Computational Microelectronics

Computational Microelectronics

Editor Siegfried Selberherr Technical University Vienna Vienna, Austria

For further volumes: http://www.springer.com/series/1263

Mahdi Pourfath

The Non-Equilibrium Green’s Function Method for Nanoscale Device Simulation

123

Mahdi Pourfath School of Electrical and Computer Engineering University of Tehran Tehran, Iran

ISSN 0179-0307 ISBN 978-3-7091-1799-6 ISBN 978-3-7091-1800-9 (eBook) DOI 10.1007/978-3-7091-1800-9 Springer Wien Heidelberg New York Dordrecht London Library of Congress Control Number: 2014943949 © Springer-Verlag Wien 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To my parents, my wife, and my daughters

Preface

This book bridges the gap between elementary quantum transport books and more rigorous graduate-level material on the quantum field theory of many-body systems. The book presents a simple, intuitive understanding of Green’s function theory and its application for the analysis of nanoelectronic devices. It attempts to explain the underlying physics with a consistent theoretical footing. This book targets graduatelevel students and researchers in electronics and physics. One of the stimulating factors for the writing of this book was the many requests I received from scientists and students who wanted to receive a copy of my dissertation, where I addressed a similar topic. This book, however, includes more materials on the underlying principles, numerical techniques, and applications. It is my hope that the inclusion of these elements will help young scientists to contribute something new to the frontiers of nanoelectronics. In this book after a short introduction in Chap. 1, the postulates of quantum mechanics are briefly presented in Chap. 2. As electrons in solids experience various scattering mechanisms, an accurate study of electron transport in solid state devices requires the knowledge and techniques of many-body theory. Chapters 3 and 4, respectively, review the basic principles of many-body systems and band theory of electrons in solids. With the aid of statical mechanics, which is discussed in Chap. 5, we relate microscopic and macroscopic quantities in many-body systems and study systems both under equilibrium and non-equilibrium condition. Next, the Green’s function formalism is presented in Chap. 6. As the exact solution of the Green’s function for a realistic system cannot be obtained, approximation methods are needed. Such approximations and the related methods are discussed in the rest of this chapter. After building a solid theoretical foundation, numerical methods for calculating Green’s functions are presented in Chap. 7. All the elements of the kinetic equations, which are the device Hamiltonian, contact self-energies, and scattering self-energies are carefully studied and efficient methods for evaluation are explained. Finally, these methods are applied to the study of electron, spin, and phonon transport in nanoribbons in Chap. 8. Additionally, device characteristics of tunneling transistors and photo-detectors are investigated using the outlined methodologies. vii

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I am deeply grateful to my family for their understanding and loving acceptance of my engagement in writing this book. Some contribution to this text, however, has come from my students: Nayereh Ghobadi, Hossein Karami-Taheri, Shoeib Babaee Touski, Zahra chagazardi, Nima Djavid, Kaveh Khaliji, Sahar Pakdel, and Mohammad Tabatabaee. I would specially like to thank Prof. Hans Kosina for his support during the preparation of this work. Finally, I owe thanks to Prof. Siegfried Selberherr for his encouragement and long lasting patience. Tehran, Iran April 2014

Mahdi Pourfath

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3

2

Review of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Measurements and Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Schrödinger Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Spinors and Pauli Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 11 11 16 20 23 24 24 26 27

3

Many-Body Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 First Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Slater Determinants and Permanents . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Operators in the First Quantization Representation . . . . . . . . . . . 3.2 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Operators in the Second Quantization Representation . . . . . . . . 3.2.3 Basis Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Field Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Quasi-particles and Collective Excitations . . . . . . . . . . . . . . . . . . . . 3.2.6 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.7 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.8 Interaction with Photons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 30 31 34 37 38 40 42 43 46 47 49 52 53

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4

Band Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Crystal Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Electrons in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Bloch States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Tight-Binding Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 The Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Phonons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Phonon Interaction Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Scattering of Bloch States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 57 59 61 62 67 72 74

5

Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1 Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2.1 Macro and Microstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2.2 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2.3 Classical and Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3.1 The Laws of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3.2 Closed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3.3 Systems in Contact with a Heat Bath . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3.4 Systems in Contact with a Heat and Particle Reservoir . . . . . . . 83 5.3.5 Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3.6 Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3.7 Connection to Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.4 Statistical Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4.1 Micro-canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4.2 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4.3 Grand-Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.5 Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5.1 Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5.2 Fermi-Dirac Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.5.3 Bose-Einstein Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.5.4 Maxwell-Boltzmann Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.6 Non-equilibrium Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.6.1 Boltzmann Transport Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.6.2 Validity of the Boltzmann Transport Equation . . . . . . . . . . . . . . . . 97 5.6.3 Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.6.4 Wigner Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.6.5 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6

Green’s Function Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Schrödinger Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 106 106 106

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6.2.3 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 The Evolution Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Imaginary Time Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Equilibrium Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Zero Temperature Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Finite Temperature Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Matsubara Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Non-equilibrium Green’s Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Non-equilibrium Ensemble Average . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Contour-Ordered Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Keldysh Contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Real-Time Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Langreth Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.6 Non-interacting Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.7 Non-interacting Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Perturbation Expansion of the Green’s Function . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 First-Order Perturbation Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Dyson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.5 Electron-Electron Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.6 Electron-Phonon Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Quantum Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 The Kadanoff-Baym Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Keldysh Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Steady-State Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Variational Derivation of Self-Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Electron-Electron Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Screened Interaction, Polarization, and Vertex Function . . . . . 6.7.3 Electron-Phonon Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.4 The Phonon Green’s Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.5 The Phonon Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.6 Approximation of the Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Relation to Observables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Electron and Hole Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Spectral Function and Local Density of States . . . . . . . . . . . . . . . . 6.8.3 Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 108 111 113 113 115 117 118 118 119 121 121 123 124 127 128 129 131 131 134 136 137 138 139 139 140 142 143 144 147 150 151 151 152 152 153 153 155

Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Basis Functions and Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Free Transverse-Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Real-Space Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Coupled Mode-Space Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Decoupled Mode-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 157 159 160 162 164

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7.2 Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Matrix Truncation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Surface Green’s Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Sancho-Rubio Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Contact Self-Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Wide-Band Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Scattering Self-Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Electron-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Acoustic Phonon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Optical Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Polar Optical Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Recursive Method for Calculating Green’s Functions. . . . . . . . . . . . . . . . . 7.4.1 Retarded and Advanced Green’s Functions . . . . . . . . . . . . . . . . . . . 7.4.2 Lesser and Greater Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Evaluation of Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Carrier Concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Transmission Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Selection of the Energy Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Confined States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Non-adaptive Energy Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Adaptive Energy Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Self-Consistent Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1 Self-Consistent Iteration Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Convergence of the Self-Consistent Simulations . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 166 168 169 172 174 175 175 177 177 178 180 181 182 184 184 185 186 187 187 188 190 192 193 194 197

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Electronic Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Transport Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Line-Edge Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Substrate Corrugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Spin Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Multi-orbital Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Transport Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Phonon Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Phonon Bandstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Phonon Green’s Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Phonon Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Ballistic Phonon Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Graphene-Based Tunneling Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Self-Consistent Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Device Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 204 205 206 211 216 216 218 219 220 221 222 223 224 230 231 235 236

Contents

8.6 CNT and GNR-Based Photodetectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Electron-Photon Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Quantum Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

241 242 245 246

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Notation

Symbols x x OO OO  IO x ex xy ˝ @t ./ rx A Aij A G G0 G r;a G? g D ˙ ˙C ˙S VOe- VOe-ph H H0 HO

Scalar Complex conjugate of x Operator Hermitian Conjugate of the operator OO Unity operator Vector Unity vector in direction x Scalar inner product Convolution Partial derivative with respect to t Gradient of x Matrix Elements of the matrix A Conjugate transpose of the matrix A Green’s function for electrons Non-interacting Green’s function Retarded and advanced Green’s function Greater and lesser Green’s function Surface or incomplete Green’s function Green’s function for phonons Self-energy Contact self-energy Scattering self-energy Electron-photon interaction potential Electron-phonon interaction potential Hamiltonian in the first quantization Single-particle or non-interacting Hamiltonian Hamiltonian in the second quantization xv

xvi

O Ob cO aO bO  cO  aO  hi r pO LO SO V A ˝ tij  u R R0 "q ˚  D0 qD E EF Ec Ev EG I J k kt q qt m M nB f  H t T

Notation

Field operator Annihilation operator for bosons Annihilation operator for fermions General annihilation operator Creation operator for bosons Creation operator for fermions General creation operator Ensemble statistical average Position operator Momentum operator Angular momentum operator Spin operator Volume Surface Volume of unit-cell Hopping parameter Spin-orbit coupling constant Lattice vibration vector Position of atoms Equilibrium position of atoms Phonon and photon polarization vector Force constant Acoustic deformation potential constant Optical deformation potential constant Inverse Debye screening length Energy Fermi energy Conduction band-edge energy Valence band-edge energy Band gap energy Current Current density Wave vector of electron Transverse wave vector of electron Wave vector of phonon Transverse wave vector of phonon Mass of electrons Mass of atoms Bose-Einstein distribution function Fermi-Dirac distribution function Density Hilbert space Time Temperature

Notation

U E D B 

xvii

Potential energy Electric field Electric displacement field Magnetic field Dielectric permittivity

Abbreviations ACF AGNR AP BTE CMOS CMS CNT DFT DOS DP FCM GNR DOS FET ITRS IR LDOS LER MOS NEGF OP POP RS SCBA TB TFET VFF VTGFET VTGNRFET ZB ZGNR

Auto-correlation function Armchair GNR Acoustic phonons Boltzmann transport equation Complementary MOS Coupled mode-space Carbon nanotube Density functional theory Density of states Deformation potential Force constant method Graphene nanoribbon Density of states Field-effect transistor International technology road-map for semiconductors Infra-red Local DOS Line-edge roughness Metal-oxide-semiconductor Non-equilibrium Green’s function Optical phonons Polar optical phonons Real-space Self-consistent Born approximation Tight-binding Tunneling FET Valence force field Vertical graphene TFET Vertical GNR TFET Zone boundary phonons Zigzag GNR