector is experimentally measured to have a responsivity Rph of 0:5 mA W1 , and its IQE ηi is in the range of 6–16 percent [11]. The low responsivity of the structure shown in Figure 7.4(b) is due to the fact that only photons incident on the photogeneration region near the electrode–graphene boundaries can lead to an external electric current; the electron–hole pairs that are generated in the rest of the graphene sheet recombine shortly after the excitation due to the lack of a builtin electric field away from the boundaries. By introducing interdigitated electrodes as shown in Figure 7.4(c), a larger photogeneration region and thus a larger responsivity is possible. The source and drain electrodes are made of different metals so that the mirror symmetry is broken; if they are made of the same metal, the excited current has equal tendency to go from source to drain or vice versa, and thus the total external current is zero. It is found that Rph can be as high as 6:1 mA W1 [13]. However, as a trade-off, fRC decreases to 16 GHz, mainly due to the high capacitance of the large area of graphene used for the device. Another factor that significantly limits the responsivity of the photodetectors shown in Figures 7.4(b) and (c) is the limited absorption of photons by a graphene sheet due to the small absorption coefficient of graphene and the small interaction length of normally incident photons with the thin graphene layer. The absorption of a single layer of graphene for normally incident light in the optical spectral region is about 2.3 percent, which fundamentally limits the responsivity of normally incident graphene-based photodetectors discussed so far. Responsivity can be increased by increasing optical absorption through increasing the length of interaction between photons and graphene. Shown in Figure 7.4(d) is a silicon waveguide embedded in an SiO2 substrate. A sheet of graphene is placed on top of the waveguide with the source and drain electrodes touching the graphene sheet on either side of the waveguide. The evanescent field of a propagating TE mode in the waveguide couples to the graphene layer and excites electron‒hole pairs. Then, the electrons and holes are separated by the built-in electric field at the electrode junction of the drain, resulting in a photocurrent. The observed responsivity is about 0:1 A W1 , and the bandwidth is about 20 GHz [14–16]. The significantly increased responsivity is due to the increased light–graphene interaction length along the waveguide. Another approach to increasing the responsivity through increasing optical absorption is to combine graphene with other materials that have a higher absorption coefficient, as shown in Figures 7.4(e). One group of materials that can be used for this purpose is the transition metal dichalcogenides (TMDCs), such as NbSe2, MoS2, WS2, and TaS2. Like graphite, these materials are composed of

224

Photonic Devices

layers that are weakly bonded by van der Waals attraction. Although TMDC monolayers and graphene are both considered 2D materials, TMDC monolayers have bandgaps in the visible region; therefore, TMDCs can only absorb visible light or light of higher frequencies. For the structure shown in Figure 7.4(e), the electron–hole pairs are mainly generated in the TMDC, whereas the two graphene sheets on its upper and lower sides act as electrodes that provide tunability and the built-in electric field. Then, depending on the direction of the electric field, the generated carriers are transferred to either the upper or the lower graphene sheet, creating a current flow via the PV effect. Because of the efficient light absorption of the TMDC, an EQE ηe as high as 30–55 percent and a high IQE ηi up to 85 percent are found [17,18]. The corresponding Rph is around 0:1 to 0:22 A W1 .

Photo-Thermoelectric Effect Thermoelectric effect is the direct conversion of heat to electricity or vice versa. The conversion from heat to voltage can be presented by the equation xð2

Δ

7.2.2

S Te  ^x dx;

V¼

ð7:3Þ

x1

where S is called the Seebeck coefficient, Te is the electron temperature, and V ¼ V2  V1 is the induced voltage between points x1 and x2 . In the case of graphene, it is found that [10] S∝

1 μ

ð7:4Þ

in the limit that kB T≪ μ, where μ is the chemical potential of graphene. The value of S as a function of μ is plotted in Figure 7.5(b). For large values of μ, SðμÞ follows the trend described by (7.4), but it quickly drops to zero near μ ¼ 0. In the case when the electrons are heated up by incident photons, the thermoelectric effect is called the photo-thermoelectric (PTE) effect. When the electrons are excited to the conduction band by absorbing photons, they quickly reach thermal equilibrium through electron–electron collisions within a few tens of femtoseconds to establish a thermal distribution characterized by an elevated electron temperature Te , which can be very different from the temperature of the lattice and that of the surrounding environment. As discussed in Chapter 2, the optical phonon scattering of electrons is very inefficient due to the high energies of intrinsic optical phonons. Therefore, the electron temperature that is higher than the lattice temperature can persist for many picoseconds without much transfer of heat to optical phonons. Figure 7.5(a) shows the PTE effect for a graphene sheet that has two regions of different chemical potentials μ1 and μ2 . The center of the graphene sheet is heated up by the incident photons. The spatial variation of the light intensity incident on the graphene

7.2 Photodetectors

S (μ )

(b)

(a)

1

μ 4

2

μ2

μ1

225

3

S1

(d)

(c)

pn(−)

Vtg

nN(+) Vtg

Pp(+) Nn(−)

Drain Source pP(−)

np(+)

Vbg Vbg

Figure 7.5 (a) Photo-thermoelectric effect. When the chemical potential μ1 and μ2 of two regions are different, the symmetry is broken and a net photocurrent is generated. (b) Seebeck coefficient as a function of chemical potential μ. (c) A photodetector based on a p‒n junction. The chemical potential of region ① is controlled by the bottom gate voltage Vbg , and the chemical potential of region ② is controlled by both Vbg and the top gate voltage Vtg . (d) Experimentally measured photovoltage Vph as a function of Vbg and Vtg for light incident on the boundary between regions ① and ② [19]. On the white arrow, the Seebeck coefficient S1 of region ① is marked in (b), and the Seebeck coefficients of region ② at different locations on the Vtg  Vbg map from 1 to 4 are marked by the corresponding numbers in (b).

surface creates a gradient in the spatial profile of the electron temperature, which leads to a flow of current away from or toward the center, depending on the sign of S, according to (7.3). Because μ1 ≠ μ2 for the two regions shown in Figure 7.5(a), the Seebeck coefficients are also different for the two regions according to (7.4) or Figure 7.5(b). The broken symmetry results in a net current flowing in a certain direction. The PTE effect is experimentally observed using a p‒n junction shown in Figure 7.5(c) [19]. The chemical potential μ1 in region ① of the graphene sheet can be adjusted by tuning the bottom gate voltage Vbg , whereas μ2 in region ② can be varied by tuning both Vbg and the local top gate voltage Vtg . Therefore, by varying the values of Vbg and Vtg , we can have six different configurations of n–p, N–n, n–N, p–n, P–p, and p–P junctions

226

Photonic Devices

between the two regions, as shown in Figure 7.5(d). Here, the capital letters N and P represent a higher doping region than those represented by the corresponding lowercase letters n and p, respectively. For example, p–P corresponds to a positively doped graphene sheet with μ2 > μ1 > 0. A spatial map of photocurrent Iph on graphene is measured using scanning photocurrent microscopy with the incident laser light focused to a spot at the boundary between regions ① and ②. At the laser spot, the photovoltage given by Vph ¼ Iph R is recorded, where R is the resistance of graphene that depends on Vbg and Vtg . As can be seen in Figure 7.5(d), there is a sixfold sign change for the measured photovoltage as indicated by the plus or minus sign in each region. To explain this sign reversal behavior, we first examine the Seebeck coefficients of the two regions along the white arrow plotted in Figure 7.5(d). Along this arrow at a fixed value of Vbg , μ1 is fixed at a value that corresponds to the Seebeck coefficient marked by S1 in Figure 7.5(b), and μ2 increases along the direction of the arrow as Vtg increases. Four locations corresponding to different values of μ2 are marked with the numbers 1, 2, 3, and 4 on the arrow; the corresponding Seebeck coefficients S2 of region ② are also marked with the numbers in Figure 7.5(b). As can be seen in Figure 7.5(b), at locations 1, 3, and 4 corresponding to n–p, N–n, and n–N regions, we have S1 > S2 , S1 < S2 , and S1 > S2 , respectively, resulting in a twofold sign reversal of the photovoltage. At location 2, the Seebeck coefficient S1 ¼ S2 ; therefore, the symmetry between regions ① and ② is maintained, and the photovoltage or photocurrent is zero as discussed earlier for Figure 7.5(a). The same procedure can be used to check other sign reversals on the map. Note that the PV effect also plays a part in the measured photovoltage as there is a built-in electric field across the boundary of regions ① and ②.

7.2.3

Bolometric Effect The bolometric effect is the change of the conductivity of a material due to heating of the material by incident photons. As shown in Figures 2.8, 2.9, and 2.11, the scattering rate increases with increasing temperature, leading to reduced carrier mobility and, as a result, reduced conductivity. The reduction of surface conductivity caused by heating of graphene is schematically illustrated in Figure 7.6(a). Unlike the PV effect, which does not need an applied voltage, the bolometric effect requires an external bias for the heating-induced conductivity reduction to be measured. The incident photons are detected when a reduction in the photocurrent due to the reduction of surface conductivity caused by photoheating of graphene is observed. Shown in Figure 7.6(b) is a simple transistor structure that is fabricated for the observation of the bolometric effect in graphene [20]. The device is biased with a fixed drain voltage Vd and an adjustable gate voltage Vg . The measured current consists of a DC current Idc and a photocurrent Iph generated by incident light. The photocurrent Iph in turn is composed of Ipv and Ibol arising from the PVeffect and the bolometric effect, respectively. The extracted Iph is plotted in Figure 7.6(c). Unlike the case of a p‒n junction, the PTE effect does not play a major role here, especially when the carrier density is approximately uniform across the graphene channel for a large Vd .

7.2 Photodetectors

227

(b)

(a)

Conductivity

Vd Heating A I ph + I dc

Drain Source

Vg

0 EF (c) I pv 0

I ph = I pv + I bol

I ph

0 Vg VDirac Figure 7.6 (a) Bolometric effect. The surface conductivity of graphene decreases with increasing heating by the incident photons. (b) Graphene transistor for measuring the bolometric effect. (c) Measured photocurrent Iph of the device shown in (b) as a function of Vg  VDirac , where Vg is the gate voltage and VDirac is the gate voltage for the Fermi energy to be tuned at the Dirac point. The PV current Ipv is modeled and plotted as the solid curve [20].

For a positive Vd , the electric field points from drain to source; therefore, according to Figure 7.4(a), Ipv flows from drain to source in the same direction as Idc . The bolometric effect, on the other hand, reduces the current in this direction, tantamount to a current Ibol flowing in the opposite direction. As can be seen in Figure 7.6(a), the bolometric effect is especially strong at a high gate voltage Vg , which creates a high carrier density by electrostatic doping through raising the chemical potential of the graphene sheet either positively or negatively depending on the sign of the gate voltage. Therefore, the magnitude of Ibol increases with the magnitude of jVg  VDirac j, where VDirac is the gate voltage for the Fermi energy to be tuned at the Dirac point. By contrast, Ipv diminishes at high values of jVg  VDirac j because the nonequilibrium carrier density generated by optical excitation at a given optical power decreases with increasing

228

Photonic Devices

jVg  VDirac j due to reduced nonequilibrium carrier temperature. As mentioned in the preceding subsection, the photogenerated hot electrons and holes thermalize among themselves within less than 100 fs to reach nonequilibrium electron and hole distributions characterized by separate chemical potentials and a nonequilibrium carrier temperature Te that is quite different from that of the equilibrium carriers created by doping, including electrostatic doping by the gate voltage. Cooling of these hot carriers to reach equilibrium with the equilibrium carriers is reached through carrier collisions and carrier recombination. A high gate voltage Vg , either positive or negative, for a large value of jVg  VDirac j leads to high electrostatic doping for a high equilibrium carrier density, which increases the collision rate, thus the cooling rate, of the photogenerated nonequilibrium hot carriers. Therefore, the nonequilibrium carrier temperature Te for a given incident optical power decreases with increasing jVg  VDirac j because of the strong electron–electron scattering [20]. Because the photogenerated carrier density depends on the nonequilibrium carrier temperature Te and the chemical potential, it is reduced at high jVg  VDirac j, resulting in a diminished Ipv . From the above discussion, it is clear that the photocurrent at high doping is primarily caused by the bolometric effect, whereas the photocurrent at low doping is primarily caused by the PV effect. Consequently, at low jVg  VDirac j, Ipv dominates and Iph is positive in the same direction as Idc ; at high jVg  VDirac j, Ibol dominates and Iph becomes negative, as shown in Figure 7.6(c). A graphene photodetector based on the bolometric effect, as shown in Figure 7.6(b), is experimentally measured to have a responsivity of 2:5 mA W1 for Vd ¼ 0:1 V [20].

7.2.4

Photogating Effect As in the case of electric gating, the photogating effect changes the carrier density of graphene through the excitation of electron–hole pairs by incident photons. The electron‒hole pairs can be generated in graphene, and then electrons or holes are transferred and trapped in nearby nanoparticles or molecules, leaving the graphene sheet and the nearby charge traps oppositely charged. The excitation of electron‒hole pairs can also take place in the nanoparticles or molecules, and then electrons or holes are transferred to the graphene sheet, as shown in Figure 7.7(a). Therefore, the carrier density of the graphene sheet can be tuned by the illumination intensity, and thus the surface conductivity of the graphene sheet is also changed. If a bias voltage is applied across the graphene sheet, a photocurrent can be detected as a function of light intensity. A graphene transistor coated with PbS quantum dots for the observation of the photogating effect is shown in Figure 7.7(b) [21]. The measured current consists of a DC part Idc , which is driven by a bias voltage Vd , and a photocurrent Iph , which increases with the light intensity. Unlike the PV effect for which the photogenerated electron‒hole pairs recombine in picoseconds, the photogating effect has a long carrier lifetime. It takes 20 ms to 1 s before charge carriers of the photogating effect recombine. If it takes τtr for a free charge carrier in the graphene channel to travel from source to

7.2 Photodetectors

(a)

229

(b)

Vd +

+

-

Vg

A I ph + I dc

Drain +

Source

+

Figure 7.7 (a) Photogating effect. (b) Graphene transistor for the observation of the photogating

effect.

drain and τ life before this free charge carrier recombines with a charge carrier that is trapped in a nearby nanoparticle, a photoconductive gain can be defined as G ¼ τ life =τtr . For a mobility of 1  103 cm2 V1 s1 , a channel length of 10 μm, and a bias voltage of Vd ¼ 1 V, the transit time τtr is of the order of 1 ns. With τ life ¼ 20 ms, we find a photoconductive gain of G ¼ 2  107 , which leads to a high responsivity of Rph ≈ 5  107 A W1 and an EQE of ηe ≈ 25 percent [21]. The photoconductive gain can be further increased for a correspondingly increased responsivity by increasing τlife , though at the expense of reducing the response speed of the photoconductive detector.

7.2.5

Terahertz Photodetectors In Chapter 6 we learned that the plasmon resonance of graphene is in the terahertz spectral region due to the low chemical potential of most extrinsic graphene. Therefore, one way to enhance the absorption rate of graphene in the terahertz region for effective graphene-based terahertz photodetection is to incorporate graphene with a structure that can provide the surface plasmon resonance modes to increase the light–matter interactions. An example is shown in Figure 7.8 for a log-periodic circular-toothed antenna on a graphene sheet. For a specific antenna measured in Reference [22], the plasmon resonance frequencies of the structure are found to be 0.4, 0.7, 1, and 1.4 THz from numerical simulation; therefore, the photodetector operates in the terahertz spectral region. Because of the high carrier scattering rate of graphene, sharp plasmon resonances are not observed. Instead, a broadband terahertz detection spectrum is obtained. Thus the terahertz photodetector operates in the overdamped region. A responsivity of Rph ≈ 100 mV W1 is extracted from the measurement of the photovoltage between the drain and the source under the illumination of a terahertz beam at a frequency of 0.3 THz [22]. One can also use a grating structure to effectively increase the light–matter interactions between graphene and the terahertz photodetector. As discussed in Chapter 6, a graphene SP mode can be excited by phase matching the parallel

230

Photonic Devices

Drain

Source

Gate Figure 7.8 Terahertz photodetector based on a log-periodic circular-toothed antenna on a graphene

sheet.

wave vector component provided by the grating structure [23]. Therefore, a grating-incorporated terahertz photodetector can effectively transfer the absorbed photon energy to the surface plasmons. As the surface plasmons are collectively oscillating electron waves, this electronic signal can be detected in terms of photocurrent or photovoltage.

7.3

Optical Modulators An optical wave is characterized by amplitude, phase, frequency, wave vector, and polarization. In optical communications, one or more of these physical quantities are modulated so that the information can be carried by the propagating optical wave, as shown in Figure 7.9. This modulated signal is then transmitted either through free space or through an optical waveguide such as one of the plasmonic waveguides discussed in Section 7.1. The modulated signal is then detected and converted into an electric signal by a photodetector, such as one of the graphene photodetectors discussed in Section 7.2. In graphene research on optical modulation, most attention is focused on amplitude modulation. Thus, the amplitude of the optical field is temporally modulated to carry a message as a function of time. Accordingly, the power of the light is temporally modulated, which is then converted into a modulated photocurrent by a photodetector. Optical modulators are discussed in this section, and terahertz modulators are discussed in Section 7.4. For an optical modulator, the signal light propagates inside an optical waveguide with graphene cladding; the waveguide can be a rectangular dielectric waveguide or a circular fiber. The benefit of using a waveguide is that the interaction of graphene and the signal light increases with the length of the waveguide. Therefore, the performance, such as the modulation depth, can be enhanced. For terahertz modulators, however, the signal light usually propagates in free space due to the lack of efficient terahertz waveguides. Therefore, increasing the modulation depth is a major challenge for terahertz modulators.

7.3 Optical Modulators

Modulator S=E

Waveguide

231

Photodetector

H 1011010

EH Modulation signal Figure 7.9 Optical communication system consisting of a modulator, a waveguide, and a photodetector. Solid and dashed curves represent optical and electric signals, respectively.

7.3.1

Waveguide-Based Optical Modulators Various graphene optical modulators for amplitude modulation of an optical wave have been proposed and reported in the literature. One example is shown in Figure 7.10(a), where the optical wave propagates in a rectangular dielectric waveguide, such as a silicon waveguide. On top of the waveguide is a sheet of graphene; the chemical potential of this graphene sheet can be tuned by an electrical modulation signal. When the chemical potential is near the Dirac point such that jμj ≤ ℏω=2, where ℏω is the photon energy of the guided optical wave, the evanescent tails of the guided optical wave couples to the graphene sheet, causing optical absorption by the graphene sheet and resulting in low transmittance. By contrast, when the chemical potential is tuned sufficiently far away from the Dirac point such that jμj > ℏω=2, the guided optical wave is transmitted through the waveguide without being absorbed by the electrons in the graphene sheet, resulting in high transmittance. For the graphene-based amplitude modulator shown in Figure 7.10(a), the modulation electrical signal that carries a message is used to tune the chemical potential μ of the graphene sheet. When μ is near the Dirac point such that jμj ≤ ℏω=2, the incident photons are absorbed by the electrons in the graphene sheet through interband transitions, as also shown in Figure 3.1; then, the transmittance is low. When μ is tuned away from the Dirac point such that jμj > ℏω=2, interband transition is prohibited because of Pauli blocking; then, the photons are not absorbed and the transmittance is high. The modulation depth m, which can be defined as how much the modulation signal affects the light output, can be expressed as m¼

ðT max  Tmin Þ=2 T max  Tmin ¼ ; Tave T max þ Tmin

ð7:5Þ

where T max , Tmin , and Tave are the maximum transmittance, the minimum transmittance, and the average transmittance, respectively. An alternative definition of the modulation depth is m¼

T max  Tmin Tmin ¼1 : T max T max

ð7:6Þ

Photonic Devices

(a) Modulation signal

Signal

μ=

(b)

μ=

2

2 Ideal Realistic

1 Transmittance

232

T =0 T =1

T =1 0 Chemical potential (modulation signal)

μ

Figure 7.10 (a) Waveguide-based graphene optical modulator. The modulation signal is coupled to the signal by modulating the chemical potential of the graphene sheet that is placed on top of the waveguide. (b) Transmittance of the optical modulator as a function of the graphene chemical potential μ tuned by the modulation signal. For an ideal optical modulator in the limits of zero temperature and infinite length, the transmittance is 1 when jμj > ℏω=2, and 0 when jμj ≤ ℏω=2. For a realistic modulator of a finite length operating at a nonzero temperature, the high transmittance never reaches 1, and the low transmittance never reaches 0.

Sometimes the modulation depth is simply defined as m ¼ 10 log

Tmin : T max

ð7:7Þ

If the reflectance is measured instead, the modulation depth is defined by replacing the transmittance T with the reflectance R in (7.5)–(7.7). For an ideal modulator in the limits of zero temperature and infinite length, the transmittance is either 0 or 1. The transmittance drops from 1 to 0, or jumps from 0 to 1 sharply at jμj ¼ ℏω=2, as shown in Figure 7.10(b). Because T max ¼ 1 and Tmin ¼ 0 for an ideal optical modulator, we have m ¼ 1, or m ¼ 100 percent, from (7.5).

7.3 Optical Modulators

233

The experimentally measured modulation depth of a realistic modulator is lower than that of an ideal modulator, as illustrated in Figure 7.10(b). As in the case of the waveguide-based photodetector shown in Figure 7.4(d), the interaction length increases with increasing waveguide length; thus the absorption of light and the modulation depth can be enhanced by increasing the waveguide length. Because a realistic modulator has a finite interaction length and operates at a nonzero temperature, the transmittance never reaches 1, nor 0, and the boundaries between low- and high-transmittance regions are fairly smooth. Therefore, the modulation depth m is usually smaller than 100 percent. A graphene-based optical modulator of the structure shown in Figure 7.10(a) has been experimentally measured to have a modulation depth of 0:1 dB for 1 μm length of the waveguide according to (7.7) [24], which gives m ¼ 1:15 percent according to (7.5), or m ¼ 2:28 percent according to (7.6), for a waveguide of 1 μm length. Like the RC-limited bandwidths of graphene-based photodetectors, the bandwidth of a graphene optical modulator is mainly limited by the RC time constant, not by the carrier transit time because of the high mobility of graphene. The bandwidth of a device that has the structure as shown in Figure 7.10(a) is experimentally measured to be about 1 GHz with a broad operation spectrum that ranges from 1.35 to 1:6 μm under ambient conditions [24].

7.3.2

Fiber-Based Optical Modulators To improve the bandwidth by avoiding the bandwidth limitation of electrical modulation imposed by the RC time constant, an all-optical approach is adopted as shown in Figure 7.11. In Figure 7.11(a), a fiber-based graphene optical modulator is schematically drawn [25]. One section of a single-mode fiber is tapered into a microfiber of 1:2 μm in diameter and about 2 mm in length, and part of the microfiber is covered by a bilayer graphene cladding of 16 μm in length. A series of optical pump pulses are sent into the fiber as the modulation signal along with a CW optical carrier wave being modulated. The wavelengths of the pump pulses and the optical carrier wave are 1064 and 1550 nm, respectively. Instead of tuning the chemical potential of graphene via electrostatic gating, the modulation is achieved by optically pumping electrons of the graphene cladding in the states at the energy level of E ¼ ℏω=2 in the valence band to the states at the energy level of E ¼ ℏω=2 in the conduction band to block further interband absorption of the optical carrier wave through Pauli blocking. This process is saturable absorption: The absorption of the optical carrier wave saturates when the power density of the optical pump is high enough such that no more electrons near the energy level of E ¼ ℏω=2 in the valence band are available for excitation and absorption. With such a high pump power, the optical carrier wave at 1550 nm cannot be absorbed any further and the transmittance of the optical carrier wave reaches its maximum, as shown in Figure 7.11(b). Therefore, one can tune the power density of the pump to modulate the transmittance of the optical carrier wave, thus imposing the message of the modulation signal carried by the pump pulses on the optical carrier wave. Usually the photon energy of the pump is higher than that of the optical carrier wave so the electrons in graphene

Photonic Devices

(a)

Microfiber

Modulation signal Modu ul

Optical carrier wave

(b)

1 Transmittance

234

0 Pump power (modulation signal) Figure 7.11 (a) Fiber-based graphene optical modulator. One section of the single-mode fiber is tapered into a microfiber with bilayer graphene cladding. (b) Transmittance of the optical modulator as a function of pump power tuned by the modulation signal. The optical carrier wave is optically modulated by the modulation signal in the form of the pump pulses.

can be excited to sufficiently high energy levels to block the absorption of the optical carrier. For the fiber-based graphene optical modulator shown in Figure 7.11(a) with the parameters mentioned above, it is found that the transmittance of the optical carrier wave at 1550 nm is 20 percent when the pump pulses are absent, and 27.6 percent with the pump pulses present [25]. Therefore, using (7.5), the modulation depth is 16 percent, or m ¼ 1 percent for 1 μm length of microfiber covered by graphene cladding. For this device, an alternative definition of the modulation depth m ¼ ðTpump  Tno pump Þ=Tno pump is also defined, which gives a modulation depth of 38 percent, where Tno pump ¼ Tmin ¼ 20 percent is the transmittance without pump, and Tpump ¼ T max ¼ 27:6 percent is the transmittance with pump. Because of the fast relaxation time of approximately 2 ps for the electrons that are excited to the conduction band by the pump pulses, the bandwidth of this modulator is around 200 GHz.

7.4 Terahertz Modulators

235

Table 7.2 Resonance modes and the corresponding substrate structures for graphene-based terahertz modulators. Structures Flat substrate

Resonance modes

References

–

[26]

Grating Metamaterial

Rings

[30] Spoof SPP modes [27‒29]

Split rings Fabry‒Pérot (FP) Grating

[31] [32‒35]

FP modes Graphene SPP modes

[36‒38] [23,39]

Terahertz wave

Modulation signal

Figure 7.12 Graphene terahertz modulator. By modulating the chemical potential of graphene, the intraband absorption changes accordingly, thus imposing the modulation signal on the terahertz wave.

7.4

Terahertz Modulators The simplest graphene terahertz modulator is shown in Figure 7.12. The graphene sheet is placed on a flat substrate, and its chemical potential is modulated through electrostatic gating. Unlike the optical modulators discussed in the preceding section, a terahertz modulator is usually not in a waveguide form; the terahertz wave propagates in free space due to the lack of low-loss terahertz waveguides. Furthermore, the physical mechanism responsible for modulating a terahertz wave is different from that for modulating an optical wave. As discussed in Chapter 4, the terahertz response of graphene is dominantly contributed by intraband transitions of electrons, whereas the response in the optical spectral region is primarily determined by interband transitions. Unlike interband transitions that can be Pauli blocked by tuning the chemical potential away from the Dirac point, as shown in Figure 7.10(b), intraband transitions cannot be effectively blocked. Therefore, for the structure shown in Figure 7.12, the terahertz wave that is being modulated cannot be completely “turned off”; the transmittance can barely be tuned, and the modulation depth is very small [26]. To enhance the modulation depth, resonance modes are introduced by adopting sophisticated structures, which are discussed below. The resonance modes associated with various substrate structures are listed in Table 7.2. By tuning the chemical potential of graphene, a resonance mode can either be excited or destroyed, resulting in enhanced

236

Photonic Devices

Resonance region

Transmittance

1

0 Chemical potential (modulation signal) Figure 7.13 Transmittance of a graphene terahertz modulator as a function of the graphene chemical

potential μ tuned by the modulation signal. The resonance is provided by a substrate structure or by a hybrid structure of graphene and the substrate.

or diminished transmittance, as illustrated in Figure 7.13. The excitation of a resonance mode does not necessarily lead to enhanced transmittance but sometimes enhanced reflectance accompanied by diminished transmittance, depending on the resonance nature of the mode [28]. In the following, terahertz modulators based on various resonance modes are discussed.

7.4.1

Spoof Surface Plasmon Polariton Modes One way to introduce resonance modes is to use a metallic substrate that has a periodic pattern, as shown in Figure 7.14(a) [29]. This type of substrate is known as a metamaterial. A metamaterial is an engineered material that cannot be found in nature. The periodic pattern of a metamaterial mimics the periodic structure of atoms or molecules of a natural material. When the wavelength of an electromagnetic wave is much larger than the structural dimensions of the periodic pattern, the details of the miniature structure of the metamaterial cannot be resolved by the wave. In this situation, a metamaterial can be well described by some macroscopic physical parameters such as refractive index and permittivity. By carefully engineering the metamaterial, an “effective” plasma frequency of the structure can be designed within the terahertz spectral region if a metal is used to construct the metamaterial. This effective plasmon resonance is called a spoof surface plasmon polariton (spoof SPP) [27‒29]. Clearly, this resonance frequency is not the intrinsic plasma frequency of the metal being used to build the metamaterial; therefore, the resonance mode “spoofs,” or mimics, the real metal SPP mode. Because the resonant field of a spoof SPP mode is sometimes confined in the gaps in the metamaterial pattern, a spoof SPP mode is sometimes also called a cavity mode. The graphene‒metamaterial hybrid devices that have been reported in the literature have structures that are similar to the one shown in Figure 7.14(a). Some of the patterns studied in the literature are shown in the insets of Figure 7.14(a); from left to right are 1D grating [23,30,39], 2D holes [27,28], rings [31], and split ring resonators (SRRs) [32‒35]. Depending on the design and the physical parameters of the structural pattern, different spoof SPP modes are excited and utilized.

7.4 Terahertz Modulators

237

(a) y d2

z h

d1

x

w Metamaterial y x Slit/Groove

Hole

Ring

SRR

(b)

d2 h

d1 w

0.5

μ2

Transmittance

(c)

μ1 0 2.5

3 Frequency (THz)

3.5

Figure 7.14 (a) Sketch of a graphene–metamaterial structure. The miniature structure of the

metamaterial can take the form of periodic slits, holes, rings, or split ring resonators (SRRs), as shown in the insets. (b) Typical field distribution of a spoof SPP mode. (c) Representative transmittance curves as a result of the excitation of a spoof SPP mode for two different chemical potentials μ1 ≠ μ2 .

The resonance frequency v of a spoof SPP mode is determined by the design parameters of the metamaterial structure. For normally incident radiation of a wavelength λ that is much larger than the structural dimensions of the metamaterial patterns, the resonance frequencies of 1D grooves with h < d1 , 1D slits with h ¼ d1 , 2D holes, and 2D rings, as shown in the insets of Figure 7.14(a), are given approximately by the relations v ≈ c=4d1 neff [27], v ≈ c=2d1 neff [28], v ≈ c=2wneff [29], and v ≈ c=2πrneff [31], respectively, where neff is the effective index of the mode, d1 is the thickness of

238

Photonic Devices

the metamaterial structure, w is the width of the hole, and r is the radius of the ring. If the ring is cut so that the symmetry is broken, the resulting SRR structure can be regarded as pffiffiffiffiffiffiffi an LC circuit with a new resonance frequency of v ≈ c=2π LC. Taking the 1D grating shown in Figure 7.14(b) as an example, the characteristic transmittance with one resonance peak is shown in Figure 7.14(c) for the excitation of the spoof SPP mode when the graphene sheet has a chemical potential of μ ¼ μ2 . By tuning the chemical potential μ, we effectively change the boundary condition on one side of the gap to enhance or suppress the excitation. If a graphene SPP mode is also excited, the graphene SPP mode and the spoof SPP mode can be coupled when their frequencies are matched. As a result, the spoof SPP mode is split into two coupled modes, as can be seen in Figure 7.14(c) for two transmittance peaks with the chemical potential tuned at μ1 . Depending on the structure, the modulation depth is found to range from 6 percent to 50 percent [31,35]. One major issue with the devices based on the excitation of spoof SPPs is that the physical dimensions of the pattern has to be large so that the resonance frequency can be in the terahertz spectral region. The typical lengths of the relevant structural parameters are in the range of 50 to 100 μm for a resonance frequency in the sub-terahertz region [31,33]. Furthermore, the resonance frequency is fixed by the structural parameters and thus cannot be tuned by adjusting the chemical potential of graphene unless a spoof SPP mode is coupled with other quasiparticles.

7.4.2

Fabry–Pérot Modes Another possible excitation for a terahertz graphene modulator is the Fabry‒ Pérot (FP) mode, as shown in Figure 7.15. A metallic grating is introduced to form an FP cavity with graphene. The periodic slits of the grating also allow the terahertz wave to go through the metallic substrate. If the grating is replaced by a gapless metallic substrate, then the reflectance is measured but not the transmittance. Note that the distance between the graphene sheet and the metallic grating has to be of the order of the terahertz wavelength so that terahertz FP modes are supported. By moving the chemical potential away from the Dirac point, the graphene sheet changes from weakly conducting to highly conducting. As a result, the boundary condition on the graphene side of the FP cavity is modified, and the corresponding resonance frequency of the FP mode changes accordingly, as shown in Figure 7.15(b). Because the field of the FP mode is more concentrated on the graphene sheet compared to the field of a structure without the FP cavity, the modulation depth can be enhanced. Indeed, a modulation depth as high as 64 percent given by (7.6) is measured [38]. As a trade-off, the operational bandwidth is sacrificed because the resonance frequencies of the FP modes are predefined by the graphene‒substrate distance. The resonance frequencies cannot be tuned effectively even with a large doping of the graphene sheet. Like a graphene terahertz modulator based on a spoof SPP, this type of device also has a large footprint. For a resonance frequency in the

7.4 Terahertz Modulators

239

(a)

d2 d1

(b)

1

Transmittance

μ1

μ2 0.5 1.5

2.5 Frequency (THz)

3.5

Figure 7.15 (a) Typical field distribution of an FP mode for a terahertz modulator consisting of a graphene sheet and a metallic grating. (b) Representative transmittance curves as a result of the excitation of an FP mode for two different chemical potentials μ1 ≠ μ2 .

sub-terahertz region, the thickness d2 of the dielectric spacer layer between the graphene sheet and the metallic metamaterial substrate ranges from around 100 μm to a few hundred micrometers [36,37].

7.4.3

Graphene Surface Plasmon Polariton Modes For the periodic structure shown in Figure 7.16, it is also possible to excite graphene SPPs, as shown in Figure 7.16(a). By increasing the chemical potential of graphene, the electron density and the conductance increase, and the resonance frequency of a graphene SPP mode, corresponding to the transmittance trough, is blueshifted, as shown in Figure 7.16(b) where the chemical potential μ2 > μ1 . Therefore, by adjusting the chemical potential of graphene, the resonance frequency can be tuned and thus the transmitted signal can be switched on or off. Compared to the devices based on other types of surface excitation, the device shown in Figure 7.16(a) has the smallest footprint because of the strong confinement of the graphene SPPs. This type of device also benefits from the large tunability of the resonance frequency through electrostatic doping [29]. For a scattering rate of 1  1012 s1 , the modulation depth is around

240

Photonic Devices

(a)

d2 d1 w

(b)

0.5

Transmittance

μ1

μ2

0 0

0.5 Frequency (THz)

1

Figure 7.16 (a) Typical field distribution of a graphene SPP mode for a terahertz modulator consisting of a graphene sheet and a metallic grating. (b) Representative transmittance curves as a result of the excitation of a graphene SPP mode for two different chemical potentials μ1 ≠ μ2 .

90 percent at 0:6 THz and even higher at 1 THz, as shown in Figure 7.16(b). This value is larger than the modulation depths of the devices based on the spoof SPPs shown in Figure 7.14, based on the FP modes shown in Figure 7.15 [37,38], and based on monolayer graphene on a flat substrate [26,35,36]. As most of the field of a graphene SPP is confined on the graphene sheet, the modulation depth is very sensitive to the quality of the graphene sheet; the modulation depth is significantly reduced if the scattering rate of the graphene sheet is too high [39].

7.4.4

Dispersion Curves of Various Terahertz Modulators The dispersion curves of the spoof SPP modes, the FP modes, and the graphene SPP modes discussed in the preceding subsections can be found by locating the resonance frequency of each mode in the transmittance spectrum. Taking the structure shown in Figure 7.16(a), for example, the dispersion map is plotted in Figure 7.17(a) as a function of d1 (right panel) and d2 (left panel) for normally incident terahertz waves by taking the parameters of w ¼ 0:1 μm and Λ ¼ 3 μm for the metallic grating. Three regions are clearly seen where the graphene SPP modes, the spoof SPP modes, and the FP modes are respectively dominant. The first two graphene SPP modes are marked by black dotted

Figure 7.17 (a) Transmittance as a function of d1 and d2 based on the structure shown in Figure 7.16(a). The parameters are w ¼ 0:1 μm and Λ ¼ 3 μm for the metallic grating, and chemical potential μ1 ¼ 100 meV and scattering rate γ ¼ 1  1012 s1 for the graphene sheet [40]. The permittivity is ϵ0 above the graphene sheet; below the graphene sheet, the grating is embedded in a dielectric medium that has a permittivity of 3:9ϵ0 . On the map, the areas of the dominant modes are separated approximately by the white dashed curves. The dotted curves roughly mark the first two modes of the graphene SPP modes, the spoof SPP modes, and the FP modes. (b,c) Electric field component Ex of two graphene SPP modes for d1 ¼ d2 ¼ 10 nm and v ¼ 0:6 THz and 1:2 THz labeled (b) and (c) in (a), respectively.

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Photonic Devices

curves that follow the transmittance troughs within the graphene SPP region. The fields of these graphene SPP modes are confined on the graphene sheet, as shown in Figures 7.17(b) and (c) for the first-order and second-order graphene SPP modes, which are labeled with (b) and (c), respectively, in Figure 7.17(a). Above 10 THz in Figure 7.17(a), the dispersion curves of graphene SPP modes are no longer observable because the Drude conductivity of graphene that supports these SPP modes drops significantly at high frequencies. As the frequency of the terahertz wave increases, excitation of the metal SPP mode becomes possible. Because ϵ m →  ∞ for a metal in the terahertz spectral region, the pffiffiffiffiffiffiffiffiffiffiffi metal SPP mode satisfies the Rayleigh condition λ ¼ ϵ 1 =ϵ 0 Λ [28], which gives a transmittance trough at v ¼ c=λ ≈ 50:64 THz seen in Figure 7.17(a) for ϵ 1 ¼ 3:9ϵ 0 and a metallic grating of a period Λ ¼ 3 μm. To reduce the frequency of the metal SPP mode to the order of 1 THz, the grating period has to be increased to the order of Λ ¼ 150 μm. As the thickness d1 of the grating increases, excitation of spoof SPP modes in the terahertz spectral region becomes possible. The first two modes are marked by the white dotted curves in the spoof SPP region. In the low-frequency region where the coupling of a spoof SPP mode with the metal SPP mode is negligible, the field of a spoof SPP mode is confined inside the structural gap in the metamaterial substrate, as shown in Figure 7.14(b). In this case, a spoof SPP mode is also called a cavity mode, which plays an important role in extraordinary optical transmission [41,42]. The spoof SPP modes can also couple with the graphene SPP modes to open up gaps in the dispersion curves, as indicated by the white arrows in Figure 7.17(a). The locations of these gaps in the dispersion curves can be tuned by adjusting the chemical potential of the graphene sheet; thus, the transmittance of the graphene‒metamaterial structure can be tuned accordingly, as shown in Figure 7.14(c). When the wavelength of the terahertz wave is comparable to the spacer thickness d2 , an FP cavity is formed between the graphene sheet and the metallic grating of the substrate, with the boundary conditions determined by the chemical potential of the graphene sheet. The mth-order resonance frequency of an FP cavity characterized by a refractive index of n2 and a thickness of d2 is given by v ¼ mc=2n2 d2, where m is a positive integer. The first two modes are plotted as dotted curves in the FP region in the left panel of Figure 7.17(a). The curves approximately agree with the simulation results. The discrepancy is caused by the finite conductance of graphene that makes the FP cavity imperfect. The transmittance is tuned by adjusting the chemical potential of the graphene sheet to achieve the functionality of the device shown in Figure 7.15. For a terahertz modulator, it is generally desirable to have a small footprint. In this respect, modulators that are based on the excitation of graphene SPP modes are better than those based on spoof SPP modes or FP modes because of the larger values of d1 and d2 required for spoof SPP modes and FP modes, respectively, than those required for graphene SPP modes, as shown in Figure 7.17(a). Also, metal SPP modes are not useful for terahertz modulators because of their high resonance frequencies, unless the period of the grating is made very large to lower the resonance frequency, which again leads to a large footprint.

7.4 Terahertz Modulators

(b)

105

100

100

95

95

A A1 (%)

R R1 (%)

(a) 105

90 85 80

243

90 85

3

4

5 6 7 Frequency (THz)

8

9

80

3

4

5 6 7 Frequency (THz)

8

9

Figure 7.18 Measured spectra of (a) reflectance R and (b) absorbance A for different carrier

densities tuned by adjusting the gate voltage for a structure shown in Figure 7.16 [30]. The curves of light to dark gray, as the arrow indicates, represent carrier densities of 1.6, 2.2, 2.9, 4.6, 6.6, 9.0, 10.3, and 11:7  1012 cm2 , respectively. The arrows also indicate the direction of frequency shift as the carrier density increases. The reflectance R1 and absorbance A1 are measured when the carrier density is at its minimum (1:6  1012 cm2 ). Reprinted with permission from M. M. Jadidi, A. B. Sushkov, R. L. Myers-Ward, et al., “Tunable terahertz hybrid metal–graphene plasmons,” Nano Letters, Vol. 15, pp. 7099–7104 (2015). Copyright 2015 American Chemical Society.

A graphene SPP modulator using a metallic grating based on the structure shown in Figure 7.16(a) is fabricated with Λ ¼ 7 μm and w ¼ 0:35 μm [30]. As shown in Figure 7.17, the plasmon resonance can be found by locating the transmittance minimum. In Reference [30], both reflectance and transmittance are recorded, and it is found that the plasmon resonance is around 5–6.5 THz depending on the chemical potential of the graphene sheet, as shown in Figure 7.18. The carrier density of graphene is electrostatically tuned throughout the terahertz range from 3 to 9 THz by applying a gate voltage. As the chemical potential is tuned away from the Dirac point, the optical conductivity increases and the excitation of surface plasmon becomes possible. Therefore, it is found that the absorption of photons is minimized, and both the reflectance and the transmittance reach their maximum values of R ¼ R max and T ¼ T max when the carrier density is low. By contrast, when the carrier density is high, the absorption of photons is maximized, and the reflectance and the transmittance reach their minimum values of R ¼ Rmin and T ¼ Tmin . Using (7.6) with T replaced by R, m¼1

Rmin ; R max

ð7:8Þ

the modulation depth is found to be roughly m ¼ 0:2, or m ¼ 20 percent. A blueshift of the resonance frequency from 5 THz to 6.5 THz is also observed when the chemical potential increases; the direction of this shift is indicated by the

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Photonic Devices

arrows shown in Figures 7.18(a) and (b). The phenomenon of blueshift is consistent with the theory. Because the period Λ is fixed, the propagation constant β of the surface plasmon is fixed, given by (6.67). Therefore, as the chemical potential increases, the optical conductivity increases and, according to (6.24), the plasmon resonance frequency ω also increases.

7.5

Saturable Absorber As we discussed in the Section 3.7 of Chapter 3, in the optical region the absorbance of N-layer graphene in the free space is approximately given by (3.132) as A0 ¼ NZ0 e σ 0 ¼ Nπα;

ð7:9Þ

pffiffiffiffiffiffiffiffiffiffiffi σ 0 ¼ e2 =4ℏ ≈ 60:8 μS is where Z0 ¼ μ0 =ϵ 0 ≈ 120π Ω is the impedance of free space, e the high-frequency surface optical conductivity of monolayer graphene defined in (3.78), and α ¼ e2 =4πϵ 0 ℏc ¼ e2 Z0 =4πℏ ≈ 1=137 is the fine-structure constant. Compared to the symbol A used in (3.132), a subscript “0” is used for A0 in (7.9) to represent the unsaturated absorbance, which will be explained shortly. From (7.9), we find the unsaturated absorbance of monolayer graphene to be about A0 ¼ 2:3 percent for N ¼ 1, consistent with the experimental measurement [43]. When the incident light has a high intensity, the optical excitation can be strong enough that the excited electron density in the conduction band becomes so high that further excitation of electrons becomes increasingly inefficient because of Pauli blocking, as shown in Figure 4.6. In such a case, the optical absorption of graphene is saturated, and (7.9) cannot be used for the absorbance of optically saturated graphene. At a sufficiently high light intensity, the graphene sheet is completely bleached and no longer absorbs the light. To account for the saturable absorption of graphene, a simple saturation model is used: Aðe nÞ ¼

A0 ; 1þe n =e n sat

ð7:10Þ

n is the photoexcited where A0 is the unsaturated absorbance given by (7.9), e surface electron–hole carrier density of the illuminated graphene sheet, and e n sat is the saturation surface carrier density. According to (7.10), we find that the maximum absorbance is the unsaturated absorbance for Að0Þ ¼ A0. When the light intensity is low such that the photoexcited electron‒hole density is negligible, the absorbance is unsaturated such that Aðe n → 0Þ ¼ A0 . When the light intensity is sufficiently high such that e n¼e n sat , the absorbance drops to one-half of A0 because Aðe n sat Þ ¼ A0 =2 from (7.10). Eventually, the absorbance approaches zero if the light intensity is very high, such that e n =e n sat ≫ 1. In the case when the incident light is a continuous wave, the photoexcited electron‒hole density reaches a constant value in a steady state. Then, we can write the rate equation as

7.5 Saturable Absorber

e ∂e n I n ¼ Aðe n Þ  ¼ 0; ∂t ℏω τ

245

ð7:11Þ

where I is the light intensity, ℏω is the photon energy, and τ ¼ γ1 is the relaxation time of the photoexcited electron–hole density. Here for simplicity and for illustration purposes, we assume that τ is independent of e n . Then, from (7.11), we obtain e n¼

Iτ Aðe n Þ: ℏω

ð7:12Þ

By plugging (7.12) into (7.10), we obtain AðIÞ ¼

A0 ; 1 þ I=Isat

ð7:13Þ

2ℏωe n sat : τA0

ð7:14Þ

where Isat ¼

As can be seen from (7.10) and (7.13), Isat is the saturation intensity such that when I ¼ Isat , we have e n¼e n sat , and the absorbance drops to one-half of A0 because AðIsat Þ ¼ Aðe n sat Þ ¼ A0 =2. In the typical experiment, however, not all the absorbance is saturable because of extrinsic factors; for example, some impurities on the graphene sample can also absorb light. Nonsaturable loss also increases due to the enhanced scattering of graphene multilayers [44]. This absorption is characterized as the nonsaturable absorption component Ans . Then, (7.13) is modified as AðIÞ ¼

A0 þ Ans : 1 þ I=Isat

ð7:15Þ

The absorbance AðIÞ of multilayer graphene is experimentally measured and shown in Figure 7.19(a). The light intensity is considered very small such that I≪ Isat and AðI → 0Þ ¼ A0 þ Ans . As discussed in Section 3.8, the absorbance of multilayer graphene in the optical region is approximately given by Nπα, where N is the number of layers, consistent with the experimental results. As the light intensity increases, the absorbance gradually is saturated as shown in Figure 7.19(b). When light intensity is high enough such that I ≫ Isat , the absorbance is fully saturated and becomes a constant given by AðI → ∞Þ ¼ Ans. It is found that the modulation depth, which is determined by the saturable component, is the largest (66.5 percent) for the samples of 2–4 layers. By fitting the curves to the experimental data using (7.15), it is found that A0 ¼ 66:5%Að0Þ, Ans ¼ 33:5%Að0Þ, and Isat ¼ 0:71 MW cm2 . Compared to singlewalled carbon nanotubes (SWNTs) and semiconductor saturable absorber mirrors (SESAMs), the saturation intensity Isat is one order of magnitude lower, while the modulation depth is 2–3 times larger [30]. However, the modulation depth decreases as the number of layers increases due to increased nonsaturable loss caused by enhanced scattering in multilayer graphene, as shown in Figure 7.19(b) [30].

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Photonic Devices

(a)

(b) 11 1.0

10 Normalized absorbance

Absorbance (

)

9 8 7 6 5 4 3 2 1

9–11 layers

0.9 7–9 layers

0.8 A0 = 66.5% A(0)

0.7 0.6

5–7 layers 4–6 layers

0.5

3–5 layers

0.4

2–4 layers

0.3 1 2 3 4 5 6 7 8 9 10 11 12 Number of layers N

0

2

4

6 8 10 12 14 16 18 20 Light intensity I (MW cm 2 )

Figure 7.19 (a) Measured absorbance of multilayer graphene, and (b) normalized absorbance of

multilayer graphene as a function of light intensity [44]. The solid dots are experimental measurement, and the dashed curves are analytical fits to the data. In (b), the data are fitted using (7.15).

Due to the ability of saturable absorbers to modulate the light intensity, they are frequently used in passively mode-locked or Q-switched lasers to generate short laser pulses. As an optical beam travels through a saturable absorber, it experiences a high loss due to the unsaturated high absorbance when its intensity is low, and a low loss due to the reduced absorbance caused by optical saturation when its intensity is high. Therefore, for a laser pulse that circulates in a laser cavity, as shown in Figure 7.20, its low-intensity leading edge tends to be more attenuated than its high-intensity peak. This attenuation can also happen to the trailing edge of the pulse if the saturable absorber recovers quickly enough, within the duration of the pulse. This mechanism allows a saturable absorber to repetitively shape and shorten a laser pulse as the pulse circulates inside the cavity until a balance is reached between this pulse-shortening effect and the pulse lengthening effects caused by dispersion and gain saturation in the laser. Short mode-locked or Q-switched laser pulses can thus be generated with properly chosen saturable absorbers. Mode-locking operation in a diode-laser-pumped Nd:GdVO4 laser was realized using graphene saturable absorber, the setup of which is schematically shown in Figure 7.21(a) [45]. The graphene saturable absorber was employed both as the saturable absorption element and the output coupler. An output power of 360 mW and a slope efficiency of 22.5 percent were recorded. The pulse duration was measured to be 16 ps with a pulse repetition rate of 43 MHz. The spectrum was measured to find the central wavelength at 1065 nm with a full width at half-maximum of 0.58 nm. These experimental results suggest that a graphene saturable absorber is highly efficient to mode lock a solid-state laser. A mode-locked fiber laser was also realized by using graphene saturable absorber; the schematic is shown in Figure 7.21(b) [44]. The large normal dispersion of graphene was

7.5 Saturable Absorber

247

Optical power

Loss

Gain

Time Figure 7.20 Modulation of optical power using a saturable absorber.

(a)

(b) Polarization controller M4 , R =

Pump laser 1480 nm WDM

Nd:GdVO 4

Isolator EDF 6.4 m

M2 R = 1.0 m

M1 R=

SMF 100 m

LD Coupler

M3 R = 0.5 m

Graphene saturable absorber

Graphene saturable absorber

10% output

Figure 7.21 Experimental setup for mode-locked (a) Nd:GdVO4 laser and (b) fiber laser, both using graphene saturable absorbers. In (a), M1 and M4 are pump-input and end mirrors, respectively; M2 and M3 are beam-folding mirrors. The graphene saturable absorber also serves as the output coupler. In (b), standard fiber-optic components are used, such as wavelength division multiplexer (WDM), erbium-doped fiber (EDF), and single-mode fiber (SMF).

compensated by an erbium-doped fiber (EDF) and an extra single-mode fiber (SMF) of 100 m length. The output power was 2 mW, and a slope efficiency of 3 percent was achieved. The repetition rate was 1.79 MHz, which matches with the cavity round-trip time. The spectrum of the laser was found to center at 1565 nm with a bandwidth of 5 nm. The pulse width was measured to be 756 fs with a signal-to-noise ratio of 65 dB, indicating the stability of the laser pulses.

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References 1. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science, Vol. 332, pp. 1291–1294 (2011). 2. I. T. Lin and J. M. Liu, “Enhanced graphene plasmon waveguiding in a layered graphene–metal structure,” Applied Physics Letters, Vol. 105, 011604 (2014). 3. L. Brey and H. A. Fertig, “Elementary electronic excitations in graphene nanoribbons,” Physical Review B, Vol. 75, 125434 (2007). 4. J. T. Kim and S. Y. Choi, “Graphene-based plasmonic waveguides for photonic integrated circuits,” Optics Express, Vol. 19, pp. 24557–24562 (2011). 5. Y. Sun, Z. Zheng, J. Cheng, and J. Liu, “Graphene surface plasmon waveguides incorporating high-index dielectric ridges for single mode transmission,” Optics Communications, Vol. 328, pp. 124–128 (2014). 6. S. Thongrattanasiri, A. Manjavacas, and F. J. García de Abajo, “Quantum finite-size effects in graphene plasmons,” ACS Nano, Vol. 6, pp. 1766–1775 (2012). 7. J. Cui, Y. Sun, L. Wang, and P. Ma, “Graphene plasmonic waveguide based on a high-index dielectric wedge for compact photonic integration,” Optik: International Journal for Light and Electron Optics, Vol. 127, pp. 152–155 (2016). 8. J. M. Liu, Photonic Devices (Cambridge University Press, 2005). 9. I. T. Lin and J. M. Liu, “Optimization of double-layer graphene plasmonic waveguides,” Applied Physics Letters, Vol. 105, 061116 (2014). 10. F. H. L. Koppens, T. Mueller, P. Avouris, et al., “Photodetectors based on graphene, other two-dimensional materials and hybrid systems,” Nature Nanotechnology, Vol. 9, pp. 780–793 (2014). 11. F. Xia, T. Mueller, Y. M. Lin, A. Valdes-Garcia, and P. Avouris, “Ultrafast graphene photodetector,” Nature Nanotechnology, Vol. 4, pp. 839–843 (2009). 12. J. E. Bowers and Y. G. Wey, “High-speed photodetectors,” in Handbook of Optics, Volume I, M. Bass, ed., 2nd ed. (McGraw-Hill, 1995). 13. T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nature Photonics, Vol. 4, pp. 297–301 (2010). 14. X. Gan, R. J. Shiue, Y. Gao, et al., “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Photonics, Vol. 7, pp. 883–887 (2013). 15. A. Pospischil, M. Humer, M. M. Furchi, et al., “CMOS-compatible graphene photodetector covering all optical communication bands,” Nature Photonics, Vol. 7, pp. 892–896 (2013). 16. X. Wang, Z. Cheng, K. Xu, H. K. Tsang, and J. B. Xu, “High-responsivity graphene/siliconheterostructure waveguide photodetectors,” Nature Photonics, Vol. 7, pp. 888–891 (2013). 17. L. Britnell, R. M. Ribeiro, A. Eckmann, et al., “Strong light–matter interactions in heterostructures of atomically thin films,” Science, Vol. 340, pp. 1311–1314 (2013). 18. W. J. Yu, Y. Liu, H. Zhou, et al., “Highly efficient gate-tunable photocurrent generation in vertical heterostructures of layered materials,” Nature Nanotechnology, Vol. 8, pp. 952–958 (2013). 19. N. M. Gabor, J. C. W. Song, Q. Ma, et al., “Hot carrier-assisted intrinsic photoresponse in graphene,” Science, Vol. 334, pp. 648–652 (2011). 20. M. Freitag, T. Low, F. Xia, and P. Avouris, “Photoconductivity of biased graphene,” Nature Photonics, Vol. 7, pp. 53–59 (2013). 21. G. Konstantatos, M. Badioli, L. Gaudreau, et al., “Hybrid graphene-quantum dot phototransistors with ultrahigh gain,” Nature Nanotechnology, Vol. 7, pp. 363–368 (2012).

References

249

22. L. Vicarelli, M. S. Vitiello, D. Coquillat, et al., “Graphene field-effect transistors as room-temperature terahertz detectors,” Nature Materials, Vol. 11, pp. 865–871 (2012). 23. X. Gu, I. T. Lin, and J. M. Liu, “Extremely confined terahertz surface plasmon-polaritons in graphene–metal structures,” Applied Physics Letters, Vol. 103, 071103 (2013). 24. M. Liu, X. Yin, E. Ulin-Avila, et al., “A graphene-based broadband optical modulator,” Nature, Vol. 474, pp. 64–67 (2011). 25. W. Li, B. Chen, C. Meng, et al., “Ultrafast all-optical graphene modulator,” Nano Letters, Vol. 14, pp. 955–959 (2014). 26. L. Ren, Q. Zhang, J. Yao, et al., “Terahertz and infrared spectroscopy of gated large-area graphene,” Nano Letters, Vol. 12, pp. 3711–3715 (2012). 27. F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surfaces with holes in them: New plasmonic metamaterials,” Journal of Optics A: Pure and Applied Optics, Vol. 7, S97 (2005). 28. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Review of Modern Physics, Vol. 82, pp. 729–787 (2010). 29. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science, Vol. 305, pp. 847–848 (2004). 30. M. M. Jadidi, A. B. Sushkov, R. L. Myers-Ward, et al., “Tunable terahertz hybrid metal–graphene plasmons,” Nano Letters, Vol. 15, pp. 7099–7104 (2015). 31. W. Gao, J. Shu, K. Reichel, et al., “High-contrast terahertz wave modulation by gated graphene enhanced by extraordinary transmission through ring apertures,” Nano Letters, Vol. 14, pp. 1242–1248 (2014). 32. F. Valmorra, G. Scalari, C. Maissen, et al., “Low-bias active control of terahertz waves by coupling large-area CVD graphene to a terahertz metamaterial,” Nano Letters, Vol. 13, pp. 3193–3198 (2013). 33. J. Li, Y. Zhou, B. Quan, et al., “Graphene–metamaterial hybridization for enhanced terahertz response,” Carbon, Vol. 78, pp. 102–112 (2014). 34. N. Papasimakis, Z. Luo, Z. X. Shen, et al., “Graphene in a photonic metamaterial,” Optics Express, Vol. 18, pp. 8353–8359 (2010). 35. R. Degl’Innocenti, D. S. Jessop, Y. D. Shah, et al., “Low-bias terahertz amplitude modulator based on split-ring resonators and graphene,” ACS Nano, Vol. 8, pp. 2548–2554 (2014). 36. B. Sensale-Rodriguez, R. Yan, M. M. Kelly, et al., “Broadband graphene terahertz modulators enabled by intraband transitions,” Nature Communications, Vol. 3, 780 (2012). 37. F. Shi, Y. Chen, P. Han, and P. Tassin, “Broadband, spectrally flat, graphene-based terahertz modulators,” Small, Vol. 11, pp. 6044–6050 (2015). 38. B. Sensale-Rodriguez, R. Yan, S. Rafique, et al., “Extraordinary control of terahertz beam reflectance in graphene electro-absorption modulators,” Nano Letters, Vol. 12, pp. 4518–4522 (2012). 39. I. T. Lin, J. M. Liu, H. C. Tsai, et al., “Family of graphene-assisted resonant surface optical excitations for terahertz devices,” Scientific Reports, Vol. 6, 35467 (2016). 40. S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, et al., “Giant intrinsic carrier mobilities in graphene and its bilayer,” Physical Review Letters, Vol. 100, 016602 (2008). 41. J. W. Yoon, J. H. Lee, S. H. Song, and R. Magnusson, “Unified theory of surface-plasmonic enhancement and extinction of light transmission through metallic nanoslit arrays,” Scientific Reports, Vol. 4, 5683 (2014). 42. Y. Ding, J. Yoon, M. H. Javed, S. H. Song, and R. Magnusson, “Mapping surface-plasmon polaritons and cavity modes in extraordinary optical transmission,” IEEE Photonics Journal, Vol. 3, pp. 365–374 (2011).

250

Photonic Devices

43. R. R. Nair, P. Blake, A. N. Grigorenko, et al., “Fine structure constant defines visual transparency of graphene,” Science, Vol. 320, p. 1308 (2008). 44. Q. Bao, H. Zhang, Y. Wang, et al., “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Advanced Functional Materials, Vol. 19, pp. 3077–3083 (2009). 45. J. L. Xu, X. L. Li, Y. Z. Wu, et al., “Graphene saturable absorber mirror for ultra-fast-pulse solid-state laser,” Optics Letters, Vol. 36, pp. 1948–1950 (2011).

Index

π band, 4, 6, 98 π bond, 1 π electron, 1, 99 π* band, 4, 6, 98 σ band, 6, 98 σ bond, 1 σ electron, 99 σ* band, 6, 98 AA stacking, 21, 63, 95 AB stacking, 15, 62, 92, 141 ABC stacking, 23 absorbance of monolayer graphene, 103, 107, 244 of multilayer graphene, 105, 245 of N-layer graphene, 244 absorption interband, 76, 78, 82, 107 intraband, 76, 78 nonsaturable, 245 optical, 76 acoustic oscillation frequency, 40 acoustic phonon, 33, 39 longitudinal, 40 transverse, 40 acoustic phonon scattering, 33, 39, 59 acoustic phonon scattering rate, 40 acoustic plasmon mode, 218 acoustic SP mode. See acoustic surface plasmon mode acoustic surface plasmon mode, 208 AFM. See atomic-force microscopy Ampère’s equation, 68, 73, 115 amplitude modulation, 230 amplitude modulator graphene-based, 231 antenna, 229 antibonding, 8 aperture SNOM, 212 apertureless SNOM, 212 a-SNOM. See apertureless SNOM atomic-force microscopy, 212 attenuation of graphene SP mode, 189, 192

attenuation constant of graphene TE mode, 196 average momentum relaxation time, 108, 178 axis extraordinary, 74 ordinary, 74 principal, 28, 29, 72, 74, 150 backward scattering, 34, 36, 142 band conduction, 6, 70, 78, 92 Dirac, 19, 20, 21 massive, 20, 62 massless, 20, 21, 62, 63 valence, 6, 70, 78, 92 π, 4, 6, 98 π*, 4, 6, 98 σ, 6, 98 σ*, 6, 98 band structure of AA-stacked graphene, 22 of ABC-stacked graphene, 25 of AB-stacked graphene, 16 of graphene, 3 of graphite, 20 Bernal stacking, 15 bilayer graphene, 233 AA-stacked, 22, 95 AB-stacked, 19, 62, 92, 141, 143 misoriented, 23 bistability, 157 Bloch function, 3 bolometric effect, 226 Boltzmann transport equation, 27, 30, 153 bond C–C, 1 π, 1 σ, 1 bonding, 8 Born approximation first-order, 39 self-consistent, 39 Bose–Einstein statistics, 40

252

Index

bound charge, 67, 114, 151, 178 bound electron, 70, 71, 74, 76, 100, 114, 115, 151, 183, 185 Bravais lattice, 3 Bravais sublattice, 3 Brillouin zone, 2, 6, 11 reduced, 20 canonical momentum, 79 carbon atom, 1 carrier concentration, 10, 13 carrier density, 54, 122, 243 carrier-density operator, 122 cavity mode, 236, 242 centrosymmetric, 151, 162 charge bound, 67, 114, 151, 178 conduction, 67, 148 free, 114 charge carrier, 67 charge density, 66 induced, 112, 114, 121, 123 surface, 116, 118, 123 charged-impurity scattering, 36 charged-impurity scattering rate, 37 chemical potential, 6, 11, 12, 78, 224, 231, 239, 242, 243 effective, 95 complex field, 68 conductance sheet, 30 specific, 30 surface, 30 conduction band, 6, 70, 78, 92 conduction charge, 67, 148 conduction current, 67, 114 conductivity, 28, 30 DC, 108 linear, 148 nonlinear, 145, 148 optical, 71, 76, 80, 113, 243 second-order, 150 sheet, 29 surface, 28, 32, 62, 64, 107, 112, 149, 183 third-order, 150 conductivity tensor, 68 continuity equation, 66 convolution integral, 70, 145 convolution relation, 148 Coulomb gauge, 79 Coulomb interaction bare, 119, 120 screened, 120 Coulomb potential, 35 critical angle, 211 crystal symmetry, 150

current conduction, 67, 114 displacement, 67, 114 external, 67 induced, 67 polarization, 67, 114 current density, 28, 66, 71, 183 electron, 31 hole, 32 induced, 113, 114, 121 linear, 148 nonlinear, 148 optical, 75 surface, 28, 75, 88, 116, 149, 151, 183 volume, 183 cutoff frequency, 218 cyclotron effective mass, 14 DC conductivity, 108 Debye layer, 55 Debye length, 55 defect scattering, 38 defect scattering rate, 38 deformation potential, 40 deformation potential energy, 40 density function theory, 98 density matrix, 80, 85, 87, 88, 120, 125, 159, 162 in the interaction picture, 87, 160 in perturbation expansion, 160 time evolution of, 87, 122, 160 density of defects, 39 density of states, 10 of AB-stacked bilayer graphene, 141 dielectric function, 133 Dirac band, 20, 21 Dirac cone, 7, 20, 23, 27, 62, 63, 95, 152 Dirac cone approximation, 100 Dirac delta function, 83 Dirac electron, 7 Dirac equation massless, 7 relativistic, 6, 14 Dirac fermion, ix, 14 Dirac point, 6, 11, 12, 14, 15, 23, 34, 39, 100, 126, 227 dispersion of acoustic plasmon mode, 218 of coupled SP–phonon modes, 199 of graphene SP mode, 189 of graphene SP–SO coupled mode, 198 of graphene surface plasmon mode, 186, 188, 190, 192, 202 of graphene surface plasmon on a metallic substrate, 203, 204 of graphene surface plasmon polariton mode, 186, 191, 209

Index

of optical plasmon mode, 218 of surface plasmon polariton mode, 182 of transverse electric mode, 195 of uncoupled SP mode, 199 of volume plasmon polariton mode, 181 dispersion curve of coupled graphene SP modes, 209 of decoupled graphene SP modes, 207, 209 of FP mode, 240 of graphene SP mode, 186, 192 of graphene SPP mode, 185, 186, 188, 240, 242 of photon, 181 of polariton, 42 of spoof SPP mode, 240 of SPP mode, 182 of volume plasmon, 181 of volume plasmon polariton, 181 RPA, 192 semiclassical, 192 dispersion map, 240 displacement current, 67, 114 drift mobility, 57 drift velocity, 57 Drude conductivity, 156, 188 Drude model, 107, 108, 178, 186, 203 Drude surface conductivity, 189 Drude surface optical conductivity, 109 EDF. See erbium-doped fiber effective mass, 14, 62, 179 cyclotron, 14 zero, 14 effective optical conductivity, 116 effective permittivity, 218 effective relaxation time, 32, 108 effective rest mass, 14 effective scattering rate, 64 effective-index method, 218 eigenenergy, 4, 6, 18, 20 eigenenergy equation, 4 eigenvalue equation, 17, 22, 185 of graphene SP–SO coupled mode, 198 of graphene surface plasmon polariton mode, 185, 189 of graphene TE mode, 194, 195 of surface optical phonon mode, 42 of surface plasmon modes of two graphene sheets, 206 elastic scattering, 34, 49 elastic scattering rate, 49 electric conductivity surface, 74 electric displacement, 66 electric field, 66 electric permittivity, 117 electric polarization, 71 surface, 116

253

electric potential energy, 113 electric susceptibility, 71 surface, 116 electric dipole approximation, 151, 164 electric dipole interaction, 151, 162, 164 electric quadrupole interaction, 151, 164 electron bound, 70, 71, 74, 100, 114, 115, 151, 183, 185 Dirac, 7 free, 70, 74, 76, 115, 151, 179 π, 1, 99 σ, 99 electron current density, 31 electron density, 11, 54, 123 surface, 116 electron mobility, 52, 57 of graphene, 52, 53 electron–electron interaction, 120 electron–hole pair, 221, 228 electron–hole recombination rate, 111 electron–phonon coupling parameter, 44 electrostatic doping, 217, 239 electrostatic gating, 233, 235 electrostatic limit, 118 energy interlayer interaction, 15, 21, 25, 92, 94, 141 intralayer interaction, 15, 21 nearest-neighbor hopping, 4 self-interaction, 4, 6 energy conservation, 83, 103 energy–momentum relation, 14 near a Dirac point, 7 relativistic, 14 EQE, 224, See external quantum efficiency erbium-doped fiber, 247 external quantum efficiency, 221 extraordinary axis, 74 extraordinary optical transmission, 242 extraordinary permittivity, 74 extraordinary refractive index, 74 extrinsic scattering mechanism, 33 Fabry–Pérot cavity, 18, 103 Fabry–Pérot mode, 238 Fermi energy, 11, 49, 55, 110 Fermi level, 6, 11, 12 Fermi velocity, 7, 14, 77, 181 Fermi’s golden rule, 34, 39, 80, 81, 90 Fermi–Dirac distribution, 11, 12 at thermal equilibrium, 31, 153, 161 fiber tip, 212 fine-structure constant, 105 first-order Born approximation, 39 flexural phonon, 54 forbidden transition, 94 form factor, 36, 122 forward scattering, 34

254

Index

Fourier-transform relation, 69 four-probe technique, 56, 60 fourth-order tensor, 150 four-wave mixing, 168, 176 FP cavity, 238, See Fabry–Pérot cavity FP mode, 240, 242, See Fabry–Pérot mode free charge, 114, 151 free charge carrier, 178 free electron, 70, 74, 76, 115, 151, 179 fundamental mode, 208, 214, 218 graphene bilayer, 19, 22, 23, 62, 92, 95, 141, 143, 233 intrinsic, 6, 29, 92, 125 monolayer, 1, 9, 15, 16, 28, 71, 76, 84, 101, 105, 111, 143, 150, 151, 194, 204 multilayer, 15, 23, 61, 63, 76, 100 single-layer, 1 trilayer, 20, 22, 63 undeformed, 29, 72 graphene nanoribbon waveguide, 216 graphene optical modulator, 231 fiber-based, 234 graphene photodetector, 228 graphene plasmonic waveguide, 218 graphene ribbon, 213 graphene saturable absorber, 246 graphene SP mode, 201, See graphene surface plasmon mode attenuation of, 189, 192 characteristics of, 186, 188 detection of, 212 dispersion of, 186, 188, 190, 203, 204 excitation of, 211, 214, 229 loss of, 193 graphene SPP mode, 185, 189, 216, 239, 240, 242 characteristics of, 185 dispersion of, 186, 191, 209 excitation of, 209, 210, 238, 242 group velocity of, 211 resonance frequency of, 239 graphene surface plasmon, 201 graphene surface plasmon mode, 188, 201 graphene surface plasmon polariton, 216 graphene terahertz modulator, 235 graphene transistor, 228 graphene waveguide, 216 graphene–metamaterial structure, 242 grating structure, 214 group velocity, 14 of carrier in graphene, 14 of graphene SPP mode, 211 H point, 20 Hall bar device, 57

Hall coefficient, 57 Hall mobility, 57 Hamilton’s equations, 79 Hamiltonian, 3 of AA-stacked multilayer gaphene, 21 of ABC-stacked multilayer gaphene, 24 of AB-stacked multilayer gaphene, 17 in an electromagnetic field, 79 interaction, 80, 81, 86, 87, 120, 121, 162 near the K point, 7 perturbation, 121, 159 Hamiltonian operator in an electromagnetic field, 80, 81 harmonic field, 68 Heaviside step function, 11, 84, 128, 138 helicity operator, 9 hexagonal symmetry, 150 high-order mode, 218 high-κ material, 47 hole current density, 32 hole density, 11 hole mobility, 52, 57 of graphene, 52 honeycomb lattice, 1, 3 impedance of free space, 244 impurity scattering, 35, 49 impurity scattering rate, 37 incident power, 221 induced charge density, 112, 114, 121, 123 induced current density, 113, 114, 121 induced polarization density, 113 induced scalar potential, 118 inelastic scattering, 34, 49 in-scattering rate, 47 interaction energy interlayer, 15, 21, 25, 92, 94, 141 intralayer, 15, 21 self-, 4, 6 interaction Hamiltonian, 80, 81, 86, 87, 120, 121, 162 interaction picture, 86, 160 interband absorption, 76, 78, 82, 107 interband scattering, 33, 192, 193 interband scattering rate, 111 interband scattering region, 191, 207 interband transition, 77, 78, 92, 125, 159, 171, 174, 231 interdigitated electrodes, 223 interlayer interaction energy, 15, 21, 25, 92, 141 internal quantum efficiency, 221 intervalley scattering, 34 intraband absorption, 76, 78 intraband scattering, 33, 34, 36 intraband scattering rate, 111 intraband scattering region, 191, 207

Index

intraband transition, 78, 107, 125, 159 intralayer interaction energy, 15, 21 intravalley scattering, 34 intrinsic graphene, 6, 29, 92, 125 intrinsic mobility, 54 intrinsic permutation operator, 167 intrinsic permutation symmetry, 167, 171 intrinsic scattering mechanism, 33 IQE, 223, 224, See internal quantum efficiency K point. See Dirac point kinetic momentum, 79 Kramers–Kronig relation, 84, 92, 128 Kretschmann configuration, 210 Kronecker delta function, 3, 83 Landau damping, 133, 139 lattice constant, 1 linear conductivity, 148 linear current density, 148 linear optical conductivity, 162 linear surface conductivity, 149, 151 linear surface susceptibility, 149, 151 linear susceptibility, 146 Liouville–von Neumann equation, 87, 160 lithium perchlorate, 55 LO phonon. See longitudinal optical phonon longitudinal acoustic phonon, 40 longitudinal mode, 179 longitudinal optical phonon, 42 longitudinal optical phonon frequency, 42 longitudinal oscillation frequency, 179 long-range scattering, 58 Lorentz force, 57, 80 Lorentzian function, 94 magnetic field, 66 magnetic induction, 66 magnetic dipole interaction, 151, 164 magnetization, 66 massive band, 20, 62 massless band, 20, 21, 62, 63 massless Dirac band, 20, 21 massless Dirac fermion, 14 Maxwell’s equations, 66, 179 mechanical exfoliation, 60 metal SP mode, 201 metal SPP mode, 201, 242 metallic grating, 243 metamaterial, 236 metamaterial structure, 237 microfiber, 233, 234 midgap state, 39 misorientation, 23 mobility, 32, 33, 53, 56, 57, 59, 229 drift, 57 electron, 52, 57

255

of graphene, 53, 58, 233 Hall, 57 hole, 52, 57 intrinsic, 54 of suspended graphene, 53 ultrahigh, 58, 59 mode-locked fiber laser, 246 mode-locked laser, 246 modulation depth, 231, 234, 235, 238, 239, 243, 245 momentum conservation, 83 momentum relaxation time, 34, 108 average, 108, 178 monolayer graphene, 1, 9, 15, 16, 28, 71, 76, 84, 101, 105, 111, 143, 150, 151, 194, 204 absorbance of, 103, 107, 244 reflectance of, 103 thickness of, 47, 71, 101, 175 transmittance of, 103, 105 multilayer graphene, 15, 23, 61, 76, 91, 100 AA-stacked, 21, 63 ABC-stacked, 23 absorbance of, 105, 245 AB-stacked, 15, 62 misoriented, 23 reflectance of, 105 transmittance of, 105 nearest-neighbor hopping energy, 4 nearest-neighbor interaction, 15, 18 near-field scanning optical microscopy, 211 neutrality point, 58 nonlinear conductivity, 145, 148 second-order, 148, 150 third-order, 148, 150, 165 nonlinear current density, 148 nonlinear optical conductivity second-order, 162 third-order, 162 nonlinear optical polarization, 145 nonlinear optical response, 145 nonlinear refractive index, 176 nonlinear surface conductivity third-harmonic, 168 third-order parametric, 171 nonlinear surface current third-order, 168 nonlinear surface current density, 162 second-order, 162 third-order, 164 nonlinear surface optical conductivity, 152 nonlinear surface susceptibility, 152 third-order, 174 nonlinear susceptibility, 145 second-order, 146, 150 third-order, 146, 150 nonlocal response, 189 nonresonant transition, 170

256

Index

nonretarded region, 183 nonsaturable absorption, 245 optical absorption, 76 optical bistability, 157 optical conductivity, 71, 76, 80, 113, 243 of AB-stacked graphene, 92 effective, 116 linear, 162 of multilayer graphene, 91 nonlinear, 162 nth-order, 151 surface, 74, 83, 89, 109, 116, 137, 151 optical current density surface, 75 optical excitation, 67 optical Kerr effect, 168, 173, 176 optical modulator, 230 graphene-based, 233 ideal, 232 optical nonlinearity, 152 second-order, 162 third-order, 164, 171 optical phonon, 33 intrinsic, 48 longitudinal, 42 surface, 42, 197 transverse, 42, 197 optical phonon frequency longitudinal, 42 transverse, 42 optical phonon scattering, 33, 41 extrinsic, 41, 42 intrinsic, 41, 48 optical plasmon mode, 218 optical polarization linear, 145 nonlinear, 145 optical saturation, 168 optical SP mode. See optical surface plasmon mode optical susceptibility, 71 nth-order, 151 surface, 73, 98, 151 optical-field-induced birefringence, 173 orbitals atomic, 1 of graphene, 1 of graphite, 1 ordinary axis, 74 ordinary permittivity, 74 ordinary refractive index, 74 Otto configuration, 210 out-of-phase vibrations, 48 out-scattering rate, 46 parametric frequency conversion, 171 Pauli blocking, 47, 94, 129, 231, 233, 244

Pauli vector, 9, 80, 159 perfect conductor, 179 permittivity, 101, 114, 179 of conductor, 183 effective, 218 electric, 117 extraordinary, 74 ordinary, 74 surface, 74, 101 permittivity tensor, 68, 98, 115 permutation symmetry intrinsic, 167 perturbation, 86 perturbation Hamiltonian, 121, 159 perturbation method, 145 phase matching, 209, 210, 214, 229 phase-matching condition, 210 phase mismatch, 209 phase velocity, 14, 128, 183 of carrier in graphene, 14 of graphene SP mode, 198 of graphene SPP mode, 188 phonon acoustic, 33, 39 flexural, 54 longitudinal acoustic, 40 longitudinal optical, 42 optical, 33 surface, 33 transverse acoustic, 40 transverse optical, 42, 197 phonon absorption, 40 phonon emission, 40 phonon frequency, 42 phonon polariton, 42 surface, 42 phonon scattering, 33, 41 acoustic, 39 optical, 41, 42, 48 SO, 43 photoconductive detector, 229 photoconductive gain, 229 photocurrent, 221, 230 photodetector, 220 graphene-based, 228, 233 terahertz, 229, 230 photogating effect, 228 photon energy, 221 photon flux, 221 absorbed, 221 photonic crystal, 176 photo-thermoelectric effect, 224 photovoltage, 230 photovoltaic effect, 221 planar waveguide, 216 plasma frequency, 178, 179 surface, 183

Index

plasma oscillation, 178 plasmon, 178 surface, 178, 183 volume, 179 plasmon mode acoustic, 218 optical, 218 plasmon polariton, 181 surface, 178, 182 volume, 181 plasmon resonance, 216, 243 plasmonic waveguide, 216, 218 of two graphene sheets, 218 plasmonic waveguide mode, 218 plasmonic wavelength, 213 p–n junction, 221 point defect, 38 point group, 150 polar crystallographic point groups, 42 polar substrate, 42 polariton dispersion, 42 polarizability. See polarizability function polarizability function, 108, 111, 112, 113 at a nonzero temperature, 138 of 2D electron gas, 142 of AB-stacked bilayer graphene, 141, 142 of monolayer graphene, 142 surface, 112, 116, 118, 124, 125 polarization, 66 polarization current, 67, 114 polarization density, 149, 179 induced, 113 surface, 75, 149, 151 polyethylene oxide, 55 polymer electrolyte, 55, 59 position operator, 123 primitive lattice vectors, 1 primitive unit cell, 2 principal axis, 28, 29, 72, 74, 150 principal surface conductivity, 74 principal surface susceptibility, 74 principal susceptibility, 72, 150 prism coupler, 210 propagation constant, 179 of graphene TE mode, 196 pseudospin, 9 PTE effect. See photo-thermoelectric effect PV effect, 224, 226, 228, See photovoltaic effect Q-switched laser, 246 quantum dot, 228 quasistatic approximation, 183, 188 random-phase approximation, 120 rate equation, 244 RC-limited bandwidth, 223, 233

257

RC time constant, 222, 233 reciprocal lattice, 2 reflectance, 232, 243 of monolayer graphene, 103 of multilayer graphene, 105 reflection coefficient, 103 reflection spectroscopy, 100 refractive index, 100 extraordinary, 74 nonlinear, 176 ordinary, 74 relaxation time, 31 effective, 32, 108 relaxation-time approximation, 31, 124, 125 resistance contact, 57 of graphene sheet, 56 sheet, 30, 56 specific, 30 surface, 30 total, 57 resistivity, 30 sheet, 30 surface, 30, 51, 56 resonance frequency, 213, 242 graphene SPP mode, 239 resonance mode, 235, 236 resonant transition, 170 response function, 113 responsivity, 221, 223 retarded region, 183 RPA. See random phase approximation RPA conductivity model, 203 RPA dispersion curve, 192 RPA surface conductivity, 203 RPA–RT surface conductivity, 189 RPA–RT surface polarizability function, 125, 190 saturable absorber, 244, 246 graphene, 246 saturable absorption, 233 saturation intensity, 245 saturation model, 244 saturation surface carrier density, 244 scalar potential, 79, 112, 121 induced, 118 scattering acoustic phonon, 33, 39, 59 backward, 34, 36, 142 charged-impurity, 36 defect, 38 elastic, 34, 49 extrinsic optical phonon, 42 forward, 34 impurity, 35, 49 inelastic, 34, 49 interband, 33, 192, 193

258

Index

scattering (cont.) intervalley, 34 intraband, 33, 34, 36 intravalley, 34 intrinsic optical phonon, 48 long-range, 58 optical phonon, 33, 41 short-range, 38, 58 SO phonon, 43, 50 scattering center, 27 scattering mechanism extrinsic, 33 intrinsic, 33 scattering potential energy, 34, 37 scattering rate, 138 acoustic phonon, 40, 53 charged-impurity, 37 defect, 38 effective, 64, 138 elastic, 49 energy-dependent, 47 impurity, 37, 53 interband, 111 intraband, 111 SO phonon, 44, 45, 47, 53 total, 49 scattering transition rate, 33 scattering wave number, 36, 45 scattering-type SNOM, 212 Schrödinger equation, 3 screened Coulomb interaction, 120 screened Coulomb potential energy 2D, 119 screening number, 48 screening wave number, 37 Thomas–Fermi, 37 second harmonic, 163 second-order optical nonlinearity, 162 second-order tensor, 150 Seebeck coefficient, 224 selection rules, 98 self-consistent Born approximation, 39 self-interaction energy, 4, 6 self-phase modulation, 176 semiclassical dispersion curve, 192 semiconductor saturable absorber mirror, 245 SESAM. See semiconductor saturable absorber mirror sheet conductance, 30 sheet conductivity, 29 sheet resistance, 30 of graphene, 56 sheet resistivity, 30 short laser pulse, 246 short-range scattering, 38, 58

single-mode fiber, 233 single-walled carbon nanotube, 245 slope efficiency, 246 SNOM. See near-field scanning optical microscopy aperture, 212 apertureless, 212 scattering-type, 212 SO phonon. See surface optical phonon SO phonon scattering, 43, 50 SO phonon scattering rate, 45 SP. See surface plasmon space group, 150 spatial nonlocality, 107, 111, 180 specific conductance, 30 specific resistance, 30 spectral broadening, 85, 170, 172, 174 spectral linewidth, 170, 172, 174, 216 spectroscopic ellipsometry, 100 split ring resonator, 236 spoof SPP mode, 236, 238, 240, 242, See spoof surface plasmon polariton mode spoof surface plasmon polariton mode, 236 SPP. See surface plasmon polariton SRR. See split ring resonator s-SNOM. See scattering-type SNOM stacking AA, 21 AB, 15 ABC, 23 Bernal, 15 static screening wave number, 141 stimulated emission, 82 surface carrier density photoexcited, 244 saturation, 244 surface charge density, 116, 118, 123 surface concentration of electrons, 11 of holes, 11 surface conductance, 30 surface conductivity, 28, 32, 62, 64, 107, 112, 183 2D, 28 DC, 107 Drude, 189 linear, 149, 151 nonlinear, 169, 171 principal, 74 RPA, 203 PRA–RT, 189 third-harmonic, 168 surface current, 183 nonlinear, 168 surface current density, 28, 88, 116, 149, 151, 183 nonlinear, 162 surface current model, 183

Index

surface electric conductivity, 74 surface electric polarization, 116 surface electric susceptibility, 116 surface electron density, 11, 116 surface hole density, 11 surface optical conductivity, 74, 83, 89, 109, 116, 137, 151 of AA-stacked graphene, 95, 97 of AB-stacked graphene, 95 high-frequency, 84 low temperature, 85, 111 of monolayer graphene, 95, 244 nonlinear, 152 at zero temperature, 91, 92 surface optical current density, 75 surface optical phonon, 42, 197 surface optical susceptibility, 73, 98, 151 surface permittivity, 74, 101 surface phonon, 33 surface phonon mode, 42 surface phonon polariton, 42 surface plasma frequency, 183 surface plasmon, 178, 183 surface plasmon mode, 183 acoustic, 208 antisymmetric, 207 graphene, 188 optical, 208 phonon-coupled, 197 symmetric, 207 surface plasmon polariton, 178, 182 surface plasmon polariton mode, 182 graphene, 185 metal, 188 spoof, 236 surface polarizability. See surface polarizability function surface polarizability function, 112, 116, 118, 124, 125 RPA–RT, 125, 190 surface polarization density, 75, 149, 151 surface resistance, 30 surface resistivity, 30, 51 graphene, 56 surface susceptibility, 112 linear, 149, 151 nonlinear, 152 principal, 74 susceptibility, 114 electric, 71 linear, 146 nonlinear, 145, 146 optical, 71 principal, 72, 150 second-order, 146, 150

259

surface, 112, 149 third-order, 146, 150 susceptibility tensor, 68 SWNT. See single-walled carbon nanotube symmetric surface plasmon mode, 207 TE mode. See transverse electric mode terahertz modulator, 230, 235, 242 terahertz photodetector, 229 terahertz waveguide, 216, 230 thermal-equilibrium distribution, 87 thermoelectric effect, 224 thickness of monolayer graphene, 47, 71, 101, 175 third-harmonic generation, 168 third-order nonlinear surface susceptibility, 174 third-order nonlinear susceptibility 3D effective, 174 third-order optical nonlinearity, 164, 171 third-order tensor, 150 Thomas–Fermi screening, 37, 51, 137, 140, 143 tight-binding model, 3 time-dependent perturbation, 87 TM mode. See transverse magnetic mode TM surface mode, 182 TMDC. See transition metal dichalcogenide TO phonon. See transverse optical phonon transition forbidden, 94 interband, 77, 78, 92, 125, 159, 171, 174, 231 intraband, 78, 107, 125, 159 nonresonant, 170 resonant, 170 transition metal dichalcogenide, 223 transition rate, 82 transit-time-limited bandwidth, 221 transmission coefficient, 103 transmittance, 231, 234, 243 of monolayer graphene, 103 of multilayer graphene, 105 transverse acoustic phonon, 40 transverse electric mode, 193 of monolayer graphene, 194 transverse magnetic mode, 185 transverse mode, 179, 181 transverse optical phonon, 42, 197 transverse optical phonon frequency, 42 trilayer graphene AA-stacked, 22 AB-stacked, 20, 63 ultrahigh mobility, 58, 59 undeformed graphene, 29, 72 uniaxial, 74 unit cell, 2

260

Index

unit polarization vector, 69 unsaturated absorbance, 244 valence band, 6, 70, 78, 92 van der Waals attraction, 224 Van Hove singularities, 100 vector potential, 79, 81 vectorial field amplitude, 69 velocity operator, 88 virtual state, 170 volume current density, 183 volume plasmon, 179 volume plasmon mode, 179 volume plasmon polariton, 181

volume plasmon polariton mode, 181 wave function of graphene, 3 waveguide graphene, 216 graphene nanoribbon, 216 graphene plasmonic, 218 planar, 216 plasmonic, 216, 218 terahertz, 216, 230 ZO mode, 48 ZO phonon, 48