Mathematics and physics for nanotechnology : technical tools and modelling

243 Pages·2019·12.863 MB·English

by  Di Sia, Paolo

Mathematics and Physics
for Nanotechnology

Mathematics and Physics
for Nanotechnology
Technical Tools and Modelling
Paolo Di Sia
Published by
Pan Stanford Publishing Pte. Ltd.
Penthouse Level, Suntec Tower 3
8 Temasek Boulevard
Singapore 038988
Email: [email protected]
Web: www.panstanford.com
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Mathematics and Physics for Nanotechnology:
Technical Tools and Modelling
All rights reserved. This book, or parts thereof, may not be reproduced in any form
Copyright © 2019 by Pan Stanford Publishing Pte. Ltd.
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or any information storage and retrieval system now known or to be invented,
without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through
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ISBN 978-981-4800-02-0 (Hardcover)
ISBN 978-0-429-02775-8 (eBook)
Contents
Preface
1. In troduc tion x1i
1.1 The Nanotechnologies World 1
1.2 Classification of Nanostructures 8
1.3 Applications of Nanotechnologies 15
1.4 Applied Mathematics and Nanotechnology 23
1.5 Spintronics, Information Technologies and
Nanotechnology 29
1.5.1 Spin Decoherence in Electronic
Materials 30
1.5.2 Transport of Polarised Spin in Hybrid
Semiconductor Structures 31
1.5.3 Spin-Based Solid State Quantum
Computing 32
1.5.4 Spin Entanglement in Solids 32
1.5.5 Optical and Electronic Control of
Nuclear Spin Polarisation 33
1.5.6 Physics of Computation 34
1.5.7 Quantum Signal Propagation in
2. Vector Analysis Nanosystems 3357
2.1 Vectors and Scalars 37
2.2 Direction Angles and Direction Cosines 39
2.3 Equality of Vectors 39
2.4 Vector Addition and Subtraction 40
2.5 Multiplication by a Scalar 41
2.6 Scalar Product 41
2.7 Vector Product 43
2.8 Triple Scalar Product 44
V
2.9 Triple Vector Product 45
2.10 Linear Vector Space 45
vi Contents
3. Vector Differentiation 49
3.1 Introduction 49
3.2 The Gradient Operator 51
3.3 Directional Derivative 53
3.4 The Divergence Operator 54
3.5 The Laplacian Operator 55
3.6 The Curl Operator 56
4. 3Co.7o rdinFaoter mSyusltaesm Ins vaonldv iInmgp tohret aNnatb Tlhae Oopreemrast or 5569
4.1 Orthogonal Curvilinear Coordinates 59
4.2 Special Orthogonal Coordinate Systems 61
4.2.1 Cylindrical Coordinates 61
4.2.2 Spherical Coordinates 62
4.3 Vector Integration and Integral Theorems 62
4.4 Gauss Theorem 64
4.5 Stokes Theorem 65
4.6 Green Theorem 66
4.7 Helmholtz Theorem 66
5. 4O.r8d inaryU sDeiffufel rIenntteigarl aElq Ruealtaitoinosn s 6771
5.1 Introduction 71
5.2 Separable Variables 73
5.3 First-Order Linear Equation 74
5.4 Bernoulli Equations 75
5.5 Second-Order Linear Equations with Constant
Coefficients 76
5.5.1 Homogeneous Linear Equations with
Constant Coefficients 77
5.5.2 Non-homogeneous Linear Equations
with Constant Coefficients 78
k
5.6 An Introduction to Differential Equations with
6. Fourier OSerrdieesr an >d 2 In tegrals 8827
6.1 Periodic Functions 87
6.2 Fourier Series 88
6.3 Euler–Fourier Formulas 88
Contents vii
6.4 Half-Range Fourier Series 89
6.5 Change of Interval 90
6.6 Parseval’s Identity 91
6.7 Integration and Differentiation of a Fourier
Series 92
6.8 Multiple Fourier Series 94
6.9 Fourier Integrals and Fourier Transforms 95
6.10 Fourier Transforms for Functions of Several
7. FunctionVsa roifa Oblnees Complex Variable 9969
7.1 Complex Numbers 99
7.2 Basic Operations with Complex Numbers 100
7.3 Polar Form of a Complex Number 100
7.4 De Moivre’s Theorem and Roots of Complex
Numbers 102
7.5 Functions of a Complex Variable 103
7.6 Limits and Continuity 104
7.7 Derivatives and Analytic Functions 104
7.8 Cauchy–Riemann Conditions 105
7.9 Harmonic Functions 106
7.10 Singular Points 106
8. 7Co.1m1 plexC oInmtepglreaxt iEolnem entary Functions 110173
8.1 Line Integrals in the Complex Plane 113
8.2 Cauchy’s Integral Theorem 115
8.3 Cauchy’s Integral Formula 116
8.4 Series Representations of Analytic Functions 117
8.5 Integration with the Residue Method 117
9. 8Pa.6r tial DEivffaelureanttioianl Eoqf uRaetailo Dnse finite Integrals 111293
9.1 Introduction 123
9.2 Linear Second-Order Partial Differential
Equations 124
9.3 Important Second-Order Partial Differential
Equations 125
viii Contents
10. Numerical Methods 127
10.1 Interpolation 127
10.2 Solutions of Equations: Graphical Method 127
10.3 Method of Linear Interpolation 129
10.4 Newton Method 130
10.5 Numerical Integration: The Rectangular Rule 131
10.6 Numerical Integration: The Trapezoidal Rule 132
11. 1Q0u.a7n tumNu Bmaseircisc afol rI nNtaengorateticohnn:o Tlohgey S impson Rule 113335
11.1 Black Body Radiation and Planck Hypothesis 135
11.2 Einstein Work on Stimulated Emission 138
11.3 De Broglie Waves 139
11.4 The Compton Effect 140
11.5 The Criterion Governing Classical and
Quantum Properties of a Particle 142
11.6 The Uncertainty Relations 142
11.7 The Reality at the Nanoscale and the Concept
12. Schrödinogf eWr aEvqeu aFtuionnct aionnd Nanotechnology 114437
12.1 The Schrödinger Equation 147
12.2 The Time-Dependent Schrödinger Equation 151
12.3 Stationary States 151
12.4 Schrödinger Equation for Quantum Wires 152
12.5 Schrödinger Equation for Quantum Dots 156
12.6 Schrödinger Equation and Drude–Lorentz-like
Model 158
12.7 Schrödinger Equation and Computational
13. MathemNaatnicoatle Mchondoellolignyg for Nanotechnology 116603
13.1 Introduction 163
13.2 The Drude Model 164
13.3 The Drude–Lorentz Model 166
13.4 About the Most Utilised Drude–Lorentz-like
Models 166
13.5 The Smith Model 169
13.6 Linear Response Theory with a New Idea 170
Contents ix
s w
13.7 Other Key Functions 173
s w
13.8 The Complex Conductivity ( ) 173
s w
13.9 Behaviour of Complex Conductivity ( ) 174
13.10 The Poles of Conductivity ( ) 175
13.11 Premises of Quantum and Relativistic
Versions of DS Model 176
13.12 Classical Results 178
13.13 Quantum Results 179
13.14 Relativistic Results 181
14. 1Pl3a.s1m5 onEixcsa manpdle Ms oodf Aelplipnlgi cation 118933
14.1 Introduction 193
14.2 Plasmons 194
14.3 Related Theoretical Models 195
14.3.1 The Mie Theory 195
14.3.2 The Gans Theory 196
14.3.3 The Discrete-Dipole Approximation
Method (DDA) 196
14.3.4 The Finite-Difference Time-Domain
Method (FDTD) 197
15. Nanodif1fu4s.3io.5n inT Ghrea pDhSe Mneo del 119979
15.1 Introduction 199
15.2 Peculiar Properties of Carbon Nanotubes 202
15.2.1 Structure 204
15.2.2 Synthesis 205
15.2.3 Electronic Properties 205
15.2.4 Mechanical Properties 207
15.2.5 Thermal Properties 207
15.2.6 Chemical and Electrochemical
Properties 208
15.2.7 Nanobiosensing Properties 208
15.3 Fields of Utilisation 209
15.4 Nanodiffusion: Classical and Quantum Results 211
15.4.1 Classical Case: Dependence on
Temperature 211