Main Semiconductors for micro and nanotechnology an introduction for engineers

Semiconductors for micro and nanotechnology an introduction for engineers

,
Semiconductors play a major role in modern microtechnology, especially in microelectronics. Since the dimensions of new microelectronic components, e.g. computer chips, now reach nanometer size, semiconductor research moves from microtechnology to nanotechnology. An understanding of the semiconductor physics involved in this new technology is of great importance for every student in engineering, especially electrical engineering, microsystem technology and physics. This textbook emphasizes a system-oriented view of semiconductor physics for applications in microsystem technology. While existing books only cover electronic device physics and are mainly written for physics students, this text gives a more hands-on approach to semiconductor physics and so avoids overloading engineering students with mathematical formulas not essential for their studies.
Year: 2002
Edition: 1
Publisher: Wiley-VCH
Language: english
Pages: 341
ISBN 10: 3527302573
ISBN 13: 9783527302574
File: PDF, 9.58 MB
 
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Semiconductors for
Micro and
Nanotechnology—
An Introduction for
Engineers

Semiconductors for
Micro and
Nanotechnology—
An Introduction for
Engineers
Jan G. Korvink and Andreas Greiner

Authors:
Prof. Dr. Jan G. Korvink
IMTEK-Institute for Microsystem
Technology
Faculty for Applied Sciences
Albert Ludwig University Freiburg
D-79110 Freiburg
Germany

Dr. Andreas Greiner
IMTEK-Institute for Microsystem
Technology
Faculty for Applied Sciences
Albert Ludwig University Freiburg
D-79110 Freiburg
Germany

This book was carefully produced. Nevertheless, authors, editors and
publisher do not warrant the information contained therein to be free of
errors. Readers are advised to keep in mind that statements, data,
illustrations, procedural details or other items may inadvertently be
inaccurate.

Library of Congress Card No.: applied for.
British Library Cataloguing-in-Publication Data:
A catalogue record for this book is available from the British Library.
Die Deutsche Bibliothek — CIP-Cataloguing-in-Publication Data
A catalogue record for this book is available from Die Deutsche
Bibliothek.
ISBN 3-527-30257-3

© WILEY-VCH Verlag GmbH, Weinheim 2002
Printed on acid-free paper
.
All rights reserved (including those of translation into other languages).
No part of this book may be reproduced in any form — by photoprinting,
microfilm, or any other means — nor transmitted or translated into
machine language without written permission from the publishers.
Registered names, trademarks, etc. used in this book, even when not
specifically marked as such, are not to be considered unprotected by law.
Printing: Strauss Offsetdruck GmbH, Mörlenbach
Bookbinding: Litges & Dopf Buchbinderei GmbH, Heppenheim
Printed in the Federal Republik of Germany

Dedicated to
Micheline Pfister,
Sean and Nicolas Vogel,
and in fond memory of
Gerrit Jörgen Korvink

Maria Cristina Vecchi,
Sarah Maria Greiner
und im Gedenken an
Gertrud Maria Greiner

Contents

Contents
Preface
Chapter 1 Introduction

7
13
15

The System Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Popular Definitions and Acronyms . . . . . . . . . . . . . . . . . . 19
Semiconductors versus Conductors and Insulators . . . . . . . . 19
The Diode Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
The Transistor Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Passive Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Microsystems: MEMS, MOEMS, NEMS, POEMS, etc. . . . . . 21

Sources of Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Summary for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
References for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . 25

Semiconductors for Micro and Nanosystem Technology

7

Chapter 2 The Crystal Lattice System
Observed Lattice Property Data

27

. . . . . . . . . . . . . . . . . . . . 29

Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Silicon Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Silicon Nitride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Gallium Arsenide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Crystal Structure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Symmetries of Crystals

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Elastic Properties: The Stressed Uniform Lattice
Statics

. . . . . . . 48

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

The Vibrating Uniform Lattice

. . . . . . . . . . . . . . . . . . . . . 64

Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Phonons, Specific Heat, Thermal Expansion . . . . . . . . . . . . . 81

Modifications to the Uniform Bulk Lattice . . . . . . . . . . . . 88
Summary for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
References for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . 92

Chapter 3 The Electronic System

95

Quantum Mechanics of Single Electrons

. . . . . . . . . . . . . 96

Wavefunctions and their Interpretation . . . . . . . . . . . . . . . . . 97
The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Free and Bound Electrons, Dimensionality Effects

. . . . 106

Finite and Infinite Potential Boxes . . . . . . . . . . . . . . . . . . . . 106
Continuous Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 115
Potential Barriers and Tunneling . . . . . . . . . . . . . . . . . . . . . 115
The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Transitions Between Electronic States . . . . . . . . . . . . . . . . . 127
Fermion number operators and number states . . . . . . . . . . 130

Periodic Potentials in Crystal

. . . . . . . . . . . . . . . . . . . . . 132

The Bloch Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Formation of Band Structure . . . . . . . . . . . . . . . . . . . . . . . . 133
Types of Band Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Effective Mass Approximation . . . . . . . . . . . . . . . . . . . . . . . 139

Summary for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 140
References for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 141

8

Semiconductors for Micro and Nanosystem Technology

Chapter 4 The Electromagnetic System 143
Basic Equations of Electrodynamics

. . . . . . . . . . . . . . . 144

Time-Dependent Potentials . . . . . . . . . . . . . . . . . . . . . . . . .149
Quasi-Static and Static Electric and Magnetic Fields . . . . .151

Basic Description of Light

. . . . . . . . . . . . . . . . . . . . . . . 158

The Harmonic Electromagnetic Plane Wave . . . . . . . . . . . .158
The Electromagnetic Gaussian Wave Packet . . . . . . . . . . . .160
Light as Particles: Photons . . . . . . . . . . . . . . . . . . . . . . . . .162

Waveguides

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Example: The Homogeneous Glass Fiber

. . . . . . . . . . . . . .166

Summary for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 167
References for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 168

Chapter 5 Statistics
Systems and Ensembles

169
. . . . . . . . . . . . . . . . . . . . . . . . . 170

Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . .171
Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174
Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . .176

Particle Statistics: Counting Particles

. . . . . . . . . . . . . . . 178

Maxwell-Boltzmann Statistics . . . . . . . . . . . . . . . . . . . . . . .178
Bose-Einstein Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . .180
Fermi-Dirac Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181
Quasi Particles and Statistics . . . . . . . . . . . . . . . . . . . . . . . .182

Applications of the Bose-Einstein Distributions . . . . . . . 183
Electron Distribution Functions . . . . . . . . . . . . . . . . . . . 184
Intrinsic Semiconductors
Extrinsic Semiconductors

. . . . . . . . . . . . . . . . . . . . . . . . . . .184
. . . . . . . . . . . . . . . . . . . . . . . . . . .187

Summary for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 190
References for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . 190

Chapter 6 Transport Theory
The Semi-Classical Boltzmann Transport Equation

191
. . . . 192

The Streaming Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193
The Scattering Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .195
The BTE for Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .197
Balance Equations for Distribution Function Moments . . .197

Semiconductors for Micro and Nanosystem Technology

9

Relaxation Time Approximation

Local Equilibrium Description

. . . . . . . . . . . . . . . . . . . . . . 201

. . . . . . . . . . . . . . . . . . . . 204

Irreversible Fluxes and Thermodynamic Forces . . . . . . . . . 205
Formal Transport Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
The Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

From Global Balance to Local Non-Equilibrium

. . . . . . 219

Global Balance Equation Systems . . . . . . . . . . . . . . . . . . . . 220
Local Balance: The Hydrodynamic Equations . . . . . . . . . . 220
Solving the Drift-Diffusion Equations . . . . . . . . . . . . . . . . . 222
Kinetic Theory and Methods for Solving the BTE . . . . . . . . 227

Summary for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 231
References for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . 231

Chapter 7 Interacting Subsystems
Phonon-Phonon

233

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Phonon Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Heat Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Electron-Electron

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

The Coulomb Potential (Poisson Equation) . . . . . . . . . . . . . 240
The Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Plasma Oscillations and Plasmons . . . . . . . . . . . . . . . . . . . 243

Electron-Phonon

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Acoustic Phonons and Deformation Potential Scattering . . 246
Optical Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Piezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Piezoelectric Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Stress Induced Sensor Effects: Piezoresistivity . . . . . . . . . . 260
Thermoelectric Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

Electron-Photon

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Intra- and Interband Effects . . . . . . . . . . . . . . . . . . . . . . . . . 268
Semiconductor Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

Phonon-Photon

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

Elasto-Optic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
Light Propagation in Crystals: Phonon-Polaritons . . . . . . . 277

Inhomogeneities

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Lattice Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Scattering Near Interfaces (Surface Roughness, . . . . . . . . . 281

10

Semiconductors for Micro and Nanosystem Technology

Phonons at Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284
The PN Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .300
Metal-Semiconductor Contacts . . . . . . . . . . . . . . . . . . . . . .313

Summary for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 324
References for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . 324

Index

Semiconductors for Micro and Nanosystem Technology

327

11

Preface

This book addresses the engineering student and practising engineer. It
takes an engineering-oriented look at semiconductors. Semiconductors
are at the focal point of vast number of technologists, resulting in great
engineering, amazing products and unheard-of capital growth. The work
horse here is of course silicon. Explaining how semiconductors like silicon behave, and how they can be manipulated to make microchips that
work—this is the goal of our book.
We believe that semiconductors can be explained consistently without
resorting 100% to the complex language of the solid state physicist. Our
approach is more like that of the systems engineer. We see the semiconductor as a set of well-defined subsystems. In an approximately top-down
manner, we add the necessary detail (but no more) to get to grips with
each subsystem: The physical crystal lattice, and charge carriers in lattice
like potentials. This elemental world is dominated by statistics, making
strange observations understandable: This is the glue we need to put the
systems together and the topic of a further chapter. Next we show the the-

Semiconductors for Micro and Nanosystem Technology

13

ory needed to predict the behavior of devices made in silicon or other
semiconducting materials, the building blocks of modern electronics.
Our book wraps up the tour with a practical engineering note: We look at
how the various sub-systems interact to produce the observable behavior
of the semiconductor. To enrich the subject matter, we tie up the theory
with concise boxed topics interspersed in the text.
There are many people to thank for their contributions, and for their help
or support. To the Albert Ludwig University for creating a healthy
research environment, and for granting one of us (Korvink) sabbatical
leave. To Ritsumeikan University in Kusatsu, Japan, and especially to
Prof. Dr. Osamu Tabata, who hosted one of us (Korvink) while on sabbatical and where one chapter of the book was written. To the ETH Zurich
and especially to Prof. Dr. Henry Baltes, who hosted one of us (Korvink)
while on sabbatical and where the book project was wrapped up. To Prof.
Dr. Evgenii Rudnyi, Mr. Takamitsu Kakinaga, Ms. Nicole Kerness and
Mr. Sadik Hafizovic for carefully reading through the text and finding
many errors. To the anonymous reviewers for their invaluable input. To
Ms. Anne Rottler for inimitable administrative support. To VCH-Wiley
for their deadline tolerance, and especially to Dr. Jörn Ritterbusch and his
team for support. To Micheline and Cristina for enduring our distracted
glares at home as we fought the clock (the calendar) to finish, and for
believing in us.
Jan G. Korvink and Andreas Greiner,
Freiburg im Breisgau,
February 2002

14

Semiconductors for Micro and Nanosystem Technology

Chapter 1 Introduction

Semiconductors have complex properties, and in the early years of the
twentieth century these were mainly discovered by physicists. Many of
these properties have been harnessed, and have been exploited in ingenious microelectronic devices. Over the years the devices have been rendered manufacturable by engineers and technologists, and have spawned
off both a multi-billion € (or $, or ¥) international industry and a variety
of other industrial mini-revolutions including software, embedded systems, the internet and mobile communications. Semiconductors still lie at
the heart of this revolution, and silicon has remained its champion, warding off the competitors by its sheer abundance, suitability for manufacturing, and of course its tremendous head-start in the field. Silicon is the
working material of an exciting, competitive world, presenting a seemingly endless potential for opportunities.

Chapter Goal The goal of this chapter is to introduce the reader to the field of semiconductors, and to the purpose and organization of the book.

Semiconductors for Micro and Nanosystem Technology

15

Introduction

Chapter
Roadmap

In this chapter we first explain the conceptual framework of the book.
Next, we provide some popular definitions that are in use in the field.
Lastly, we indicate some of the sources of information on new inventions.

1.1 The System Concept
This book is about semiconductors. More precisely, it is about semiconductor properties and how to understand them in order to be exploited for
the design and fabrication of a large variety of microsystems. Therefore,
this book is a great deal about silicon as a paradigm for semiconductors.
This of course implies that it is also about other semiconductor systems,
namely for those cases where silicon fails to show the desired effects due
to a lack of the necessary properties or structure. Nevertheless, we will
not venture far away from the paradigmatic material silicon, with its
overwhelming advantage for a wide field of applications with low costs
for fabrication. To quote the Baltes theorem [1.1]:
To prove your idea, put in on silicon.
Since you will need circuitry, make it with CMOS.
If you want to make it useful, get it packaged.
The more expensive fabrication becomes, the less attractive the material
is for the design engineer. Designers must always keep in mind the cost
and resources in energy and personnel that it takes to handle materials
that need additional nonstandard technological treatment. This is not to
say that semiconductors other than silicon are unimportant, and there are
many beautiful applications. But most of today’s engineers encounter silicon CMOS as a process with which to realize their ideas for microscopic
systems. Therefore, most of the emphasis of this book lies in the explanation of the properties and behavior of silicon, or better said, “the semiconductor system silicon”.

16

Semiconductors for Micro and Nanosystem Technology

The System Concept

A semiconductor can be viewed as consisting of many subsystems. For
one, there are the individual atoms, combining to form a chunk of crystalline material, and thereby changing their behavior from individual systems to a composite system. The atomic length scale is still smaller than
typical length scales that a designer will encounter, despite the fact that
sub-nanometer features may be accessible through modern experimental
techniques. The subsystems that this book discusses emerge when silicon
atoms are assembled into a crystal with unique character. We mainly discuss three systems:

• the particles of quantized atom vibration, or phonons;
• the particles of quantized electromagnetic radiation, or photons;
• the particles of quantized charge, or electrons.
There are many more, and the curious reader is encouraged to move on to
other books, where they are treated more formally. The important feature
of these subsystems is that they interact. Each interaction yields effects
useful to the design engineer.
Why do we emphasize the system concept this much? This has a lot to do
with scale considerations. In studying nature, we always encounter scales
of different order and in different domains. There are length scales that
play a significant role. Below the nanometer range we observe single
crystal layers and might even resolve single atoms. Thus we become
aware that the crystal is made of discrete constituents. Nevertheless, on a
micrometer scale—which corresponds to several thousands of monatomic layers—the crystal appears to be a continuous medium. This
means that at certain length scales a homogenous isotropic continuum
description is sufficient. Modern down-sizing trends might force us to
take at least anisotropy into account, which is due to crystal symmetry, if
not the detailed structure of the crystal lattice including single defects.
Nanotechnology changes all of this. Here we are finally designing at the
atomic length scale, a dream that inspired the early twentieth century.

Semiconductors for Micro and Nanosystem Technology

17

Introduction

Almost everything becomes quantized, from thermal and electrical resistance to interactions such as the Hall effect.
Time scale considerations are at least as important as length scales. They
are governed by the major processes that take place within the materials.
The shortest technological time scales are found in electron-electron
scattering processes and are on the order of a few femtoseconds, followed by the interaction process of lattice vibrations and electronic systems with a duration of between a few hundreds of femtoseconds to a
picosecond. Direct optical transitions from the conduction band to the
valence band lie in the range of a few nanoseconds to a few microseconds. For applications in the MHz (106 Hz) and GHz (109 Hz) regime the
details of the electron-electron scattering process are of minor interest
and most scattering events may be considered to be instantaneous. For
some quantum mechanical effects the temporal resolution of scattering is
crucial, for example the intra-collisional field effect.
The same considerations hold for the energy scale. Acoustic electron
scattering may be considered elastic, that is to say, it doesn’t consume
energy. This is true only if the system’s resolution lies well above the few
meV of any scattering process. At room temperature ( 300 K) this is a
good approximation, because the thermal energy is of the order of 25.4
meV. The level of the thermal energy implies a natural energy scale, at
which the band gap energy of silicon of about 1.1 eV is rather large. For
high energy radiation of several keV the band gap energy again is negligible.
The above discussion points out the typical master property of a composite system: A system reveals a variety of behavior at different length
(time, energy, …) scales. This book therefore demands caution to be able
to account for the semiconductor as a system, and to explain its building
blocks and their interactions in the light of scale considerations.

18

Semiconductors for Micro and Nanosystem Technology

Popular Definitions and Acronyms

1.2 Popular Definitions and Acronyms
The microelectronic and microsystem world is replete with terminology
and acronyms. The number of terms grows at a tremendous pace, without
regard to aesthetics and grammar. Their use is ruled by expedience. Nevertheless, a small number have survived remarkably long. We list only a
few of the most important, for those completely new to the field.

1.2.1 Semiconductors versus Conductors and Insulators
A semiconductor such as silicon provides the technologist with a very
special opportunity. In its pure state, it is almost electrically insulating.
Being in column IV of the periodic table, it is exceptionally balanced,
and comfortably allows one to replace the one or other atom of its crystal
with atoms from column III or V (which we will term P and N type doping). Doing so has a remarkable effect, for silicon then becomes conductive, and hence the name “semiconductor”. Three important features are
easily controlled. The density of “impurity” atoms can vary to give a tremendously wide control over the conductivity (or resistance) of the bulk
material. Secondly, we can decide whether electrons, with negative
charge, or holes, with positive charge, are the dominant mechanism of
current flow, just by changing to an acceptor or donor atom, i.e., by
choosing P or N type doping. Finally, we can “place” the conductive
pockets in the upper surface of a silicon wafer, and with a suitable geometry, create entire electronic circuits.
If silicon is exposed to a hot oxygen atmosphere, it forms amorphous silicon dioxide, which is a very good insulator. This is useful to make
capacitor devices, with the SiO 2 as the dielectric material, to form the
gate insulation for a transistor, and of course to protect the top surface of
a chip.
Silicon can also be grown as a doped amorphous film. In this state we
lose some of the special properties of semiconductors that we will
explore in this book. In the amorphous form we almost have metal-like
Semiconductors for Micro and Nanosystem Technology

19

Introduction

behavior, and indeed, semiconductor foundries offer both real metals
(aluminium, among others) and polysilicon as “metallic” layers.

1.2.2 The Diode Family
The simplest device that one can make using both P and N doping is the
diode. The diode is explained in Section 7.6.4. The diode is a one-way
valve with two electrical terminals, and allows current to flow through it
in only one direction. The diode provides opportunities for many applications. It is used to contact metal wires to silicon substrates as a Shottkey
diode. The diode can be made to emit light (LEDs). Diodes can detect
electromagnetic radiation as photo-detectors, and they form the basis of
semiconductor lasers. Not all of these effects are possible using silicon,
and why this is so is also explained later on.

1.2.3 The Transistor Family
This is the true fame of silicon, for it is possible to make this versatile
device in quantities unheard of elsewhere in the engineering world.
Imagine selling a product with more than 10 9 working parts! Through
CMOS (complimentary metal oxide semiconductor) it is possible to create reliable transistors that require extraordinary little power (but remember that very little times 10 9 can easily amount to a lot). The trend in
miniaturization, a reduction in lateral dimensions, increase in operation
speed, and reduction in power consumption, is unparalleled in engineering history. Top that up with a parallel manufacturing step that does not
essentially depend on the number of working parts, and the stage is set
for the revolution that we have witnessed.
The transistor is useful as a switch inside the logic gates of digital chips
such as the memories, processors and communications chips of modern
computers. It is also an excellent amplifier, and hence found everywhere
where high quality analog circuitry is required. Other uses include magnetic sensing and chemical sensing.

20

Semiconductors for Micro and Nanosystem Technology

Popular Definitions and Acronyms

1.2.4 Passive Devices
In combination with other materials, engineers have managed to miniaturize every possible discrete circuit component known, so that it is possible to create entire electronics using just one process: resistors,
capacitors, inductors and interconnect wires, to name the most obvious.
For electromagnetic radiation, waveguides, filters, interferometers and
more have been constructed, and for light and other forms of energy or
matter, an entirely new industry under the name of microsystems has
emerged, which we now briefly consider.

1.2.5 Microsystems: MEMS, MOEMS, NEMS, POEMS, etc.
In North America, the acronym MEMS is used to refer to micro-electromechanical systems, and what is being implied are the devices at the
length scale of microelectronics that include some non-electrical signal,
and very often the devices feature mechanical moving parts and electrostatic actuation and detection mechanisms, and these mostly couple with
some underlying electrical circuitry. A highly successful CMOS MEMS,
produced by Infineon Technologies, is shown in Figure 1.1. The device,
a)

Figure 1.1. MEMS device. a) Infineon’s surface micromachined
capacitive pressure sensor with
interdigitated signal conditioning
Type KP120 for automotive BAP
and MAP applications. b) SEM
photograph of the pressure sensor
cells compared with a human hair.
Image © Infineon Technologies,
Munich [1.2].

b)

Semiconductors for Micro and Nanosystem Technology

21

Introduction

placed in a low-cost SMD package, is used in MAP and BAP tire pressure applications. With an annual production running to several millions,
it is currently sold to leading automotive customers [1.2].
MEMS has to date spawned off two further terms that are of relevance to
us, namely MOEMS, for micro-opto-electro-mechanical systems, and
NEMS, for the inevitable nano-electro-mechanical systems.
MOEMS can include entire miniaturized optical benches, but perhaps the
most familiar example is the digital light modulator chip sold by Texas
Instruments, and used in projection display devices, see Figure 1.2.

Figure 1.2. MOEMS device.
This 30 µm by 30 µm device is
a single pixel on a chip that has as
many pixels as a modern computer
screen display. Each mirror is
individually addressable, and
deflects light from a source to a
systems of lenses that project the
pixel onto a screen. Illustration ©
Texas Instruments Corp., Dallas
[1.3].

As for NEMS, the acronym of course refers to the fact that a critical
dimension is now no longer the large micrometer, but has become a factor 1000 smaller. The atomic force microscope cantilever [1.4] may
appear to be a MEMS-like device, but since it resolves at the atomic
diameter scale, it is a clear case of NEMS, see Figure 1.3. Another example is the distributed mirror cavity of solid-state lasers made by careful
epitaxial growth of many different semiconductor layers, each layer a
few nanometers thick. Of major commercial importance is the submicron microchip electronic device technology. Here the lateral size of a
transistor gate is the key size, which we know has dropped to below 100

22

Semiconductors for Micro and Nanosystem Technology

Popular Definitions and Acronyms

Figure 1.3. NEMS devices.
Depicted are two tips of an atomic
force microscope, made in CMOS,
and used to visualize the force
field surrounding individual
atoms. Illustration © Physical
Electronics Laboratory, ETH Zurich, Switzerland [1.4].

nm in university and industrial research laboratories. Among NEMS we
count the quantum wire and the quantum dot, which have not yet made it
to the technological-commercial arena, and of course any purposefullydesigned and functional molecular monolayer film.
POEMS, or polymer MEMS, are microstructures made of polymer materials, i.e., they completely depart from the traditional semiconductorbased devices. POEMS are usually made by stereo micro-lithography
through a photo-polymerization process, by embossing a polymer substrate, by milling and turning, and by injection moulding. This class of
devices will become increasingly important because of their potentially
low manufacturing cost, and the large base of materials available.
In Japan, it is typical to refer to the whole field of microsystems as
Micromachines, and manufacturing technology as Micromachining. In
Europe, the terms Microsystems, Microtechnology or Microsystem
Technology have taken root, with the addition of Nanosystems and the
inevitable Nanosystem Technology following closely. The European
naming convention is popular since it is easily translated into any of a
large number of languages (German: Mikrosystemtechnik, French:
Microtechnique, Italian: tecnologia dei microsistemi, etc.).

Semiconductors for Micro and Nanosystem Technology

23

Introduction

1.3 Sources of Information
We encourage every student to regularly consult the published literature,
and in particular the following journals:
IOP

• Journal of Micromechanics and Microengineering
• Journal of Nanotechnology

IEEE

• Journal of Microelectromechanical Systems
• Journal of Electron Devices
• Journal of Sensors
• Journal of Nanosystems

Other

• Wiley-VCH Sensors Update, a journal of review articles
• MYU Journal of Sensors and Materials
• Elsevier Journal of Sensors and Actuators
• Springer Verlag Journal of Microsystem Technology
• Physical Review
• Journal of Applied Physics
Of course there are more sources than the list above, but it is truly impossible to list everything relevant. Additional sources on the world-wideweb are blossoming (see e.g. [1.5]), as well as the emergence of standard
texts on technology, applications and theory. A starting point is best taken
from the lists of chapter references. Two useful textbook references are
Sze’s book on the physics of semiconductor devices [1.6] and Middelhoek’s book on silicon sensors [1.7].

1.4 Summary for Chapter 1
Silicon is a very important technological material, and understanding its
behavior is a key to participating in the largest industry ever created. To

24

Semiconductors for Micro and Nanosystem Technology

References for Chapter 1

understand the workings of the semiconductor silicon, it helps to
approach it as a system of interacting subsystems.
The subsystems comprise the crystal lattice and its quantized vibrations—the phonons, electromagnetic radiation and its quantized form—
the photons, and the loosely bound quantized charges—the electrons.
The interactions between these systems is a good model with which to
understand most of the technologically useful behavior of silicon. To
understand the ensuing topics, we require a background in particle statistics. To render the ideas useful for exploitation in devices such as diodes,
transistors, sensors and actuators, we require an understanding of particle
transport modelling.
These topics are now considered in more detail in the six remaining
chapters of the book.

1.5 References for Chapter 1
1.1
1.2
1.3
1.4

1.5
1.6
1.7

Prof. Dr. Henry Baltes, Private communication
Prof. Dr. Christofer Hierold, Private communication.
See e.g. http://www.dlp.com
D. Lange, T. Akiyama, C. Hagleitner, A. Tonin, H. R. Hidber, P.
Niedermann, U. Staufer, N. F. de Rooij, O. Brand, and H. Baltes,
Parallel Scanning AFM with On-Chip Circuitry in CMOS Technology, Proc. IEEE MEMS, Orlando, Florida (1999) 447-452
See e.g. http://www.memsnet.org/
S. M. Sze, Physics of Semiconductor Devices, 2nd Ed., John Wiley
and Sons, New York (1981)
Simon Middelhoek and S. A. Audet, Silicon Sensors, Academic
Press, London (1989)

Semiconductors for Micro and Nanosystem Technology

25

Chapter 2 The Crystal
Lattice System

In this chapter we start our study of the semiconductor system with the
crystals of silicon ( Si ) , adding some detail on crystalline silicon dioxide
( SiO 2 ) and to a lesser extent on gallium arsenide ( GaAs ) . All three are
regular lattice-arrangements of atoms or atoms. For the semiconductors
silicon and gallium arsenide, we will consider a model that completely
de-couple the behavior of the atoms from the valence electrons, assuming
that electronic dynamics can be considered as a perturbation to the lattice
dynamics, a topic dealt with in Chapter 3. For all the electrons of the
ionic crystal silicon dioxide, as well as the bound electrons of the semiconductors, we here assume that they obediently follow the motion of the
atoms.
We will see that by applying the methods of classical, statistical and
quantum mechanics to the lattice, we are able to predict a number of
observable constitutive phenomena of interest—i.e., we are able to
explain macroscopic measurements in terms of microscopic crystal lattice mechanics. The effects include an approximation for the elastic coef-

Semiconductors for Micro and Nanosystem Technology

27

The Crystal Lattice System

ficients of continuum theory, acoustic dispersion, specific heat, thermal
expansion and heat conduction. In fact, going beyond our current goals, it
is possible to similarly treat dielectric, piezoelectric and elastooptic
effects. However, the predictions are of a qualitative nature in the majority of cases, mainly because the interatomic potential of covalently
bonded atoms is so hard to come by. In fact, in a sense the potential is
reverse engineered, that is, using measurements of the crystal, we fit
parameters that improve the quality of the models to make them in a
sense “predictive”.
Chapter Goal Our goal for this chapter is to explain the observed crystal data with preferably a single comprehensive model that accounts for all effects.
Chapter
Roadmap

Our road map is thus as follows. We start by stating some of the relevant
observable data for the three materials Si , SiO 2 and GaAs , without
more than a cursory explanation of the phenomena.
Our next step is to get to grips with the concept of a crystal lattice and
crystal structure. Beyond this point, we are able to consider the forces
that hold together the static crystal. This gives us a method to describe
the way the crystal responds, with stress, to a strain caused by stretching
the lattice. Then we progress to vibrating crystal atoms, progressively
refining our method to add detail and show that phonons, or quantized
acoustic pseudo “particles”, are the natural result of a dynamic crystal
lattice.
Considering the phonons in the lattice then leads us to a description of
heat capacity. Moving away from basic assumptions, we consider the
anharmonic crystal and find a way to describe the thermal expansion. The
section following presents a cursory look at what happens when the regular crystal lattice is locally deformed through the introduction of foreign
atoms. Finally, we leave the infinitely-extended crystal model and briefly
consider the crystal surface. This is important, because most microsystem devices are build on top of semiconductor wafers, and so are repeat-

28

Semiconductors for Micro and Nanosystem Technology

Observed Lattice Property Data

edly subject to the special features and limitations that the surface
introduces.

2.1 Observed Lattice Property Data
Geometric
Structure

The geometric structure of a regular crystal lattice is determined using xray crystallography techniques, by recording the diffraction patterns of xray photons that have passed through the crystal. From such a recorded
pattern (see Figure 2.1 (a)), we are able to determine the reflection planes
formed by the constituent atoms and so reconstruct the relative positions
of the atoms. This data is needed to proceed with a geometric (or, strictly
speaking, group-theoretic) characterization of the crystal lattice’s symmetry properties. We may also use an atomic force microscope (AFM) to
map out the force field that is exerted by the constituent atoms on the surface of a crystal. From such contour plots we can reconstruct the crystal
structure and determine the lattice constants. We must be careful, though,
because we may observe special surface configurations in stead of the
actual bulk crystal structure, see Figure 2.1 (b).

Elastic
Properties

The relationship between stress and strain in the linear region is via the
elastic property tensor, as we shall shortly derive in Section 2.3.1. To
measure the elastic parameters that form the entries of the elastic property tensor, it is necessary to form special test samples of exact geometric
shape that, upon mechanical loading, expose the relation between stress
and strain in such a way that the elastic coefficients can be deduced from
the measurement. The correct choice of geometry relies on the knowledge of the crystal’s structure, and hence its symmetries, as we shall see
in Section 2.2.1. The most common way to extract the mechanical properties of crystalline materials is to measure the direction-dependent
velocity of sound inside the crystal, and by diffracting x-rays through the
crystal (for example by using a synchrotron radiation source).

Semiconductors for Micro and Nanosystem Technology

29

The Crystal Lattice System

(a)

(b)

Figure 2.1. (a) The structure of a silicon crystal as mapped out by x-ray crystallography
in a Laue diagram. (b) The famous 〈 111〉 7 × 7 -reconstructed surface of silicon as
mapped out by an atomic force microscope. The bright dots are the Adatoms that screen
the underlying lattice. Also see Figure 2.30.

Dispersion
Curves

Dispersion curves are usually measured by scattering a neutron beam (or
x-rays) in the crystal and measuring the direction-dependent energy lost
or gained by the neutrons. The absorption or loss of energy to the crystal
is in the form of phonons.

Thermal
Expansion

Thermal expansion measurements proceed as for the stress-strain measurement described above. A thermal strain is produced by heating the
sample to a uniform temperature. Armed with the knowledge of the elastic parameters, the influence of the thermal strain on the velocity of
sound may then be determined.
The material data presented in the following sections and in Table 2.1
was collected from references [2.5, 2.6]. Both have very complete tables
of measured material data, together with reference to the publications
where the data was found.

30

Semiconductors for Micro and Nanosystem Technology

Crystalline
Silicon
Semiconductors for Micro and Nanosystem Technology

Property

Si

LPCVD
Poly-Si

Atomic Weight

Si: 28.0855

Si: 28.0855 .

Process

Gallium
Arsenide

α -Quartz

Si 3 N 4

GaAs

Si: 28.0855

Si: 28.0855

Si: 28.0855

O: 15.994

O: 15.994

N: 14.0067

Ga: 69.72
As: 74.9216

Thermal
Oxide SiO 2

Carbon-like,
Cubic
symmetry

Polycrystalline

Amorphous

Trigonal
symmetry

Amorphous

Zinc-blende
Cubic
symmetry

Density ( kg ⁄ m )

2330

2330

< 2200

2650

3100

5320

Elastic moduli
( Gpa )
Y: Young modulus
B: Bi-axial modulus
S: Shear modulus
C i : Tensor

Pure:
C 11 : 166

Y: 130 – 174
S: 69

Y: 72 – 75

Y: 87

Y: 97 – 320
B: 249 - 311

Pure:
C 11 : 118.1

Crystal class/
Symmetry
3

coefficients.

C 12 : 63.9

C 12 : 53.2

C 44 : 79.6

C 44 : 59.4

p-Type:
C 11 : 80.5
C 12 : 115
C 44 : 52.8

n-Type:
C 11 : 97.1
C 12 : 54.8
C 44 : 172

Observed Lattice Property Data

Table 2.1. Lattice properties of the most important microsystem base materialsa.

31

Crystalline
Silicon
Property
Semiconductors for Micro and Nanosystem Technology

Hardness ( Gpa )

Process

Gallium
Arsenide

α -Quartz

Si 3 N 4

GaAs

14.4 – 18

8.2

-

-

5.43 ,
polycrystal

Amorphous

X i 4.9127

Amorphous

5.65

Si

LPCVD
Poly-Si

Thermal
Oxide SiO 2

〈 100〉 :

10.5 – 12.5

5.1 – 13 ;
〈 111〉 : 11.7

Lattice parameter
( ) X, C: axes

5.43

Melting point ( °C )

1412

1412 .

1705

-

1902

-

Poisson ratio ν̃

〈 100〉 : 0.28

0.2 – 0.3

0.17 - 0.22

0.169

0.26

0.31

C 5.4046

〈 111〉 : 0.36

Specific heat
C p ( J ⁄ kgK

702.24

702.24

740

740

750

350

Thermal
conductivity
( W ⁄ mK )

150

150

1.1 - 1.5

1.4

18

46 ( 12 )

Thermal expansion
coefficient

2.33 ×10

XY cut 14.3
Z cut 7.8

2.7 ×10

α (T

–1

)

–6

–6

2.33 ×10

–6

0.4 ×10

–6

0.55 ×10

-

a. The tabulated values for amorphous process materials are foundry-dependent and are provided only as an indication of typical values. Also, many of the measurements on crystalline materials are for doped samples and
hence should be used with care. Properties depend on the state of the material, and a common choice is to
describe them based on the temperature and the pressure during the measurements. For technological work, we
require the properties under operating conditions, i.e., at room temperature and at 1 atmosphere of pressure.

–6

–6

6.86 ×10

The Crystal Lattice System

32

Table 2.1. Lattice properties of the most important microsystem base materialsa.

Observed Lattice Property Data

2.1.1 Silicon
A semiconductor quality Silicon ingot is a gray, glassy, face-centered
crystal. The element is found in column IV of the periodic table. It has
the same crystal structure as diamond, as illustrated in Figure 2.2. Silicon

(a)

(b)

(c)

(d)

(e)

Figure 2.2. The diamond-like structure of the Silicon crystal is caused by the tetrahedral
3
arrangement of the four sp bond-forming orbitals of the silicon atom, symbolically
shown (a) as stippled lines connecting the ball-like atomic nuclei in this tetrahedral
repeating unit. (b) When the atoms combine to form a crystal, we observe a structure that
may be viewed as a set of two nested cubic lattices, or (c) a single face-centered cubic lat3
tice with a basis. (d) The structure of the sp hybrid bonds. (e) The fcc unit cell can also
be viewed as four tetrahedra.

has a temperature dependent coefficient of thermal expansion in K
described by
Semiconductors for Micro and Nanosystem Technology

–1

33

The Crystal Lattice System

α(T )
–3

–4

= ( 3.725 ( 1 – exp ( – 5.88 ×10 ( T – 124 ) ) ) + 5.548 ×10 T ) ×10
and a lattice parameter (the interatomic distance in
temperature as

–6

) that varies with

a(T )
–5

–9

(2.1)

= 5.4304 + 1.8138 ×10 ( T – 298.15 ) + 1.542 ×10 ( T – 298.15 )

2

(2.2)

We will later take a more detailed look at the thermal strain ε ( T – T 0 ) =
α ( T ) ( T – T 0 ) . Both α ( T ) and a ( T ) are plotted in Figure 2.3. Both of
5.440

4 ×10

–6

2 ×10

–6

5.438
5.436
5.434

400

800
0

Temperature in K

0

600
1200
Temperature in K

Figure 2.3. The thermal expansion properties of Silicon. Shown on the left is the temperature dependence of the lattice parameter (the size of a unit cell), and to its right is the
temperature dependence of the thermal expansion coefficient. The typical engineering
temperature range for silicon electronic devices is indicated by the background gray
boxes.

these properties are also dependent on the pressure experienced by the
material, hence we should write α ( T , p ) and a ( T , p ) . It is important to
note that “technological” silicon is doped with foreign atoms, and will in
general have material properties that differ from the values quoted in
Table 2.1, but see [2.6] and the references therein. Silicon’s phonon dispersion diagram is shown in Figure 2.4.

34

Semiconductors for Micro and Nanosystem Technology

Observed Lattice Property Data

ω(k )

X
U

W
K

Γ L

k
Γ

∆

X U, K Σ

Γ

Λ

k
L L

K

W

X

Figure 2.4. The measured and computed dispersion diagrams of crystalline silicon. The
vertical axis represents the phonon frequency, the horizontal axis represents straight-line
segments in k-space between the main symmetry points of the Brillouin zone, which is
shown as an insert. Figure adapted from [2.5].

Silicon is the carrier material for most of today’s electronic chips and
microsystem (or microelectromechanical system (MEMS), or micromachine) devices. The ingot is sliced into wafers, typically 0.5 mm thick
and 50 to 300 mm in diameter. Most electronic devices are manufactured in the first 10 µ m of the wafer surface. MEMS devices can extend
all the way through the wafer. Cleanroom processing will introduce foreign dopant atoms into the silicon so as to render it more conducting.
Other processes include subtractive etching steps, additive deposition
steps and modifications such as oxidation of the upper layer of silicon.
Apart from certain carefully chosen metal conductors, the most common
materials used in conjunction with silicon are oven-grown or deposited
thermal silicon oxides and nitrides (see the following sections), as well as
poly-crystalline silicon.

Semiconductors for Micro and Nanosystem Technology

35

The Crystal Lattice System

IC process quality LPCVD poly-crystalline silicon (Poly-Si) has properties that depend strongly on the foundry of origin. It is assumed to be isotropic in the plane of the wafer, and is mainly used as a thin film thermal
and electrical conductor for electronic applications, and as a structural
and electrode material for MEMS devices.

2.1.2 Silicon Dioxide
Crystalline silicon dioxide is better known as fused quartz. It is unusual
to obtain quartz from a silicon-based process, say CMOS, because the
production of crystalline quartz usually requires very high temperatures
that would otherwise destroy the carefully produced doping profiles in
the silicon. Semiconductor-related silicon dioxide is therefore typically
amorphous.
Since quartz has a non-cubic crystal structure, and therefore displays useful properties that are not found in high-symmetry cubic systems such as
silicon, yet are of importance to microsystems, we also include it in our
discussion. We consider α -quartz, one of the variants of quartz that is

Figure 2.5. Three perspective views of the trigonal unit cell of α -Quartz. From left to
right the views are towards the origin along the X 1 , X 2 and the C axes. The six large
spheres each with two bonds represent oxygen atoms, the three smaller spheres each with
four tetrahedral bonds represent silicon atoms.

36

Semiconductors for Micro and Nanosystem Technology

Observed Lattice Property Data
o

stable below 573 C , with trigonal crystal symmetry. The unit cell of the
quartz crystal is formed by two axes, called X 1 and X 2 , at 60° to each
other, see Figure 2.5. Quartz is non-centro-symmetric and hence piezoelectric. It also has a handedness as shown in Figure 2.6.

Figure 2.6. α -Quartz is found as
either a right or a left-handed
structure, as indicated by the thick
lines in the structure diagram that
form a screw through the crystal.
In the figure, 8 unit cells are
arranged in a ( 2 × 2 × 2 ) block.

2.1.3 Silicon Nitride
Crystalline silicon nitride (correctly known as tri-silicon tetra-nitride) is
not found on silicon IC wafers because, as for silicon dioxide, very high
temperatures are required to form the pure crystalline state, see
Figure 2.7. These temperature are not compatible with silicon foundry
processing. In fact, on silicon wafers, silicon nitride is usually found as
an amorphous mixture that only approaches the stochiometric relation of
Si 3 N 4 , the specific relation being a strong function of process parameters and hence is IC-foundry specific. In the industry, it is variously
referred to as “nitride”, “glass” or “passivation”, and may also contain
amounts of oxygen.

2.1.4 Gallium Arsenide
Crystalline gallium arsenide (GaAs) is a “gold-gray” glassy material with
the zinc-blende structure. When bound to each other, both gallium and
arsenic atoms form tetrahedral bonds. In the industry, GaAs is referred to
as a III-V (three-five), to indicate that it is a compound semiconductor
Semiconductors for Micro and Nanosystem Technology

37

The Crystal Lattice System

Figure 2.7. Silicon nitride appears
in many crystalline configurations.
The structure of α and of β
Si 3 N 4 are based on vertical
stacks of SiN 4 tetrahedra, as
shown in (I) and (II) respectively.

(I)

(II)

whose constituents are taken from the columns III and V of the periodic
table. The atoms form into a zinc-blende crystal, structurally similar to
the diamond-like structure of silicon, but with gallium and arsenic atoms
alternating, see Figure 2.8.

Figure 2.8. The zinc-blende structure of gallium arsenide. The two
atom types are represented by
spheres of differing diameter. Also
see Figure 2.2.

38

Semiconductors for Micro and Nanosystem Technology

Crystal Structure

Gallium arsenide is mainly used to make devices and circuits for the allimportant opto-electronics industry, where its raw electronic speed or the
ability to act as an opto-electronic lasing device is exploited. It is not
nearly as popular as silicon, though, mainly because of the prohibitive
processing costs. Gallium arsenide has a number of material features that
differ significantly from Silicon, and hence a reason why we have
included it in our discussion here. Gallium arsenide’s phonon dispersion
diagram is shown in Figure 2.9.
ω(k )

W
K

X
U
Γ L

k
Γ

∆

X

X U, K

Σ

Γ

Λ

L

Figure 2.9. The measured and computed dispersion diagrams of crystalline gallium arsenide. The vertical axis represents the phonon frequency, the horizontal axis represents
straight-line segments in k-space between the main symmetry points of the Brillouin zone,
which is shown as an insert. Figure adapted from [2.5].

2.2 Crystal Structure
As we have seen, crystals are highly organized regular arrangements of
atoms or ions. They differ from amorphous materials, which show no
regular lattice, and poly-crystalline materials, which are made up of adjacent irregularly-shaped crystal grains, each with random crystal orienta-

Semiconductors for Micro and Nanosystem Technology

39

The Crystal Lattice System

tion. From observations and measurements we find that it is the regular
crystalline structure that leads to certain special properties and behavior
of the associated materials. In this section we develop the basic ideas that
enable us to describe crystal structure analytically, so as to exploit the
symmetry properties of the crystal in a systematic way.

2.2.1 Symmetries of Crystals
Consider a regular rectangular arrangement of points on the plane. The
points could represent the positions of the atoms that make up a hypothetical two-dimensional crystal lattice. At first we assume that the atoms
are equally spaced in each of the two perpendicular directions, say by a
pitch of a and b . More general arrangements of lattice points are the
rule.
Translational
Invariance

Consider a vector a that lies parallel to the horizontal lattice direction
and with magnitude equal to the pitch a . Similarly, consider vector b in
the other lattice direction with magnitude equal to the pitch b . Then,
starting at point p i , we can reach any other lattice point q j with
q j = α j a + β j b , where α j and β j are integers. Having reached another
interior point q j , the vicinity is the same as for point p i , and hence we
say that the lattice is invariant to translations of the form α j a + β j b , see
the example in Figure 2.10.

Figure 2.10. In this 2-dimensional
infinitely-extending regular lattice
the 2 lattice vectors are neither
perpendicular nor of equal length.
Given a starting point, all lattice
points can be reached through
q j = α j a + β j b . The vicinities of
p i and q i are similar.

40

q j = 2a + 5b
qj

pi

b
a

Semiconductors for Micro and Nanosystem Technology

b

a

Crystal Structure

Rotational
Symmetry

If we consider the vicinity of an interior point p i in our lattice, and let us
assume that we have very many points in the lattice, we see that by rotating the lattice in the plane about point p i by an angle of 180° , the vicinity of the point p i remains unchanged. We say that the lattice is invariant
to rotations of 180° . Clearly, setting a = b and a ⊥ b makes the lattice
invariant to rotations of 90° as well. Rotational symmetry in lattices are
due to rotations that are multiples of either 60° , 90° or 180° , see
Figure 2.11. If the underlying lattice has a rotational symmetry, we will
expect the crystal’s material properties to have the same symmetries.

180°

90°

60°

Figure 2.11. Illustrations of lattice symmetries w.r.t. rotations.

Bravais
Lattice

With the basic idea of a lattice established, we now use the concept of a
Bravais lattice to model the symmetry properties of a crystal’s structure.
A Bravais lattice is an infinitely extending regular three-dimensional
array of points that can be constructed with the parametrized vector
q j = α ja + β jb + γ jc

(2.3)

where a , b and c are the non-coplanar lattice vectors and α j , β j and
γ j are arbitrary (positive and negative) integers. Note that we do not
Semiconductors for Micro and Nanosystem Technology

41

The Crystal Lattice System

assume that the vectors a , b and c are perpendicular. The Bravais lattice
is inherently symmetric with respect to translations q j : this is the way we
construct it. The remainder of the symmetries are related to rotations and
reflections. There are 7 crystal systems: hexagonal, trigonal, triclinic,
monoclinic, orthorhombic, tetragonal and cubic. In addition, there are
14 Bravais lattice types, see Figure 2.12. These are grouped into the following six lattice systems in decreasing order of geometric generality (or
increasing order of symmetry): triclinic, monoclinic, orthorhombic, tetragonal, Hexagonal and cubic. The face-centered cubic (fcc) diamondlike lattice structure of silicon is described by the symmetric arrangement
of vectors shown in Figure 2.13 (I). The fcc lattice is symmetric w.r.t. 90o
rotations about all three coordinate axes. Gallium arsenide´s zinc-blende
bcc-structure is similarly described as for silicon. Because of the presence of two constituent atoms, GaAs does not allow the same translational symmetries as Si.
Primitive
Unit Cell

We associate with a lattice one or more primitive unit cells. A primitive
unit cell is a geometric shape that, for single-atom crystals, effectively
contains one lattice point. If the lattice point is not in the interior of the
primitive cell, then more than one lattice point will lie on the boundary of
the primitive cell. If, for the purpose of illustration, we associate a sphere
with the lattice point, then those parts of the spheres that overlap with the
inside of the primitive cell will all add up to the volume of a single
sphere, and hence we say that a single lattice point is enclosed. Primitive
cells seamlessly tile the space that the lattice occupies, see Figure 2.14.

Wigner-Seitz
Unit Cell

The most important of the possible primitive cells is the Wigner-Seitz
cell. It has the merit that it contains all the symmetries of the underlying
Bravais lattice. Its definition is straightforward: The Wigner-Seitz cell of
lattice point p i contains all spatial points that are closer to p i than to any
other lattice point q j . Its construction is also straight-forward: Considering lattice point p i , connect p i with its neighbor lattice points q j . On
each connection line, construct a plane perpendicular to the connecting
line at a position halfway along the line. The planes intersect each other

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Crystal Structure

TC

OP

T

OI

MI

MP

OC

OF

HP

HR

TI
TP

CF
CI
CP
Figure 2.12. The fourteen Bravais lattice types. CP: Cubic P; CI: Body-centered cubic;
CF: Face-centered cubic; a = b = c , α = β = γ = 90° . TP: Tetragonal; TI: Bodycentered tetragonal; a = b ≠ c , α = β = γ = 90° . HP: Hexagonal; HR: Hexagonal R;
a = b ≠ c , α = β = 90° , γ = 120° (this not a separate Bravais lattice type). OP:
Orthorhombic; OI: Body-centered orthorhombic; OF: Face-centered orthorhombic; OC:
C-orthorhombic; a ≠ b ≠ c , α = β = γ = 90° . MP: Monoclinic; MI: Face-centered
monoclinic; a ≠ b ≠ c , α = β = 90° ≠ γ . T: Trigonal; a = b = c , α ≠ β ≠ γ . TC: Triclinic; a ≠ b ≠ c , α ≠ β ≠ γ . a, b, c refer to lattice pitches; α, β, γ to lattice vector
angles.

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The Crystal Lattice System

b
b

c

c

a

a
Figure 2.13. The equivalent lattice vectors sets that define the structures of the diamondlike fcc structure of Si and the zinc-blende-like structure of GaAs . The vectors are either
the set: a = ( 2, 0, 0 ) , b = ( 0, 2, 0 ) , c = ( 0.5, 0.5, 0.5 ) or the set:
a = ( 0.5, 0.5, – 0.5 ) , b = ( 0.5, – 0.5, 0.5 ) , c = ( 0.5, 0.5, 0.5 ) .

Figure 2.14. Illustrations of some
of the many primitive unit cells for
a 2-dimensional lattice. Of these,
only (g) is a Wigner-Seitz cell. Figure adapted from [2.10].

(f)
(e)

(c)
(b)
(a)

(g)

(d)

and, taken together, define a closed volume around the lattice point p i .
The smallest of these volumes is the Wigner-Seitz cell, illustrated in
Figure 2.15.
Reciprocal
Lattice

The spatial Bravais crystal lattice is often called the direct lattice, to refer
to the fact that we can associate a reciprocal lattice with it. In fact, in
studying the properties of the crystal lattice, most data will be referred to

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Crystal Structure

(I)

(II)
(III)

Figure 2.15. The Wigner-Seitz cell for the bcc lattice. (I) shows the cell without the atomic
positions. (II) includes the atomic positions so as to facilitate understanding the method
of construction. (III) The Wigner-Seitz cell tiles the space completely.

the reciprocal lattice. Let us consider a plane wave (see Box 2.1), traversing the lattice, of the form
e

ik • r

.

(2.4)

We notice two features of this expression: the vectors k and r are symmetrical in the expression (we can swap their positions without altering
the expression); and the crystal is periodic in r space. The vector r is a
position in real space; the vector k is called the position vector in reciprocal space, and often also called the wave vector, since it is “defined”
using a plane wave. Next, we consider waves that have the same periodicity as the Bravais lattice. The Bravais lattice points lie on the regular
grid described by the vectors q , so that a periodic match in wave amplitude is expected if we move from one grid position to another, and therefore
e

ik • ( r + q )

= e

ik • r

.

(2.5)

This provides us with a condition for the wave vector k for waves that
have the same spatial period as the atomic lattice, because we can cancel
ik • r
out the common factor e
to obtain
e

ik • q

0

= 1 = e .

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(2.6)

45

The Crystal Lattice System

Equation (2.6) generates the k (wave) vectors from the q (lattice position) vectors, and we see that these are mutually orthogonal. We summarize the classical results of the reciprocal lattice:
Reciprocal
Lattice
Properties

• The reciprocal lattice is also a Bravais lattice.
• Just as the direct lattice positions are generated with the primitive unit
vectors q j = α j a + β j b + γ j c , the reciprocal lattice can also be so
generated using k j = δ j d + ε j e + ζ j f . The relation between the two
primitive vector sets is
b×c
d = 2π -----------------------a ⋅ (b × c)

(2.7a)

c×a
e = 2π -----------------------a ⋅ (b × c)

(2.7b)

a×b
f = 2π -----------------------a ⋅ (b × c)

(2.7c)

• The reciprocal of the reciprocal lattice is the direct lattice.
• The reciprocal lattice also has a primitive cell. This cell is called the
Brillouin zone after its inventor.

• The volume of the direct lattice primitive cell is v = a ⋅ ( b × c ) .
• The

volume

of the reciprocal lattice
V = d ⋅ ( e × f ) . From (2.7c) we see that
2π
d = ------ ( b × c )
v

2π
e = ------ ( c × a )
v

primitive

2π
f = ------ ( a × b )
v

cell

is

(2.8)

and hence that
3

( 2π )
V = ------------v

(2.9)

Miller Indices The planes formed by the lattice are identified using Miller indices.
These are defined on the reciprocal lattice, and are defined as the coordinates of the shortest reciprocal lattice vector that is normal to the plane.
Thus, if we consider a plane passing through the crystal, the Miller indi-

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Crystal Structure

Box 2.1. Plane waves and wave-vectors.
We describe a general plane wave with
i ( k ⋅ r – ωt )

ψ ( r, t ) = Ae
.
(B 2.1.1)
Recalling the relation between the trigonometric
functions and the exponential function,
iθ

(B 2.1.2)
e = cos ( θ ) + i sin ( θ ) .
we see that ψ is indeed a wave-like function, for
θ parametrizes an endless circular cycle on the
complex plane.
I

e

θ

iθ

counts the number cycles completed along a spatial segment.
If we fix the time t (“freezing” the wave in space
and time), then moving along a spatial direction
we will experience a wave-like variation of the
amplitude of ψ as k ⋅ r . If we now move in a
direction perpendicular to the propagation of the
wave k , then we will experience no amplitude
modulation. Next, staying in the perpendicular
direction to the wave propagation at a fixed position, if the time is again allowed to vary, we will
now experience an amplitude modulation as the
wave moves past us.

R

Figure B2.1.1: The exponential function
describes a cycle in the complex plane.
Equation (B 2.1.1) is a powerful way of describing
a plane wave. Two vectors, the spatial position
vector r and the wave vector, or reciprocal position vector k , are position arguments to the exponential function. The cyclic angle that it the wave
has rotated through is the complete factor
( k ⋅ r – ωt ) . Since r measures the distance along
an arbitrary spatial direction and k points along
the propagation direction of the wave, k ⋅ r gives
the component of this parametric angle. k measures rotation angle per distance travelled (a full
cycle of 2π is one wavelength), so that k ⋅ r

x

t

Figure B2.1.2: The appearance of a 1dimensional wave plotted for t and x as
parameters. If t is kept stationary, moving in the
x-direction is accompanied by a wave-like
variation.

ces are the components of a k -space vector (the vector is normal to the
plane) that fulfil k ⋅ r = Constant , where r lies on the plane of interest.
As an example, consider a cubic lattice with the Miller indices ( m, n, p )
of a plane that lies parallel to a face of the cube. The indices determine
the plane-normal k -space vector k = mb 1 + nb 2 + pb 3 . The plane is
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The Crystal Lattice System

k ⋅ r = Constant
= 2π ( r 1 m + r 2 n + r 3 p ) .
fixed
by
Since
k ⋅ a 1 = 2πm , k ⋅ a 2 = 2πn and k ⋅ a 3 = 2πp , the space axis interr 1 = Constant ⁄ 2πm ,
cepts for the plane are therefore
r 2 = Constant ⁄ 2πn and r 3 = Constant ⁄ 2πp .
Silicon
Reciprocal
Lattice Shape

We have seen that the silicon crystal can be represented by a face-centered cubic lattice with a basis. This means that its reciprocal lattice is a
body-centered lattice with a basis. This has the following implications.
The Wigner-Seitz cell for the silicon direct lattice is a rhombic dodecahedron, whereas the first Brillouin zone (the Wigner-Seitz cell of the reciprocal lattice) is a truncated dodecahedron.

2.3 Elastic Properties: The Stressed Uniform
Lattice
In a broad sense the geometry of a crystal’s interatomic bonds represent
the “structural girders” of the crystal lattice along which the forces act
that keep the crystal intact. The strength of these directional interatomic
forces, and the way in which they respond to small geometrical perturbations; these are the keys that give a crystal lattice its tensorial elastic
properties and that enable us to numerically relate applied stress to a
strain response. In this section we derive the Hooke law for crystalline,
amorphous and poly-crystalline materials based on lattice considerations.

2.3.1 Statics
Atomic Bond
Model

It is well known that atoms form different types of bonds with each other.
The classification is conveniently viewed as the interaction between a
pair of atoms:

• Ionic—a “saturated” bond type that is characterized by the fact that
one atom ties up the electrons participating in the bond in its outermost shell. This leaves the two atoms oppositely charged. The cou-

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Elastic Properties: The Stressed Uniform Lattice

lombic (electrostatic) force between the net charges of the constituent
“ions” form the bond. This bond is sometimes termed localized,
because the electrons are tightly bound to the participating atoms.

• van der Waals—a very weak bonding force that is often termed the
fluctuating dipole force because it is proportional to an induced dipole
between the constituent atoms, and this effective dipole moment has a
non-vanishing time average.

• Valency—another localized electron-pair bond. Here, the electronpair of the participating atoms form a hybrid orbital that is equally
shared by the two atoms, hence the term covalent. Clearly, ionic and
covalent bonds are the two limiting cases of a similar phenomena, so
that a bond in-between these limits can also be expected. Valency
bonds are quite strong, and account for the hardness and brittle nature
of the materials.

• Metallic—the electrons participating in the bonding are non-localized. Typically, the number of valence electrons at a point is exceeded
by the number of nearest-neighbor atoms. The electrons are therefore
shared by many atoms, making them much more mobile, and also
accounts for the ductility of the material.
a)

b)

1
σ 1 = --- ( s + p x + p y + p z )
2
1
σ 3 = --- ( s – p x + p y – p z )
2

c)

1
σ 2 = --- ( s + p x – p y – p z )
2
1
σ 4 = --- ( s – p x – p y + p z )
2

Figure 2.16. The geometry of a tetrahedral bond for Carbon-like atoms can be repre3
sented by a hybrid sp -function as a superposition of 2s and 2p-orbitals. a) s-orbital. b)
3
p-orbital. c) s p -orbital.
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The Crystal Lattice System

The semiconductors that we consider are covalently bonded. We now
give a qualitative description of the tetrahedral covalent bond of the
atoms of an fcc crystal. Solving the Schrödinger equation (see Box 2.2
Box 2.2. The stationary Schrödinger equation and the Hamiltonian of a Solid [2.4].
In principle, all basic calculations of solid state
properties are computed with the Schrödinger
equation
Hϕ = Eϕ
(B 2.2.1)
The Hamiltonian operator H defines the dynamics and statics of the model, ϕ represents a state
of the model, and E is the scalar-valued energy.
Thus the Schrödinger equation is an Eigensystem
equations with the energy plying the role of an
eigenvalue and the state the role of an eigen-function.
The Hamiltonian is usually built up of contributions from identifiable subcomponents of the system. Thus, for a solid-state material we write that
H = H e + H i + H ei + H x

(B 2.2.2)

i.e., the sum of the contributions from the electrons, the atoms, their interaction and interactions
with external influences (e.g., a magnetic field).
Recall that the Hamiltonian is the sum of kinetic
and potential energy terms. Thus, for the electrons
we have
H e = T e + U ee
2

=

pα

1

e

2

+ ------------ ′ ------------------∑ ------2m 8πε 0 ∑ r α – r β
α

The second term is the Coulombic potential
energy due to the electron charges. Note that the
sum is primed: the sum excludes terms where
α = β.
For the atoms the Hamiltonian looks similar to
that of the electron
H i = T i + U ii
2

=

Pα

1

+ --- ′V ( R – R β )
∑ -------2M 2 ∑ i α
α

(B 2.2.4)

αβ

For the electron-atom interaction we associate
only a potential energy
H ei = U ei =

∑ V ei ( rα – Rβ )

(B 2.2.5)

αβ

Equation (B 2.2.1) is hardly ever solved in all its
generality. The judicious use of approximations
and simplifications have yielded not only tremendous insight into the inner workings of solid state
materials, but have also been tremendously successful in predicting complex phenomena.
We will return to this topic in the next chapter,
where we will calculate the valence band structure
of silicon to remarkable accuracy.

(B 2.2.3)

αβ

and Chapter 3) for a single atom yields the orthogonal eigenfunctions
that correspond to the energy levels of the atom, also known as the orbitals. The spherical harmonic functions shown in Table 2.2 are such eigenfunctions. The tetrahedral bond structure of the Si atom can be made,
3
through a superposition of basis orbitals, to form the hybrid s p orbital

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Elastic Properties: The Stressed Uniform Lattice

m

2

Table 2.2. The spherical harmonic functions Y l ( θ, φ ) , Also see Table 3.1 and
Figure 3.9 in Chapter 3.
l

m Plot

l

m Plot

0

0

1

l

m Plot

0

1

1

2

0

2

3

0

4

0

l

m Plot

1

2

2

3

1

3

2

3

3

4

1

4

2

4

3

that we associate with its valence electrons. The construction is illustrated in Figure 2.16.
The valence electron orbitals may overlap to form bonds between atoms.
The interpretation is straightforward: a valence electron’s orbital represents the probability distribution of finding that electron in a specific
region of space. In a first approximation, we let the orbitals simply overlap and allow them to interfere to form a new shared orbital. Bonding
takes place if the new configuration has a lower energy than the two separate atom orbitals. The new, shared, hybrid orbital gives the electrons of

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51

The Crystal Lattice System

the two binding atoms a relatively high probability of occupying the
space between the atoms.
Figure 2.17 shows what happens when atoms bond to form a diamond-

a

a

E(r)
r

a

(I)

(II)

r=a

Figure 2.17. When atoms with a tetrahedral bond structure form a covalently-bonded
crystal lattice, the valence electrons are localized about the nearest-neighbor atoms. We
visualize these (I) as the overlap positions of the sp3-hybrid orbitals, shown here for four
atoms in a diamond-like lattice. We assume that the distance a between the atomic cores,
the lattice constant, is the position of minimum energy for the bond (II).

like crystal. The energy of the atomic arrangement is lowered to a minimum when all atoms lie approximately at a separation equal to the lattice
constant a (the lattice constant is the equilibrium distance that separates
3
atoms of a lattice). The s p orbitals of neighboring atoms overlap to
form new shared orbitals. The electrons associated with these new states
are effectively shared by the neighboring atoms, but localized in the
bond. The bonding process does not alter the orbitals of the other electrons significantly.
The potential energy plot of Figure 2.17 (II) is necessarily only approximate, yet contains the necessary features for a single ionic bond. In fact,
it is a plot of the Morse potential, which contains a weaker attractive and
a very strong, repulsive constituent, localized at the core

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Elastic Properties: The Stressed Uniform Lattice

E ( r ) = D e ( 1 – exp [ – β ( r – r eq ) ] )

2

(2.10)

The potential energy for bond formation D e and the equilibrium bond
length r eq depend on the constituent atoms, and may be obtained by
experiment. The parameter β controls the width of the potential well,
i.e., the range of the interparticle forces. For covalent crystals, such as silicon, this picture is too simplistic. The tetrahedral structure of the orbitals
forms bonds that can also take up twisting moments, so that, in addition
to atom pair interactions, also triplet and perhaps even larger sets of interacting atoms should be considered. A three-atom interaction model that
accounts well for most thermodynamic quantities of Si appears to be the
Stillinger-Weber potential [2.8], which, for atom i and the interaction
with its nearest neighbors j , k , l and m is
1
E Si-Si(r i, r j, r k, r l, r m) = --- [ E 2(r ij) + E 2(r ik) + E 2(r il) + E 2(r im) ]
2
(2.11)
+ E 3(r ij, r ik, θ jik) + E 3(r im, r ik, θ mik) + E 3(r il, r im, θ lim)
+ E 3(r ij, r im, θ jim) + E 3(r ij, r il, θ jil) + E 3(r il, r ik, θ lik)
In this model, the two-atom interaction is modelled as
E 2(r ij)
 EG ( H e
= 


–p

– r ij ⁄ a

–q

–1

) exp ( ( r ij ⁄ a – c ) )

0

r ij ⁄ a < c

(2.12)

r ij ⁄ a > c

and the three-atom interaction by
E 3(r ij, r ik, θ jik)
–1

r ik
r
 –1
 Eλ exp γ   ----ij- – c +  ----- – c 
a
a


= 
1 2
×  cos ( θ jik ) + ---


3


0


, r ij ⁄ a < c
(2.13)

otherwise

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The Crystal Lattice System

For silicon, this model works well with the following parameters:
G = 7.0495563 , H = 0.60222456 , p = 4 , q = 0 , λ = 21 ,
γ = 1.2 , the cut-off radius c = 1.8 , the “lattice” constant
– 19
a = 0.20951nm and the bond energy E = 6.9447 ×10 J ⁄ ion . Note
that in the above r ij = r i – r j .
Linearization

We assume that the atoms of the crystal always remain in the vicinity of
their lattice positions, and that the distance they displace from these positions is “small” when measured against the lattice constant a , see
Figure 2.18. This is a reasonable assumption for a solid crystal at typical
k
uk
rk = Rk + uk

i

Rk

Figure 2.18. Instantaneous snapshot of the atom positions of a regular square lattice with
respect to their average lattice site positions. On the right is shown the relation between
the lattice position vector R k , the atom position vector r k and the atom displacement
vector u k for atom k. The shading around atom i indicates how the inter-atom interaction
strength falls off as a function of distance.

operating temperatures; the covalent and ionic bonds are found to be
strong enough to make it valid.

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Elastic Properties: The Stressed Uniform Lattice

Linearized
Potential
Energy

To illustrate the derivation of the elastic energy in terms of the strain and
the elastic constants of the crystal, we will use an interatomic potential
that only involves two-atom interactions. For a more general derivation,
see e.g. [2.1]. Denote the relative vector position of atom i by u i , and its
absolute position by r i = R i + u i , where R i localizes the regular lattice
site. For a potential binding energy E between a pair of atoms as
depicted in Figure 2.17 (II), we form the potential energy of the whole
crystal with
N

1
U = --2

N

∑∑

N

1
E ( r i – r j ) = --2

i=1j=1

N

∑ ∑ E(R – R
i

j

+ ui – u j )

i=1j=1
N

1
= --2

.

N

∑ ∑ E(R

ij

(2.14)

+ u ij )

i=1j=1

where R ij = R i – R j and u ij = u i – u j . The factor 1 ⁄ 2 arises because
the double sum counts each atom pair twice. We expand the energy about
the lattice site R ij using the Taylor expansion for vectors
1
2
E ( R ij + u ij ) = E ( R ij ) + ( u ij ⋅ ∇ )E ( R ij ) + --- ( u ij ⋅ ∇ ) E ( R ij ) + … (2.15)
2
because of the assumption that the atom displacements u i and hence
n
their differences u ij are small. The terms ( u ij ⋅ ∇ ) E ( R ij ) must be read
n
as ( u ij ⋅ ∇ ) operating n times on the position-dependent energy E
evaluated at the atom position R ij . Applying (2.15) to (2.14) we obtain
N

1
U = --2

N

N

∑∑

1
E ( R ij ) + --2

i=1j=1
N
N

1
+ --4

∑ ∑ (u

N

∑∑u

ij

• ∇E ( R ij )

i=1j=1

(2.16)

2

ij

• ∇ ) ( E ( R ij ) ) + …

i=1j=1

The first term in (2.16) is a constant for the lattice, and is denoted by U o .
The second term is identically zero, because the energy gradient is evaluated at the rest position of each atom where by definition is must be zero

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The Crystal Lattice System

N

1
--2

N

∑∑u

ij

• ∇E ( R ij ) = 0

(2.17)

i=1j=1

This leaves us with the zeroth and second and higher order terms
N

1
U = U o + --4

N

∑ ∑ (u

2

ij

• ∇ ) E ( R ij ) + …

(2.18)

i=1j=1

The first term in (2.18) is the energy associated with the atoms at the lattice positions, i.e., at rest, and represents the datum of energy for the
crystal. The second term is the harmonic potential energy, or the smalldisplacement potential energy. When it is expanded, we obtain
N

1
h
U = --4

N

∑∑u

ij

• ∇( ∇E ( R ij ) ) • u ij

(2.19)

i=1j=1

Elasticity
Tensor

Equation (2.19) illustrates two terms, the deformation felt between two
sites u ij , and the second derivative of the interatomic potential at zero
deformation ∇( ∇E ( R ij ) ) . The second derivative is the “spring constant”
of the lattice, and the deformation is related to the “spring extension”. We
can now go a step further and write the harmonic potential energy in
terms of the elastic constants and the strain.
Moving towards a continuum view, we will write the quantities in (2.19)
in terms of the position R i alone. Consider the term
∇[ ∇E ( R ij ) ] = ∇[ ∇E ( R i – R j ) ] . We expect E ( R ij ) to fall off rapidly
away from R i when R j is far removed, so that for the site R i , most of
the terms in (2.19) in the sum over j are effectively zero. To tidy up the
notation, we denote D ( R i ) αβ as the components of the dynamical tensor
D ( R i ) = ∇[ ∇E ( R ij ) ] . The remainder of the sites R j are close to R i ,
making R ij small, so that we are justified in making an expansion of the
components of D ( R ij ) about R i
D ( R ij ) αβ = D ( R i ) αβ + ∇ { D ( R i ) αβ } • R ij + …

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(2.20)

Elastic Properties: The Stressed Uniform Lattice

for which we only keep the leading term. We now apply a similar series
expansion to the term u j , because we expect it to vary little in the vicinity of R i , and from now on we do not consider atom sites that are far
away, to obtain
u j = u i + R ij • ∇u i(R i) + …

(2.21)

Taking the first right-hand-side term of (2.20) and the first two terms of
(2.21), and inserting these into (2.19), we obtain
N

1
h
U = – --4

N

∑ ∑ (R

T

ij

• ∇u i ) • D ( R i ) • ( R ij • ∇u i )

(2.22)

i=1j=1

The result of (2.22) can now be rewritten again in terms of the original
quantities, to give
N

1
h
U = – --4

∑

N

T

( ∇u i ) :

i=1

∑ {R

ij D ( R i )R ij } : ∇u i

j=1

(2.23)

N

1
= – --2

∑ ( ∇u ) :F(R ): ∇u
T

i

i

i

i=1

The rank four elastic material property tensor F(R i) in the vicinity of the
lattice site R i is defined as
N

1
F(R i) = --2

∑ (R

ij D ( R i )R ij )

(2.24)

j=1

in terms of the crystal constituent positions and the resulting net interatom binding energy. We expect this tensor to be translationally invariant
with respect to the lattice. Therefore, we can now move from a discrete
crystal description to the continuum, by considering the crystal as a collection of primitive cells of volume V with an average “density” of elastic material property in any particular unit cell to be E = F ⁄ V , so that
we can replace the sum in (2.23) by an integral to obtain

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The Crystal Lattice System

1
T
h
U = – --- ( ∇u i ) • E • ∇u i dV
2

∫

(2.25)

Ω

Strain

We now wish to bring the gradient of deformation ∇u , a vector-valued
field over the crystal, in relation to the strain, which is a rank two tensorvalued field. We consider Figure 2.19, where we follow the behavior of
x
P′
dx′
Q′′
X

P

dx
Q

Figure 2.19. An arbitrary body before and after deformation. The points P and Q are
“close” to each other.

two points P and Q before and after deformation. The points are chosen
to be close to each other, and we assume that the body has deformed elastically, i.e., without cracks forming, and without plastically yielding. To a
very good approximation, the movement of each point in the body can
then be written as a linear transformation
P′ = α o + ( δ + α ) • P

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(2.26)

Elastic Properties: The Stressed Uniform Lattice

where α o represents a pure translation and α a rotation plus stretch.
Next, we consider what happens to the line element d x under the deformation field u
T

T

d x′ = Q′ – P′ = α o + ( δ + α ) • Q – ( α o + ( δ + α ) • P )
T

(2.27)

= d x + α • d x = d x + du
T

We can rewrite du via the chain rule as du = ( ∇u ) • d x , thereby
making the association α = ∇u , to find that we can now rewrite the
expression for d x' only in terms of the original quantities that we measure
T

d x' = d x + ( ∇u ) • d x

(2.28)

We are interested in how the line element deforms (stretches), and for
this we form an expression for the square of its length
2

T

T

T

T

T

( d x' ) = ( d x ) • d x + [ ( ∇u ) • d x ] • d x + ( d x ) • [ ( ∇u ) • d x ]
T

T

T

+ [ ( ∇u ) • d x ] • [ ( ∇u ) • d x ]
T

T

(2.29)

T

= ( d x ) • [ δ + ∇u + ( ∇u ) + ( ∇u ) • ∇u ] • d x
T

T

Consider the argument [ δ + ∇u + ( ∇u ) + ( ∇u ) • ∇u ] . If the stretch is
small, which we have assumed, then we can use 1 + s ≅ 1 + s ⁄ 2 ,
which defines the strain ε by
1
1
T
T
T
ε = --- [ ∇u + ( ∇u ) + ( ∇u ) • ∇u ] ≈ --- [ ∇u + ( ∇u ) ]
2
2

(2.30)

The components of the small strain approximation may be written as
1 ∂u α ∂u β
+
ε αβ = --2 ∂ xβ ∂ xα

(2.31)

The small strain is therefore a symmetric rank two tensor.

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59

The Crystal Lattice System

Crystal
Energy in
Terms of
Strain

We return our attention to equation (2.25), where our goal is to rewrite
h

the harmonic crystal energy U in terms of the crystal’s strain. Clearly,
h
U should not be dependent on our choice of coordinate axes, and is
therefore invariant with respect to rigid-body rotations. We now choose a
deformation field u = θ × R which simply rotates the crystal atoms
from their lattice positions by a constant defined angle θ . The gradient of
this field is ∇u = ∇θ × R + θ × ∇R = 0 , so that the energy associated
with pure rotations is clearly zero. The gradient of u can always be written as the sum of a symmetric and an anti-symmetric part
T

T

∇u = ∇u s + ∇u a = [ ∇u + ( ∇u ) ] ⁄ 2 + [ ∇u – ( ∇u ) ] ⁄ 2
= (ε + κ)
Note that ε = ε
obtain

T

(2.32)

T

and κ = – κ . We substitute ε and κ into (2.25) to

1
h
T
T
T
T
U = – --- ( ε :E:ε + ε :E:κ + κ :E:ε + κ :E:κ ) dV
2

∫

(2.33)

Ω

for the harmonic energy of the crystal.
Independent
Elastic
Constants

The elastic tensor E has inherent symmetries that can be exploited to
simplify (2.33). We first write an expression for the components of the
tensor E, centered at site i, in cartesian coordinates
N

E(R i) αβγµ

1
= ------2V

∑  ( R

j=1

 ∂

ij ) α  -------∂x β

∂E ( R i ) 
----------------- ( R ij ) γ

∂x µ 

(2.34)

Consider the argument of the sum. Clearly, E is symmetric with respect
to the indices β and µ , since the order of differentiation of the energy
with respect to a spatial coordinate is arbitrary. Furthermore, swapping
T
T
α and γ also has no effect. Now look at the terms ε :E:κ and κ :E:ε
in (2.33). They vanish because, due the above symmetries of E ,
T
T
ε :E:κ = – κ :E:ε . Finally, because the anti-symmetric strain κ repreT
sents a pure rotation, the last term κ :E:κ in (2.33) must also vanish. As

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Elastic Properties: The Stressed Uniform Lattice

a consequence, the tensor E must be symmetric with respect to its first
and second pairs of indices as well, because it now inherits the symmetry
of the symmetric strain. Thus we obtain the familiar expression for the
elastic energy as
1
h
T
U = – --- [ ε :E(R):ε ] dV
2

∫
V

N

 3
1  T 1
( R – R j ) D ( R ) ( R – R j ) :ε dR
= – ---  ε : ------2V
2 

V
j=1

(2.35)

∑

∫

Of the possible 81 independent components for the fourth rank tensor E,
only 21 remain. We can organize these into a 6 × 6 matrix C if the following six index pair associations are made:
E ( ij ) ( kl ) ≡ C ( m ) ( n )
11 → 1
23 → 4

ij → m
22 → 2
31 → 5

kl → n
33 → 3 

12 → 6 

(2.36)

In order to obtain analog relations using the reduced index formalism, we
have to specify the transformation of the stress and strain as well. For the
stress we can use the same index mapping as in (2.36)
σ ij ≡ s m
11 → 1
23 → 4

ij → m

22 → 2
31 → 5

33 → 3 

12 → 6 

(2.37)

For the strain, however, we need to combine the index mapping with a
scaling
e 1 = ε 11

e 2 = ε 22

e 3 = ε 33

e 4 = 2ε 23

e 5 = 2ε 31

e 6 = 2ε 12

(2.38)

In this way, we retain the algebraic form for the energy terms

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61

The Crystal Lattice System

1
1
T
3
T
3
h
U = – --- [ e • C(R) • e ] dR = – --- [ e • s ] dR
2
2

∫

∫

V

(2.39)

V

The further reduction of the number of independent elastic coefficients
now depends on the inherent symmetries of the underlying Bravais lattice. Silicon has a very high level of symmetry because of its cubic structure. The unit cell is invariant to rotations of 90° about any of its
coordinate axes. Consider a rotation of the x-axis of 90° so that x → x ,
y → z and z → – y . Since the energy will remain the same, we must have
that C 22 = C 33 , C 55 = C 66 and C 21 = C 31 . By a similar argument for
rotations about the other two axes, we obtain that C 11 = C 22 = C 33 ,
C 21 = C 31 = C 23 and C 44 = C 55 = C 66 . Since the other matrix
entries experience an odd sign change in the transformed coordinates, yet
symmetry of C is required, they must all be equal to zero. To summarize,
Si and other cubic-symmetry crystals have an elasticity matrix with the
following structure (for coefficient values, consult Table 2.1) but with
only three independent values:
C 11 C 12 C 12
C 12 C 11 C 12
C =

C 12 C 12 C 11

(2.40)
C 44
C 44
C 44

C can be inverted to produce S with exactly the same structure and the
following relation between the constants:
–1

–1

( S 11 – S 12 ) = ( C 11 – C 12 ) 

–1

( S 11 + 2S 12 ) = ( C 11 + 2C 12 )

S 44 = C 44

(2.41)

The bulk modulus B and compressibility K of the cubic material is
given by

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Elastic Properties: The Stressed Uniform Lattice

1
1
B = --- ( C 11 + 2C 12 ) = ---3
K

(2.42)

Isotropic materials have the same structure as (2.40) but with only two
independent constants. In the literature, however, no less that three notations are common. Often engineers use the Young modulus E and Poisson ratio ν , whereas we find the Lamé constants λ and µ often used in
the mechanics literature. The relations between the systems are summarized by:
C 11 – C 12

C 44 = ---------------------- = µ

2

Eν
C 11 = λ + 2µ
C 12 = -------------------------------------- 
( 1 + ν ) ( 1 – 2ν )


E
E(1 – ν)
C 11 = -------------------------------------- 
C 44 = -------------------2(1 + ν)
( 1 + ν ) ( 1 – 2ν ) 

µ ( 3λ + 2µ )
λ
E = ---------------------------ν = --------------------
λ+µ
2(λ + µ)

C 11 – 2C 44 
C 44 ( 3C 11 – 4C 44 )
ν = ------------------------------- 
E = ------------------------------------------C 11 – C 44
2 ( C 11 – C 44 ) 
C 12 = λ

(2.43)

The α -Quartz crystal has the following elastic coefficient matrix structure (for coefficient values, consult Table 2.1):
C 11 C 12 C 12 C 14
C 12 C 11 C 12 – C 14
C Quartz =

C 12 C 12 C 11
C 14 – C 14

(2.44)
C 44
C 44
C 44

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63

The Crystal Lattice System

2.4 The Vibrating Uniform Lattice
The elastic, spring-like nature of the interatomic bonds, together with the
massive atoms placed at regular intervals; these are the items we isolate
for a model of the classical mechanical dynamics of the crystal lattice
(see Box 2.3 for brief details on Lagrangian and Hamiltonian mechanics). Here we see that the regular lattice displays unique new features
unseen elsewhere: acoustic dispersion is complex and anisotropic, acoustic energy is quantized, and the quanta, called phonons, act like particles
carrying information and energy about the lattice.

2.4.1 Normal Modes
As we saw in Section 2.3, an exact description of the forces between the
atoms that make up a crystal are, in general, geometrically and mathematically very complex. Nevertheless, certain simplifications are possible here and lead both to an understanding of what we otherwise observe
in experiments, and often to a fairly close approximation of reality. We
assume that:

• The atoms that make up the lattice are close to their equilibrium positions, so that we may use a harmonic representation of the potential
binding energy about the equilibrium atom positions. The spatial gradient of this energy is then the position-dependent force acting on the
atom, which is zero when each atom resides at its equilibrium position.

• The lattice atoms interact with their nearest neighbors only.
• The lattice is infinite and perfect. This assumption allows us to limit
our attention to a single Wigner-Seitz cell by assuming translational
symmetry.

• The bound inner shell electrons move so much faster than the crystal
waves that they follow the movement of the more massive nucleus
that they are bound to adiabatically.

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The Vibrating Uniform Lattice

Box 2.3. A Brief Note on Hamiltonian and Lagrangian Mechanics.
Hamilton’s Principle. Hamilton’s principle states
that the variation of the action A, also called the
variational indicator, is a minimum over the path
chosen by a mechanical system when proceeding
from one known configuration to another
δA

t2
t1

= 0

(B 2.3.1)

The Action. The action is defined in terms of the
lagrangian L and the generalized energy sources
Ξ j ξ j of the system
A =

t2

∫t

1

n

L+

∑ Ξ jξ j

dt

(B 2.3.2)

j=1

where Ξ j is the generalized force and ξ j the generalized displacement
The Lagrangian. The mechanical Lagrangian of
a system is the difference between its kinetic coenergy T ∗ ( ξ˙ j, ξ j, t ) (the kinetic energy expressed
in terms of the system’s velocities) and its potential energy U ( ξ j, t )
L M ( ξ˙ j, ξ j, t ) = T ∗ ( ξ˙ j, ξ j, t ) – U ( ξ j, t ) (B 2.3.3)
In general, the lagrangian for a crystal is written in
terms of three contributions: the mechanical (or
matter), the electromagnetic field and the fieldmatter interaction
L =

∫ ( L M + LF + LI ) dV

(B 2.3.4)

V

1
L M = --- ( ρu̇ + ε • C • ε )
2

(B 2.3.5)

1
Ȧ
L F = --- ( E + B )
L I =  ---- – qψ̇
c

2
Generalized Energy Sources. This term groups
all external or non-conservative internal sources

(sinks) of energy in terms of generalized forces
Ξ j and the generalized displacements ξ j .
Generalized Displacements and Velocities. For
a system, we establish the m independent scalar
m
degrees of freedom d ∈ ℜ required to describe
its motion in general. Then we impose the p constraint equations B • d = 0 that specify the
m
required kinematics (the admissible path in ℜ ).
This reduces the number of degrees of freedom by
p, and we obtain the n = m – p generalized scan
m
lar displacements ξ ∈ ℜ ⊃ ℜ of the system.
The generalized velocities are simply the time rate
of change of the generalized displacements,
ξ̇ = dξ ⁄ dt .
Lagrange’s Equations. An immediate consequence of (B 2.3.2) is that the following equations
hold for the motion
d  ∂L  ∂L
= Ξj
 –
d t  ∂ ξ̇ j ∂ ξ j

(B 2.3.6)

These are known as the Lagrange equations of
motion. For a continuum, we can rewrite (B 2.3.6)
for the Lagrange density as
d
d ∂L
∂L
∂L
----- -------- = -------- – ---------- -----------------------------dt ∂ ẋ j
∂ x j d Xk ∂(∂ x j ⁄ ∂ Xk)

(B 2.3.7)

In the continuum crystal lagrangian the variables
X represent the material coordinates of the undeformed crystal; x the spatial coordinates of the
deformed crystal. The Lagrange equations are
most convenient, because they allow us to add
detail to the energy expressions, so as to derive the
equations of motion thereafter in a standard way.

• The valence electrons form a uniform cloud of negative space charge
that interacts with the atoms.

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65

The Crystal Lattice System

Normal
Modes

Normal modes are the natural eigen-shapes of the mechanical system.
We already know these from musical instruments: for example from the
shapes of a vibrating string (one-dimensional), the shapes seen on the
stretched surface of a vibrating drum (two-dimensional), or a vibrating
bowl of jelly (three-dimensional). Normal modes are important, because
they are orthogonal and span the space of the atomic movements and
thereby describe all possible motions of the crystal. To obtain the normal
modes of the crystal, we will assume time-dependent solutions that have
the same geometric periodicity of the crystal.
We will now derive an expression for the equations of motion of the crysh
tal from the mechanical Lagrangian L M = T ∗ – U (see Box 2.3.) From
h
T
the previous section we have that U = 1 ⁄ 2 ( ε :E:ε ) dV . The kinetic
V
co-energy is added up from the contributions of the individual atoms

∫

n

1
T ∗ = --2

∑ mṙ

2
i

(2.45)

i=1

As for the potential energy, this sum can also be turned into a volume
integral, thereby making the transition to a continuum theory. We consider a primitive cell of volume V
n

1
T ∗ = --2

∑

1
2
m
---- ṙ V = --V i
2

n

∑ ρṙ V ≅ --2- ∫ ρṙ dV
2
i

i=1

i=1

1

2
i

(2.46)

V

Thus we have for the mechanical Lagrangian that
LM =

∫L

M dV

V

LM

1
2
T
= --- ( ρṙ i – ε :E:ε ) dV , with
2

∫
V

1
2
T
= --- ( ρṙ i – ε :E:ε )
2

(2.47)

The continuum Lagrange equations read [2.7]
∂L
d
d ∂L
∂L
----- -------- = -------- – --------- ----------------------------dt ∂ẋ j
∂x j d X k ∂ ( ∂x j ⁄ ∂X k )

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(2.48)

The Vibrating Uniform Lattice

where we have chosen the coordinates X to describe the undeformed
configuration of the crystal, and x the coordinates in the deformed crystal, see Figure 2.19.
1D
Monatomic
Dispersion
Relation

The complexity in applying the lagrangian formulation to a general 3D
crystal can be avoided by considering a 1D model system that demonstrates the salient features of the more involved 3D system. A large 1D
lattice of N identical bound atoms are arranged in the form of a ring by
employing the Born-von Karmann boundary condition, i.e.,
u ( N + 1 ) = u ( N ) . From equation (2.31), the strain in the 1D lattice is
simply
1
du
T
ε 11 = --- ( ∇u + ∇u ) = ------2
dX

(2.49)

which gives a potential energy density of
1 du 2
U = --- E  -------
2  dX 

(2.50)

where u is the displacement of the atom from its lattice equilibrium site
and E is the linear Young modulus of the interatomic bond. Note that we
only consider nearest-neighbour interactions. The kinetic co-energy density is
1 2
*
T = --- ρu̇
2

(2.51)

where the mass density is ρ = m ⁄ a for an atomic mass m and interatomic spacing a . The lagrangian density for the chain is the difference
between the kinetic co-energy density and the potential energy density,
*
L = T – U . Note that u = X – x and hence that ∇u = Id – ∇x and
u̇ = – ẋ . We insert the lagrangian density in equation (2.48) to obtain
d du
ρu̇˙ = – E -------  -------
dX  dX 

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(2.52)

67

The Crystal Lattice System

Since the potential energy is located in the bonds and not the lattice site,
and therefore depends on the positions of the neighboring lattice sites,
2
2
the term d u ⁄ d X is replaced by its lattice equivalent for the lattice site
2
i ( 2u i – u i – 1 – u i + 1 ) ⁄ a . This finite difference formula expresses the
fact that the curvature of u at the lattice site i depends on the next-neighbor lattice positions. This step is necessary for a treatment of waves with
a wavelength of the order of the interatomic spacing. If we use the simpler site relation, we only obtain the long wavelength limit of the dispersion relation, indicated by the slope lines in Figure 2.20. We now look for
solutions, periodic in space and time, of the form
u ( X, t )

X = ia

= exp [ j ( kX – ωt ) ]

X = ia

(2.53)

which we insert into equation (2.52) and cancel the common exponential
E
2
– ρω = – ----2- ( 2 – exp [ kia ] – exp [ – kia ] )
a

(2.54)

Reorganizing equation (2.54), we obtain
ω =

ka
E
2E ( 1 – cos [ ka ] )
---------------------------------------- = 2 --------2 sin -----2
2
ρa
ρa

(2.55)

Equation (2.55) is plotted in Figure 2.20 on the left, and is the dispersion
relation for a monatomic chain. The curve is typical for an acoustic wave
in a crystalline solid, and is interpreted as follows. In the vicinity where
ω is small, the dispersion relation is linear (since sin [ ka ⁄ 2 ] ≈ ka ⁄ 2 )
and the wave propagates with a speed of E ⁄ ρ as a linear acoustic
wave. As the frequency increases, the dispersion relation flattens off,
causing the speed of the wave ∂ω ⁄ ∂k to approach zero (a standing wave
resonance).
1D Diatomic
Dispersion
Relation

Crystals with a basis, i.e., crystals with a unit cell that contains different
atoms, introduce an important additional feature in the dispersion curve.
We again consider a 1D chain of atoms, but now consider a unit cell containing two different atoms of masses m and M .

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The Vibrating Uniform Lattice

unit cell = a

unit cell = a

slope = 0
ω(k)

v

ω(k)

e=

v

slope =

Optical
branch

slop

Acoustic
branch
Acoustic
branch
π
--a

π
– --a

X

∆

Γ

∆

π
--a

π
– --a

k

X

∆

X

Γ

∆

k

X

Figure 2.20. The monatomic and diatomic one-dimensional chain lattices lend themselves
to analytical treatment. Depicted are the two computed dispersion curves with schematic
chains and reciprocal unit cell indicated as a gray background box. Γ is the symmetry
point at the origin, ∆ the symmetry point at the reciprocal cell boundary. Only nearestneighbor interactions are accounted for.

The kinetic co-energy is added up from the contributions of the individual atoms
1
T ∗ = --2

n, 2

∑ m u̇

2
α iα

(2.56)

iα

The index α counts over the atoms in an elementary basis cell, and the
index i counts over the lattice cells. Similarly, the bond potential energy
is dependent on the stretching of the inter-atom bonds
1
U = --4

n, 2, n, 2

∑

iα, βj

E αiβj
2
----------- ( u iα – u jβ )
a

Semiconductors for Micro and Nanosystem Technology

(2.57)

69

The Crystal Lattice System

The restriction to next-neighbor interactions and identical interatomic
force constants yields the following equations of motion when the
expressions (2.56) and (2.57) are inserted into the Lagrange equations
E
mu̇˙im = – --- ( 2u im – u iM – u ( i – 1 )M )
a

(2.58a)

E
Mu̇˙iM = – --- ( 2u iM – u ( i + 1 )m – u im )
a

(2.58b)

At rest the atoms occupy the cell positions (due to identical static forces)
( i – 1 ⁄ 4 )a and ( i + 1 ⁄ 4 )a . Hence we choose an harmonic atom displacement ansatz for each atom of the form


1
1
u im = -------- c m exp  j ka  i – --- – ωt 


4
m



(2.59a)



1
1
u iM = --------- c M exp  j ka  i + --- – ωt 


4
M



(2.59b)

The second ansatz can be written in terms of u im
u iM =

ka
m cM
-----  ------  exp  j ------ u im
 2
M  cm 

(2.60)

Equations (2.59b) and (2.60) are now inserted into (2.58a). Eliminating
common factors and simplifying, we obtain an equation for the amplitudes c m and c M
ka
2E
2E
2
 ----------– ω m – ------------ cos  ------
a m
 a M
 2

cm

= 0

(2.61)

c
ka
2E
2E
2
– ----------- cos  ------  ------------ – ω M M




2
a m
a M
This is an eigensystem equation for ω , and its non-trivial solutions are
obtained by requiring the determinant of the matrix to be zero. Performing this, we obtain

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The Vibrating Uniform Lattice

4
ka 2
E 1 1
1 1 2
2
ω = ---  ---- + ----- −
+  ---- + ----- – --------- sin  ------


mM
2
a m M
m M

(2.62)

The two solutions are plotted in Figure 2.20 on the right.
Discussion

The case when the two masses of the unit-cell atoms differ only by a
small amount, i.e., M = m + µ with µ small, is instructive. The optical
and acoustic branches approach then each other at the edge of the reciprocal cell, i.e., at k = −
+ π ⁄ a . As the mass difference µ goes to zero, the
lattice becomes a monatomic lattice with lattice constant a ⁄ 2 , so that the
branches touch each other at the reciprocal cell edge.
The interpretation of the two branches is as follows. For the lower
branch, all the atoms move in unison just as for an acoustic wave, hence
the name acoustic branch. In fact, it appears as a center-of-mass oscillation. For the upper or optical branch, the center of mass is stationary, and
the atoms of a cell only move relative to each other. Its name refers to the
fact that for ionic crystals, this mode is often excited by optical interactions.

2D Square
Lattice
Dispersion
Relation

The next construction shows the richness in structure that appears in the
dispersion relation when an additional spatial dimension and one level of
interaction is added to the 1D monatomic lattice. It serves as an illustration that the anisotropy of the interatomic binding energy enables more
involved crystal vibrational modes and hence additional branches in the
dispersion curves. At the same time it shows that most of the essential
features of the expected structure is already clear from the simple 1D
models.
The model considers a 2D monatomic square la