Main Phonons in nanostructures

Phonons in nanostructures

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This book focuses on the theory of phonon interactions in nanoscale structures with particular emphasis on modern electronic and optoelectronic devices. A key goal is to describe tractable models of confined phonons and how these are applied to calculations of basic properties and phenomena of semiconductor heterostructures. The level of presentation is appropriate for undergraduate and graduate students in physics and engineering with some background in quantum mechanics and solid state physics or devices.
Year: 2001
Edition: 1st
Publisher: Cambridge University Press
Language: english
Pages: 290
ISBN 10: 0521792797
ISBN 13: 9780521792790
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Phonons in nanostructures
This book focuses on the theory of phonon interactions in nanoscale structures
with particular emphasis on modern electronic and optoelectronic devices.
The continuing progress in the fabrication of semiconductor nanostructures with
lower dimensional features has led to devices with enhanced functionality and
even to novel devices with new operating principles. The critical role of phonon
effects in such semiconductor devices is well known. There is therefore a
pressing need for a greater awareness and understanding of confined phonon
effects. A key goal of this book is to describe tractable models of confined
phonons and how these are applied to calculations of basic properties and
phenomena of semiconductor heterostructures.
The level of presentation is appropriate for undergraduate and graduate students
in physics and engineering with some background in quantum mechanics and
solid state physics or devices. A basic understanding of electromagnetism and
classical acoustics is assumed.

ii

D R M ICHAEL A. S TROSCIO earned a Ph.D. in physics from Yale University
and held research positions at the Los Alamos Scientific Laboratory and the
Johns Hopkins University Applied Physics Laboratory, before moving into the
management of federal research and development at a variety of US government
agencies. Dr Stroscio has served as a policy analyst for the White House Office
of Science and Technology Policy and as Vice Chairman of the White House
Panel on Scientific Communication. He has taught physics and electrical
engineering at several universities including Duke University, the North
Carolina State University and the University of California at Los Angeles. Dr
Stroscio is currently the Senior Scientist in the Office of the Director at the US
Army Research Office (ARO) as well as an Adjunct Professor at both Duke
University and the North Carolina State University. He has authored about 500
publications, presentations and patents covering a wide variety of topics in the
physical sciences and electronics. He is the author of Quantum Heterostructures:
Microelectronics and Optoelectronics and the joint editor of two World Scientific
books entitled Quantum-based Electronic Devices and Systems and Advances in
Semiconductor Lasers and Applications to Optoelectronics. He is a Fellow of
both the Institute of Electrical and Electronics Engineers (IEEE) and the
American Association for the Advancement of Science and he was the 1998
recipient of the IEEE Harry Diamond Award.
D R D UTTA earned a Ph.D. in physics from the University of Cincinnati; she was
a research associate at Purdue University and at City College, New York, as well
as a visiting scientist at Brookhaven National Laboratory before assuming a
variety of government posts in research and development. Dr Dutta was the
Director of the Physics Division at the US Army’s Electronics Technology and
Devices Laboratory as well as at the Army Research Laboratory prior to her
appointment as the Associate Director for Electronics in the Army Research
Office’s Engineering Sciences Directorate. Dr Dutta recently assumed a senior
executive position as ARO’s Director of Research and Technology Integration.
She has over 160 publications, 170 conference presentations, 10 book chapters,
and has had 24 US patents issued. She is the joint editor of two World Scientific
books entitled Quantum-Based Electronic Devices and Systems and Advances in
Semiconductor Lasers and Applications to Optoelectronics. She is an Adjunct
Professor of the Electrical and Computer Engineering and Physics departments
of North Carolina State University and has had adjunct appointments at the
Electrical Engineering departments of Rutgers University and the University of
Maryland. Dr Dutta is a Fellow of both the Institute of Electrical and Electronics
Engineers (IEEE) and the Optical Society of America, and she was the recipient
in the year 2000 of the IEEE Harry Diamond Award.

Phonons in
Nanostructures
Michael A. Stroscio and Mitra Dutta
US Army Research Office, US Army Research Laboratory

         
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
  
The Edinburgh Building, Cambridge CB2 2RU, UK
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
Ruiz de Alarcón 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http://www.cambridge.org
© Michael A. Stroscio and Mitra Dutta 2004
First published in printed format 2001
ISBN 0-511-03187-4 eBook (Adobe Reader)
ISBN 0-521-79279-7 hardback

Mitra Dutta dedicates this book to her parents
Dhiren N. and Aruna Dutta
and
Michael Stroscio dedicates this book to
his friend and mentor Morris Moskow and
his friend and colleague Ki Wook Kim

Contents

Preface xi

Chapter 1 Phonons in nanostructures 1
1.1

Phonon effects: fundamental limits on carrier mobilities and dynamical

1.2

Tailoring phonon interactions in devices with nanostructure components 3

processes 1

Chapter 2 Phonons in bulk cubic crystals 6
2.1

Cubic structure 6

2.2

Ionic bonding – polar semiconductors 6

2.3

Linear-chain model and macroscopic models 7

2.3.1

Dispersion relations for high-frequency and low-frequency modes 8

2.3.2

Displacement patterns for phonons 10

2.3.3

Polaritons 11

2.3.4

Macroscopic theory of polar modes in cubic crystals 14

Chapter 3 Phonons in bulk würtzite crystals 16
3.1

Basic properties of phonons in würtzite structure 16

3.2

Loudon model of uniaxial crystals 18

3.3

Application of Loudon model to III-V nitrides 23

Chapter 4 Raman properties of bulk phonons 26
4.1

Measurements of dispersion relations for bulk samples 26

4.2

Raman scattering for bulk zincblende and würtzite structures 26

vii

Contents

viii

4.2.1

Zincblende structures 28

4.2.2

Würtzite structures 29

4.3

Lifetimes in zincblende and würtzite crystals 30

4.4

Ternary alloys 32

4.5

Coupled plasmon–phonon modes 33

Chapter 5 Occupation number representation 35
5.1

Phonon mode amplitudes and occupation numbers 35

5.2

Polar-optical phonons: Fröhlich interaction 40

5.3

Acoustic phonons and deformation-potential interaction 43

5.4

Piezoelectric interaction 43

Chapter 6 Anharmonic coupling of phonons 45
6.1

Non-parabolic terms in the crystal potential for ionically bonded atoms 45

6.2

Klemens’ channel for the decay process LO → LA(1) + LA(2) 46

6.3

LO phonon lifetime in bulk cubic materials 47

6.4

Phonon lifetime effects in carrier relaxation 48

6.5

Anharmonic effects in würtzite structures: the Ridley channel 50

Chapter 7 Continuum models for phonons 52
7.1

Dielectric continuum model of phonons 52

7.2

Elastic continuum model of phonons 56

7.3

Optical modes in dimensionally confined structures 60

7.3.1

Dielectric continuum model for slab modes: normalization of

7.3.2

Electron–phonon interaction for slab modes 66

interface modes 61
7.3.3

Slab modes in confined würtzite structures 71

7.3.4

Transfer matrix model for multi-heterointerface structures 79

7.4

Comparison of continuum and microscopic models for phonons 90

7.5

Comparison of dielectric continuum model predictions with Raman

7.6

Continuum model for acoustic modes in dimensionally confined structures 97

measurements 93
7.6.1

Acoustic phonons in a free-standing and unconstrained layer 97

7.6.2

Acoustic phonons in double-interface heterostructures 100

7.6.3

Acoustic phonons in rectangular quantum wires 105

7.6.4

Acoustic phonons in cylindrical structures 111

7.6.5

Acoustic phonons in quantum dots 124

Contents

Chapter 8 Carrier–LO-phonon scattering 131
8.1

Fröhlich potential for LO phonons in bulk zincblende and würtzite

8.1.1

Scattering rates in bulk zincblende semiconductors 131

8.1.2

Scattering rates in bulk würtzite semiconductors 136

8.2

Fröhlich potential in quantum wells 140

8.2.1

Scattering rates in zincblende quantum-well structures 141

8.2.2

Scattering rates in würtzite quantum wells 146

structures 131

8.3

Scattering of carriers by LO phonons in quantum wires 146

8.3.1

Scattering rate for bulk LO phonon modes in quantum wires 146

8.3.2

Scattering rate for confined LO phonon modes in quantum wires 150

8.3.3

Scattering rate for interface-LO phonon modes 154

8.3.4

Collective effects and non-equilibrium phonons in polar quantum wires 162

8.3.5

Reduction of interface–phonon scattering rates in metal–semiconductor

8.4

Scattering of carriers and LO phonons in quantum dots 167

structures 165

Chapter 9 Carrier–acoustic-phonon scattering 172
9.1

Carrier–acoustic-phonon scattering in bulk zincblende structures 172

9.1.1

Deformation-potential scattering in bulk zincblende structures 172

9.1.2

Piezoelectric scattering in bulk semiconductor structures 173

9.2

Carrier–acoustic-phonon scattering in two-dimensional structures 174

9.3

Carrier–acoustic-phonon scattering in quantum wires 175

9.3.1

Cylindrical wires 175

9.3.2

Rectangular wires 181

Chapter 10 Recent developments 186
10.1

Phonon effects in intersubband lasers 186

10.2

Effect of confined phonons on gain of intersubband lasers 195

10.3

Phonon contribution to valley current in double-barrier structures 202

10.4

Phonon-enhanced population inversion in asymmetric double-barrier quantum-well

10.5

Confined-phonon effects in thin film superconductors 208

10.6

Generation of acoustic phonons in quantum-well structures 212

lasers 205

Chapter 11 Concluding considerations 218
11.1

Pervasive role of phonons in modern solid-state devices 218

11.2

Future trends: phonon effects in nanostructures and phonon engineering 219

ix

x

Contents

Appendices 221
Appendix A: Huang–Born theory 221
Appendix B: Wendler’s theory 222
Appendix C: Optical phonon modes in double-heterointerface structures 225
Appendix D: Optical phonon modes in single- and double-heterointerface würtzite
structures 236
Appendix E: Fermi golden rule 250
Appendix F: Screening effects in a two-dimensional electron gas 252
References 257
Index 271

Preface

This book describes a major aspect of the effort to understand nanostructures,
namely the study of phonons and phonon-mediated effects in structures with
nanoscale dimensional confinement in one or more spatial dimensions. The necessity for and the timing of this book stem from the enormous advances made in the
field of nanoscience during the last few decades.
Indeed, nanoscience continues to advance at a dramatic pace and is making
revolutionary contributions in diverse fields, including electronics, optoelectronics,
quantum electronics, materials science, chemistry, and biology. The technologies
needed to fabricate nanoscale structures and devices are advancing rapidly. These
technologies have made possible the design and study of a vast array of novel
devices, structures and systems confined dimensionally on the scale of 10 nanometers or less in one or more dimensions. Moreover, nanotechnology is continuing
to mature rapidly and will, no doubt, lead to further revolutionary breakthroughs
like those exemplified by quantum-dot semiconductor lasers operating at room
temperature, intersubband multiple quantum-well semiconductor lasers, quantumwire semiconductor lasers, double-barrier quantum-well diodes operating in the
terahertz frequency range, single-electron transistors, single-electron metal-oxide–
semiconductor memories operating at room temperature, transistors based on carbon
nanotubes, and semiconductor nanocrystals used for fluorescent biological labels,
just to name a few!
The seminal works of Esaki and Tsu (1970) and others on the semiconductor
superlattice stimulated a vast international research effort to understand the fabrication and electronic properties of superlattices, quantum wells, quantum wires, and
quantum dots. This early work led to truly revolutionary advances in nanofabrication

xi

xii

Preface

technology and made it possible to realize band-engineering and atomic-level
structural tailoring not envisioned previously except through the molecular and
atomic systems found in nature. Furthermore, the continuing reduction of dimensional features in electronic and optoelectronic devices coupled with revolutionary
advances in semiconductor growth and processing technologies have opened many
avenues for increasing the performance levels and functionalities of electronic and
optoelectronic devices. Likewise, the discovery of the buckyball by Kroto et al.
(1985) and the carbon nanotube by Iijima (1991) led to an intense worldwide
program to understand the properties of these nanostructures.
During the last decade there has been a steady effort to understand the optical
and acoustic phonons in nanostructures such as the semiconductor superlattice,
quantum wires, and carbon nanotubes. The central theme of this book is the
description of the optical and acoustic phonons in these nanostructures. It deals
with the properties of phonons in isotropic, cubic, and hexagonal crystal structures
and places particular emphasis on the two dominant structures underlying modern
semiconductor electronics and optoelectronics – zincblende and würtzite. In view
of the successes of continuum models in describing optical phonons (Fasol et al.,
1988) and acoustic phonons (Seyler and Wybourne, 1992) in dimensionally confined
structures, the principal theoretical descriptions presented in this book are based
on the so-called dielectric continuum model of optical phonons and the elastic
continuum model of acoustic phonons. Many of the derivations are given for the
case of optical phonons in würtzite crystals, since the less complicated case for
zincblende crystals may then be recovered by taking the dielectric constants along
the c-axis and perpendicular to the c-axis to be equal.
As a preliminary to describing the dispersion relations and mode structures for
optical and acoustic phonons in nanostructures, phonon amplitudes are quantized in
terms of the harmonic oscillator approximation, and anharmonic effects leading to
phonon decay are described in terms of the dominant phonon decay channels. These
dielectric and elastic continuum models are applied to describe the deformationpotential, Fröhlich, and piezoelectric interactions in a variety of nanostructures
including quantum wells, quantum wires and quantum dots. Finally, this book
describes how the dimensional confinement of phonons in nanostructures leads to
modifications in the electronic, optical, acoustic, and superconducting properties of
selected devices and structures including intersubband quantum-well semiconductor
lasers, double-barrier quantum-well diodes, thin-film superconductors, and the thinwalled cylindrical structures found in biological structures known as microtubulin.
The authors wish to acknowledge professional colleagues, friends and family
members without whose contributions and sacrifices this work would not have been
undertaken or completed. The authors are indebted to Dr C.I. (Jim) Chang, who is
both the Director of the US Army Research Office (ARO) and the Deputy Director
of the US Army Research Laboratory for Basic Science, and to Dr Robert W. Whalin
and Dr John Lyons, the current director and most recent past director of the US Army

Preface

Research Laboratory; these leaders have placed a high priority on maintaining an
environment at the US Army Research Office such that it is possible for scientists
at ARO to continue to participate personally in forefront research as a way of
maintaining a broad and current knowledge of selected fields of modern science.
Michael Stroscio acknowledges the important roles that several professional
colleagues and friends played in the events leading to his contributions to this
book. These people include: Professor S. Das Sarma of the University of Maryland;
Professor M. Shur of the Rensselaer Polytechnic Institute; Professor Gerald J. Iafrate
of Notre Dame University; Professors M.A. Littlejohn, K.W. Kim, R.M. Kolbas, and
N. Masnari of the North Carolina State University (NCSU); Dr Larry Cooper of the
Office of Naval Research; Professor Vladimir Mitin of the Wayne State University;
Professors H. Craig Casey Jr, and Steven Teitsworth of Duke University; Professor
S. Bandyopadhyay of the University of Nebraska; Professors G. Belenky, Vera B.
Gorfinkel, M. Kisin, and S. Luryi of the State University of New York at Stony
Brook; Professors George I. Haddad, Pallab K. Bhattacharya, and Jasprit Singh and
Dr J.-P. Sun of the University of Michigan; Professors Karl Hess and J.-P. Leburton
at the University of Illinois; Professor L.F. Register of the University of Texas at
Austin; Professor Viatcheslav A. Kochelap of the National Academy of Sciences of
the Ukraine; Dr Larry Cooper of the Office of Naval Research; and Professor Paul
Klemens of the University of Connecticut. Former graduate students, postdoctoral
researchers, and visitors to the North Carolina State University who contributed
substantially to the understanding of phonons in nanostructures as reported in this
book include Drs Amit Bhatt, Ulvi Erdogan, Daniel Kahn, Sergei M. Komirenko,
Byong Chan Lee, Yuri M. Sirenko, and SeGi Yu. The fruitful collaboration of Dr
Rosa de la Cruz of the Universidad Carlos III de Madrid during her tenure as a
visiting professor at Duke University is acknowledged gratefully. The authors also
acknowledge gratefully the professionalism and dedication of Mrs Jayne Aldhouse
and Drs Simon Capelin and Eoin O’Sullivan, of Cambridge University Press, and
Dr Susan Parkinson.
Michael Stroscio thanks family members who have been attentive during the
periods when his contributions to the book were being written. These include:
Anthony and Norma Stroscio, Mitra Dutta, as well as Gautam, Marshall, and
Elizabeth Stroscio. Moreover, eight-year-old Gautam Stroscio is acknowledged
gratefully for his extensive assistance in searching for journal articles at the North
Carolina State University.
Mitra Dutta acknowledges the interactions, discussions and work of many
colleagues and friends who have had an impact on the work leading to this book.
These colleagues include Drs Doran Smith, K.K. Choi, and Paul Shen of the Army
Research Laboratory, Professor Athos Petrou of the State University of New York
at Buffalo, and Professors K.W. Kim, M.A. Littlejohn, R.J. Nemanich, Dr Leah
Bergman and Dimitri Alexson of the North Carolina State University, as well
as Professors Herman Cummins, City College, New York, A.K. Ramdas, Purdue

xiii

xiv

Preface

University and Howard Jackson, University of Cincinnati, her mentors in various
facets of phonon physics. Mitra Dutta would also like to thank Dhiren Dutta, without
whose encouragement she would never have embarked on a career in science, as well
as Michael and Gautam Stroscio who everyday add meaning to everything.
Michael Stroscio and Mitra Dutta

Chapter 1

Phonons in nanostructures
There are no such things as applied sciences, only applications of
sciences.
Louis Pasteur, 1872

1.1

Phonon effects: fundamental limits on carrier
mobilities and dynamical processes

The importance of phonons and their interactions in bulk materials is well known to
those working in the fields of solid-state physics, solid-state electronics, optoelectronics, heat transport, quantum electronics, and superconductivity.
As an example, carrier mobilities and dynamical processes in polar semiconductors, such as gallium arsenide, are in many cases determined by the interaction of
longitudinal optical (LO) phonons with charge carriers. Consider carrier transport
in gallium arsenide. For gallium arsenide crystals with low densities of impurities
and defects, steady state electron velocities in the presence of an external electric
field are determined predominantly by the rate at which the electrons emit LO
phonons. More specifically, an electron in such a polar semiconductor will accelerate
in response to the external electric field until the electron’s energy is large enough for
the electron to emit an LO phonon. When the electron’s energy reaches the threshold
for LO phonon emission – 36 meV in the case of gallium arsenide – there is a
significant probability that it will emit an LO phonon as a result of its interaction
with LO phonons. Of course, the electron will continue to gain energy from the
electric field.
In the steady state, the processes of electron energy loss by LO phonon emission
and electron energy gain from the electric field will come into balance and the
electron will propagate through the semiconductor with a velocity known as the
saturation velocity. As is well known, experimental values for this saturated drift
velocity generally fall in the range 107 cm s−1 to 108 cm s−1 . For gallium arsenide
this velocity is about 2 × 107 cm s−1 and for indium antimonide 6 × 107 cm s−1 .
1

2

1 Phonons in nanostructures

For both these polar semiconductors, the process of LO phonon emission plays a
major role in determining the value of the saturation velocity. In non-polar materials
such as Si, which has a saturation velocity of about 107 cm s−1 , the deformationpotential interaction results in electron energy loss through the emission of phonons.
(In Chapter 5 both the interaction between polar-optical-phonons and electrons –
known as the Fröhlich interaction – and the deformation-potential interaction will
be defined mathematically.)
Clearly, in all these cases, the electron mobility will be influenced strongly by the
interaction of the electrons with phonons. The saturation velocity of the carriers in
a semiconductor provides a measure of how fast a microelectronic device fabricated
from this semiconductor will operate. Indeed, the minimum time for the carriers to
travel through the active region of the device is given approximately by the length
of the device – that is, the length of the so-called gate – divided by the saturation
velocity. Evidently, the practical switching time of such a microelectronic device
will be limited by the saturation velocity and it is clear, therefore, that phonons play
a major role in the fundamental and practical limits of such microelectronic devices.
For modern integrated circuits, a factor of two reduction in the gate length can be
achieved in many cases only through building a new fabrication facility. In some
cases, such a building project might cost a billion dollars or more. The importance
of phonons in microelectronics is clear!
A second example of the importance of carrier–phonon interactions in modern
semiconductor devices is given by the dynamics of carrier capture in the active
quantum-well region of a polar semiconductor quantum-well laser. Consider the
case where a current of electrons is injected over a barrier into the quantum-well
region of such a laser. For the laser to operate, an electron must lose enough energy
to be ‘captured’ by the quasi-bound state which it must occupy to participate in
the lasing process. For many quantum-well semiconductor lasers this means that
the electron must lose an energy of the order of a 100 meV or more. The energy
loss rate of a carrier – also known as the thermalization rate of the carrier – in
a polar-semiconductor quantum well is determined by both the rate at which the
carrier’s energy is lost by optical-phonon emission and the rate at which the carrier
gains energy from optical-phonon absorption. This latter rate can be significant
in quantum wells since the phonons emitted by energetic carriers can accumulate
in these structures. Since the phonon densities in many dimensionally confined
semiconductor devices are typically well above those of the equilibrium phonon
population, there is an appreciable probability that these non-equilibrium – or ‘hot’
– phonons will be reabsorbed. Clearly, the net loss of energy by an electron in such
a situation depends on the rates for both phonon absorption and phonon emission.
Moreover, the lifetimes of the optical phonons are also important in determining the
total energy loss rate for such carriers. Indeed, as will be discussed in Chapter 6, the
longitudinal optical (LO) phonons in GaAs and many other polar materials decay
into acoustic phonons through the Klemens’ channel. Furthermore, over a wide

1.2 Devices with nanostructure components

range of temperatures and phonon wavevectors, the lifetimes of longitudinal optical
phonons in GaAs vary from a few picoseconds to about 10 ps (Bhatt et al., 1994).
(Typical lifetimes for other polar semiconductors are also of this magnitude.) As
a result of the Klemens’ channel, the ‘hot’ phonons decay into acoustic phonons in
times of the order of 10 ps. The LO phonons undergoing decay into acoustic phonons
are not available for absorption by the electrons and as a result of the Klemens’
channel the electron thermalization is more rapid than it would be otherwise; this
phenomenon is referred to as the ‘hot-phonon-bottleneck effect’.
The electron thermalization time is an important parameter for semiconductor
quantum-well lasers because it determines the minimum time needed to switch the
laser from an ‘on’ state to an ‘off’ state; this occurs as a result of modulating the
electron current that leads to lasing. Since the hot-phonon population frequently
decays on a time scale roughly given by the LO phonon decay rate (Das Sarma
et al., 1992), a rough estimate of the electron thermalization time – and therefore
the minimum time needed to switch the laser from an ‘on’ state to an ‘off’ state –
is of the order of about 10 ps. In fact, typical modulation frequencies for gallium
arsenide quantum-well lasers are about 30 GHz. The modulation of the laser at
significantly higher frequencies will be limited by the carrier thermalization time
and ultimately by the lifetime of the LO phonon. The importance of the phonon in
modern optoelectronics is clear.
The importance of phonons in superconductors is well known. Indeed, the
Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity is based on the
formation of bosons from pairs of electrons – known as Cooper pairs – bound
through the mediating interaction produced by phonons. Many of the theories
describing the so-called high-critical-temperature superconductors are not based on
phonon-mediated Cooper pairs, but the importance of phonons in many superconductors is of little doubt. Likewise, it is generally recognized that acoustic phonon
interactions determine the thermal properties of materials.
These examples illustrate the pervasive role of phonons in bulk materials.
Nanotechnology is providing an ever increasing number of devices and structures
having one, or more than one, dimension less than or equal to about 100 ångstroms.
The question naturally arises as to the effect of dimensional confinement on the
properties on the phonons in such nanostructures as well as the properties of the
phonon interactions in nanostructures. The central theme of this book is the description of the optical and acoustic phonons, and their interactions, in nanostructures.

1.2

Tailoring phonon interactions in devices with
nanostructure components

Phonon interactions are altered unavoidably by the effects of dimensional confinement on the phonon modes in nanostructures. These effects exhibit some similarities

3

4

1 Phonons in nanostructures

to those for an electron confined in a quantum well. Consider the well-known
wavefunction of an electron in a infinitely deep quantum well, of width L z in the
z-direction. The energy eigenstates n (z) may be taken as plane-wave states in the
directions parallel to the heterointerfaces and as bound states in an infinitely deep
quantum well in the z-direction:

eik ·r 2
sin k z z,
(1.1)
n (z) = √
Lz
A
where r and k are the position vector and wavevector components in a plane
parallel to the interfaces, k z = nπ/L z , and n = 1, 2, 3, . . . labels the energy
eigenstates, whose energies are
E n (k ) =

h̄ 2 (k )2
h̄ 2 π 2 n 2
.
+
2m
2m L 2z

(1.2)

A is the area of the heterointerface over which the electron wavefunction is
normalized. Clearly, a major effect of dimensional confinement in the z-direction
is that the z-component of the bulk continuum wavevector is restricted to integral
multiples of π/L z . Stated in another way, the phase space is restricted.
As will be explained in detail in Chapter 7, the dimensional confinement of
phonons results in similar restrictions in the phase space of the phonon wavevector
q. Indeed, we shall show that the wavevectors of the optical phonons in a dielectric
layer of thickness L z are given by qz = nπ/L z (Fuchs and Kliewer, 1965) in analogy
to the case of an electron in an infinitely deep quantum well. In fact, Fasol et al.
(1988) used Raman scattering techniques to show that the wavevectors qz = nπ/L z
of optical phonons confined in a ten-monolayer-thick AlAs/GaAs/AlAs quantum
well are so sensitive to changes in L z that a one-monolayer change in the thickness
of the quantum well is readily detectable as a change in qz ! These early experimental
studies of Fasol et al. (1988) demonstrated not only that phonons are confined in
nanostructures but also that the measured phonon wavevectors are well described by
relatively simple continuum models of phonon confinement.
Since dimensional confinement of phonons restricts the phase space of the
phonons, it is certain that carrier–phonon interactions in nanostructures will be
modified by phonon confinement. As we shall see in Chapter 7, the so-called dielectric and elastic continuum models of phonons in nanostructures may be applied
to describe the deformation-potential, Fröhlich, and piezoelectric interactions in a
variety of nanostructures including quantum wells, quantum wires, and quantum
dots. These interactions play a dominant role in determining the electronic, optical
and acoustic properties of materials (Mitin et al., 1999; Dutta and Stroscio, 1998b;
Dutta and Stroscio, 2000); it is clearly desirable for models of the properties
of nanostructures to be based on an understanding of how the above-mentioned
interactions change as a result of dimensional confinement. To this end, Chapters

1.2 Devices with nanostructure components

8, 9 and 10 of this book describe how the dimensional confinement of phonons in
nanostructures leads to modifications in the electronic, optical, acoustic, and superconducting properties of selected devices and structures, including intersubband
quantum-well semiconductor lasers, double-barrier quantum-well diodes, thin-film
superconductors, and the thin-walled cylindrical structures found in the biological
structures known as microtubulin. Chapters 8, 9, and 10 also provide analyses of the
role of collective effects and non-equilibrium phonons in determining hot-carrier
energy loss in polar quantum wires as well as the use of metal–semiconductor
structures to tailor carrier–phonon interactions in nanostructures. Moreover, Chapter
10 describes how confined phonons play a critical role in determining the properties
of electronic, optical, and superconducting devices containing nanostructures as
essential elements. Examples of such phonon effects in nanoscale devices include:
phonon effects in intersubband lasers; the effect of confined phonons on the gain
of intersubband lasers; the contribution of confined phonons to the valley current in
double-barrier quantum-well structures; phonon-enhanced population inversion in
asymmetric double-barrier quantum-well lasers; and confined phonon effects in thin
film superconductors.

5

Chapter 2

Phonons in bulk cubic crystals
The Creator, if He exists, has a special preference for beetles.
J.B.S. Haldane, 1951

2.1

Cubic structure

Crystals with cubic structure are of major importance in the fields of electronics and
optoelectronics. Indeed, zincblende crystals such as silicon, germanium, and gallium
arsenide may be regarded as two face-centered cubic (fcc) lattices displaced relative
to each other by a vector (a/4, a/4, a/4), where a is the size of the smallest unit of
the fcc structure. Figure 2.1 shows a lattice with the zincblende structure.
A major portion of this book will deal with phonons in cubic crystals. In
addition, we will describe the phonons in so-called isotropic media, which are
related mathematically to cubic media as explained in detail in Section 7.2. The
remaining portions of this book will deal with crystals of würtzite structure, defined
in Chapter 3. More specifically, the primary focus of this book concerns phonons
in crystalline structures that are dimensionally confined in one, two, or three
dimensions. Such one-, two-, and three-dimensional confinement is realized in
quantum wells, quantum wires, and quantum dots, respectively. As a preliminary
to considering phonons in dimensionally confined structures, the foundational case
of phonons in bulk structures will be treated. The reader desiring to supplement this
chapter with additional information on the basic properties of phonons in bulk cubic
materials will find excellent extended treatments in a number of texts including
Blakemore (1985), Ferry (1991), Hess (1999), Kittel (1976), Omar (1975), and
Singh (1993).

2.2

Ionic bonding – polar semiconductors

As is well known, the crystal structure of silicon is the zincblende structure shown
in Figure 2.1. The covalent bonding in silicon does not result in any net transfer
of charge between silicon atoms. More specifically, the atoms on the two displaced
6

2.3 Linear-chain model and macroscopic models

7

face-centered cubic (fcc) lattices depicted in Figure 2.1 have no excess or deficit
of charge relative to the neutral situation. This changes dramatically for polar
semiconductors like gallium arsenide, since here the ionic bonding results in charge
transfer from the Group V arsenic atoms to the Group III gallium atoms: Since
Group V atoms have five electrons in the outer shell and Group III atoms have
three electrons in the outer shell, it is not surprising that the gallium sites acquire
a net negative charge and the arsenic sites a net positive charge. In binary polar
semiconductors, the two atoms participating in the ionic bonding carry opposite
charges, e∗ and −e∗ , respectively, as a result of the redistribution of the charge
associated with polar bonding. In polar materials such ionic bonding is characterized
by values of e∗ within an order of magnitude of unity. In the remaining sections of
this chapter, it will become clear that e∗ is related to the readily measurable or known
ionic masses, phonon optical frequencies, and high-frequency dielectric constant of
the polar semiconductor.

2.3

Linear-chain model and macroscopic models

The linear-chain model of a one-dimensional diatomic crystal is based upon a system
of two atoms with masses, m and M, placed along a one-dimensional chain as
depicted in Figure 2.2. As for a diatomic lattice, the masses are situated alternately
along the chain and their separation is a. On such a chain the displacement of
one atom from its equilibrium position will perturb the positions of its neighboring
atoms.
Figure 2.1. Zincblende
crystal. The white spheres
and black spheres lie on
different fcc lattices.

Figure 2.2. One-dimensional linear-chain representation of a diatomic lattice.

2 Phonons in bulk cubic crystals

8

In the simple linear-chain model considered in this section, it is assumed that
only nearest neighbors are coupled and that the interaction between these atoms is
described by Hooke’s law; the spring constant α is taken to be that of a harmonic
oscillator. This model describes many of the basic properties of a diatomic lattice.
However, as will become clear in Chapter 6, it is essential to supplement the socalled ‘harmonic’ interactions with anharmonic interactions in order to describe the
important process of phonon decay.
2.3.1

Dispersion relations for high-frequency and
low-frequency modes

To model the normal modes of this system of masses, the atomic displacements
along the direction of the chain – the so-called longitudinal displacements of each
of the two types of atoms – are taken to be
u 2r = A1 ei(2rqa−ωt)

(2.1)

u 2r +1 = A2 ei[(2r +1)qa−ωt]

(2.2)

and

where q is the phonon wavevector and ω is its frequency. In the nearest-neighbor
approximation, these longitudinal displacements satisfy
m(d 2 u 2r /dt 2 ) = −α(u 2r − u 2r −1 ) − α(u 2r − u 2r +1 )
= α(u 2r +1 + u 2r −1 − 2u 2r )

(2.3)

and
M(d 2 u 2r +1 /dt 2 ) = −α(u 2r +1 − u 2r ) − α(u 2r +1 − u 2r +2 )
= α(u 2r +2 + u 2r − 2u 2r +1 ).

(2.4)

The signs in the four terms on the right-hand sides of these equations are
determined by considering the relative displacements of neighboring atoms. For
example, if the positive displacement of u 2r is greater than that of u 2r −1 there is
a restoring force −α(u 2r +1 − u 2r ). Hence
−mω2 A1 = α A2 (eiqa + e−iqa ) − 2α A1

(2.5)

−Mω2 A2 = α A1 (eiqa + e−iqa ) − 2α A2 .

(2.6)

and

Eliminating A1 and A2 ,





1
1
1 2 4 sin2 qa 1/2
1
2
+
±α
+
−
.
ω =α
m
M
m
M
mM

(2.7)

This relationship between frequency and wavevector is commonly called a dispersion relation. The higher-frequency solution is known as the optical mode

2.3 Linear-chain model and macroscopic models

9

since, for many semiconductors, its frequency is in the terahertz range, which
happens to coincide with the infrared portion of the electromagnetic spectrum.
The lower-frequency solution is known as the acoustic mode. More precisely, since
only longitudinal displacements have been modeled, these two solutions correspond
to the longitudinal optical (LO) and longitudinal acoustic (LA) modes of the
linear-chain lattice. Clearly, the displacements along this chain can be described
in terms of wavevectors q in the range from −π/2a to π/2a. From the solution
for ω, it is evident that over this Brillouin zone the LO modes have a maximum
frequency [2α(1/m + 1/M)]1/2 at the center of the Brillouin zone and a minimum
frequency (2α/m)1/2 at the edge of the Brillouin zone. Likewise, the LA modes have
a maximum frequency (2α/M)1/2 at the edge of the Brillouin zone and a minimum
frequency equal to zero at the center of the Brillouin zone.
In polar semiconductors, the masses m and M carry opposite charges, e∗ and
−e∗ , respectively, as a result of the redistribution of the charge associated with
polar bonding. In polar materials such ionic bonding is characterized by values of
e∗ equal to 1, to an order-of-magnitude. When there is an electric field E present
in the semiconductor, it is necessary to augment the previous force equation with
terms describing the interaction with the charge. In the long-wavelength limit of the
electric field E, the force equations then become
−mω2 u 2r = m(d 2 u 2r /dt 2 ) = α(u 2r +1 + u 2r −1 − 2u 2r ) + e∗ E
= α(ei2qa + 1)u 2r −1 − 2αu 2r + e∗ E

(2.8)

and
−Mω2 u 2r +1 = M(d 2 u 2r +1 /dt 2 ) = α(u 2r +2 + u 2r − 2u 2r +1 ) − e∗ E
= α(1 + e−i2qa )u 2r +2 − 2αu 2r +1 − e∗ E.

(2.9)

Regarding the phonon displacements, in the long-wavelength limit there is no
need to distinguish between the different sites for a given mass type since all atoms
of the same mass are displaced by the same amount. In this limit, q → 0. Denoting
the displacements on even-numbered sites by u 1 and those on odd-numbered sites
by u 2 , in the long-wavelength limit the force equations reduce to
−mω2 u 1 = 2α(u 2 − u 1 ) + e∗ E

(2.10)

−Mω2 u 2 = 2α(u 1 − u 2 ) − e∗ E.

(2.11)

and

Adding these equations demonstrates that −mω2 u 1 − Mω2 u 2 = 0 and it is clear
that mu 1 = −Mu 2 ; thus


m
2
(2.12)
−mω u 1 = 2α − u 1 − u 1 + e∗ E
M

2 Phonons in bulk cubic crystals

10

and



m
−Mω u 2 = 2α u 1 + u 1 − e∗ E;
M
2

(2.13)

accordingly,
−(ω2 − ω02 )u 1 = e∗ E/m

(2.14)

−(ω2 − ω02 )u 2 = −e∗ E/M

(2.15)

and

where ω02 = 2α(1/m + 1/M) is the resonant frequency squared, in the absence of
Coulomb effects; that is, for e∗ = 0. The role of e∗ in shifting the phonon frequency
will be discussed further in the next section.
Clearly, the electric polarization P produced by such a polar diatomic lattice is
given by


N e∗ u
N e∗ (u 1 − u 2 )
1
N e∗2
1
1
P=
=
=
+
E,
(∞)
(∞)
(∞) (ω02 − ω2 ) m
M

(2.16)

where u = u 1 − u 2 , N is the number of pairs per unit volume, and e∗ is as
defined previously. This equation may be rewritten to show that it describes a driven
oscillator:


1
1
+
E.
(2.17)
(ω02 − ω2 )u = e∗
m
M
2.3.2

Displacement patterns for phonons

As discussed in subsection 2.3.1, in the limit q → 0 the displacements, u 1 and u 2 ,
of the optical modes satisfy −mu 1 = Mu 2 and the amplitudes of the two types of
mass have opposite signs. That is, for the optical modes the atoms vibrate out of
phase, and so with their center of mass fixed. For the acoustic modes, the maximum
frequency is (2α/M)1/2 . This maximum frequency occurs at the zone edge so that,
near the center of the zone, ω is much less than (2α/M)1/2 . From subsection 2.3.1,
the ratio A2 /A1 may be expressed as
2α cos qa
2α − mω2
A2
=
=
,
2
A1
2α cos qa
2α − Mω

(2.18)

and it is clear that the ratio of the displacement amplitudes is approximately equal
to unity for acoustic phonons near the center of the Brillouin zone. Thus, in contrast
to the optical modes, the acoustic modes are characterized by in-phase motion of

2.3 Linear-chain model and macroscopic models

11

the different masses m and M. Typical mode patterns for zone-center acoustic
and optical modes are depicted in Figures 2.3(a), (b). The transverse modes are
illustrated here since the longitudinal modes are more difficult to depict graphically.
The higher-frequency optical modes involve out-of-plane oscillations of adjacent
ions, while the lower-frequency acoustic modes are characterized by motion of
adjacent ions on the same sinusoidal curve.

2.3.3

Polaritons

In the presence of a transverse electric field, transverse optical (TO) phonons of
a polar medium couple strongly to the electric field. When the wavevectors and
frequencies of the electric field are in resonance with those of the TO phonon, a
coupled phonon–photon field is necessary to describe the system. The quantum of
this coupled field is known as the polariton. The analysis of subsection 2.3.1 may
be generalized to apply to the case of transverse displacements. In particular, for a
transverse field E, the oscillator equation takes the form
2
− ω2 )P =
(ωTO

N e∗2
(∞)



1
1
+
m
M


E,

(2.19)

2 = 2α(1/m + 1/M) since
where ω02 of subsection 2.3.1 has been designated ωTO
the resonant frequency in the absence of Coulomb effects, e∗2 = 0, corresponds to

Figure 2.3. Transverse displacements of heavy ions (large disks) and light ions
(small disks) for (a) transverse acoustic modes, and (b) transverse optical modes
propagating in the q-direction.

12

2 Phonons in bulk cubic crystals

the transverse optical frequency. As will become apparent later in this section, the
LO phonon frequency squared differs from the TO phonon frequency squared by an
amount proportional to e∗2 .
According to the electromagnetic wave equation, ∂ 2 D/∂t 2 = c2 ∇ 2 E, where
D = E + 4π P, the dispersion relation describing the coupling of the field E of the
electromagnetic wave to the electric polarization P of the TO phonon is
c2 q 2 E = ω2 (E + 4π P)

(2.20)

or, alternatively,
4πω2 P = (c2 q 2 − ω2 )E,

(2.21)

where waves of the form ei(qr −ωt) have been assumed. The driven oscillator
equation and the electromagnetic wave equation have a joint solution when the
determinant of the coefficients of the fields E and P vanishes,




4π ω2
ω2 − c2 q 2




=0
 N e∗2 1
(2.22)
1

2
2
−(ωTO − ω ) 
 (∞) m + M
At q = 0, there are two roots: ω = 0 and


N e∗2 1
1
2
2
2
= ωLO
.
+
ω = ωTO + 4π
(∞) m
M

(2.23)

The dielectric function (ω) is then given by
4π Pe (ω) 4π P(ω)
D(ω)
=1+
+
E(ω)
E(ω)
E(ω)


4π
1
N e∗2 1
4π Pe (ω)
+ 2
+
,
=1+
E(ω)
M
(ωTO − ω2 ) (∞) m

(ω) =

(2.24)

where the polarization due to the electronic contribution, Pe (ω), has been included
as well as the polarization associated with the ionic contribution, P(ω).
As is customary, the dielectric constant due to the electronic response is denoted
by
(∞) = 1 +

4π Pe (ω)
,
E(ω)

(2.25)

and it follows that


N e∗2 1
1
4π
.
+
(ω) = (∞) + 2
M
(ωTO − ω2 ) (∞) m
The so-called static dielectric constant (0) is then given by

(2.26)

2.3 Linear-chain model and macroscopic models



4π N e∗2 1
1
.
(0) = (∞) + 2
+
M
ωTO (∞) m

13

(2.27)

From these last two results it follows straightforwardly that
(ω) = (∞) +

2
[ (0) − (∞)]ωTO

2 − ω2 )
(ωTO
(0) − (∞)
.
= (∞) +
2
1 − ω2 /ωTO

(2.28)

From electromagnetic theory it is known that the dielectric function (ω) must
vanish for any longitudinal electromagnetic disturbance to propagate. Accordingly,
the frequency of the LO phonons, ωLO , must be such that (ωLO ) = 0; from the last
equation, this condition implies that
(ωLO ) = 0 = (∞) +

(0) − (∞)
2 /ω2
1 − ωLO
TO

(2.29)

or, equivalently,

ωLO =

(0)
(∞)

1/2
ωTO .

(2.30)

It then follows that
(0) − (∞)
(ωLO /ωTO )2 (∞) − (∞)
= (∞) +
2
2
1 − ω2 /ωTO
1 − ω2 /ωTO


(ωLO /ωTO )2 − 1
= (∞) 1 +
2
1 − ω2 /ωTO

(ω) = (∞) +

2 − ω2
ωTO

= (∞)
= (∞)

2 − ω2
ωTO
2 − ω2
ωLO
2 − ω2
ωTO

,

+

2 − ω2
ωLO
TO
2 − ω2
ωTO

(2.31)

or alternatively
ω2 − ω2
(ω)
.
= LO
2 − ω2
(∞)
ωTO

(2.32)

In the special case where ω = 0, this relation reduces to the celebrated Lyddane–
Sachs–Teller relationship
ω2
(0)
.
= LO
2
(∞)
ωTO

(2.33)

When ω = ωLO the dielectric constant vanishes, (ωLO ) = 0; as stated above,
this condition is familiar from electromagnetics as a requirement for the propagation

2 Phonons in bulk cubic crystals

14

of a longitudinal electromagnetic wave. That is, a longitudinal electromagnetic wave
propagates only at frequencies where the dielectric constant vanishes; accordingly,
ωLO is identified as the frequency of the LO phonon. From the relation


N e∗2 1
1
2
2
+ 4π
,
+
= ωLO
ωTO
(∞) m
M
it follows that ωTO = ωLO for zone-center phonons in materials with e∗ = 0;
this is just as observed in non-polar materials such as silicon. In polar materials
such as GaAs there is a gap between ωTO and ωLO , associated with the Coulomb
energy density arising from e∗ . When ω = ωTO , (ωTO )−1 = 0 and the pole in (ω)
reflects the fact that electromagnetic waves with the frequency of the TO phonon are
absorbed. Throughout the interval (ωTO , ωLO ), (ω) is negative and electromagnetic
waves do not propagate.
2.3.4

Macroscopic theory of polar modes in cubic crystals

As was apparent in subsections 2.3.1 and 2.3.3, polar-optical phonon vibrations
produce electric fields and electric polarization fields that may be described in
terms of Maxwell’s equations and the driven-oscillator equations. Loudon (1964)
advocated a model of optical phonons based on these macroscopic fields that has
had great utility in describing the properties of optical phonons in so-called uniaxial
crystals such as würtzite crystals. The Loudon model for uniaxial crystals will be
developed more fully in Chapters 3 and 7. In this section, the concepts underlying
the Loudon model will be discussed in the context of cubic crystals.
From the pair of Maxwell’s equations,
∇ ×E+

1 ∂B
=0
c ∂t

and

∇ ×B−

1 ∂D
= J,
c ∂t

(2.34)

it follows that
∇ × (∇ × E) +

1 ∂ 2D
1 ∂(∇ × B)
= ∇(∇ · E) − ∇ 2 E + 2 2 = 0,
c
∂t
c ∂ t

(2.35)

where the source current, J, has been taken to equal zero. Then since ∇ · D =
∇ · E + 4π∇ · P = 4πρ = 0, it follows that
−4π∇(∇ · P) − ∇ 2 E +

1 ∂ 2P
1 ∂ 2E
+
4π
= 0.
c2 ∂ 2 t
c2 ∂ 2 t

(2.36)

Assuming that P and E both have spatial and time dependences of the form
this last result takes the form

ei(q·r−ωt) ,

E=

−4π [q(q · P) − ω2 P/c2 ]
.
q 2 − ω2 /c2

(2.37)

2.3 Linear-chain model and macroscopic models

15

The condition q · P = 0 corresponds to the transverse wave; in this case,
E=

−4πω2 P/c2
.
q 2 − ω2 /c2

(2.38)

From Appendix A, E and P are also related through


2
1 [ (0) − (∞)]ωTO
+ [ (∞) − 1] E;
P=
2 − ω2
4π
ωTO

(2.39)

thus
2
[ (0) − (∞)]ωTO
q 2 − ω2 /c2
=
+ [ (∞) − 1],
2 − ω2
ω2 /c2
ωTO

(2.40)

or, equivalently,
2
(0) − ω2 (∞)
ωTO
q 2 c2
=
.
2 − ω2
ω2
ωTO

(2.41)

For longitudinal waves, q · P = q P, so that q = (q/P)P, and it follows that
4π ω2 P/c2
4πq Pq
− 2
q 2 − ω2 /c2
q − ω2 /c2
4π ω2 P/c2
4πq 2 P
= 2
−
q − ω2 /c2
q 2 − ω2 /c2

E=

=

q2

4π
− ω2 /c2

ω2
− q2 P
c2

= −4π P.

Then

(2.42)



2
1 [ (0) − (∞)]ωTO
P=
+ [ (∞) − 1] E
2 − ω2
4π
ωTO


2
[ (0) − (∞)]ωTO
+ [ (∞) − 1] P
=−
2 − ω2
ωTO

(2.43)

or, equivalently,

ω = ωTO

(0)
(∞)

1/2
= ωLO ,

and the Lyddane–Sachs–Teller relation is recovered once again! In Chapter 3, we
shall return to the Loudon model to describe uniaxial crystals of the würtzite type.

Chapter 3

Phonons in bulk würtzite crystals
Next when I cast mine eyes and see that brave vibration, each
way free; O how that glittering taketh me.
Robert Herrick, 1648

3.1

Basic properties of phonons in würtzite structure

The GaAlN-based semiconductor structures are of great interest in the electronics
and optoelectronics communities because they possess large electronic bandgaps
suitable for fabricating semiconductor lasers with wavelengths in the blue and
ultraviolet as well as electronic devices designed to work at elevated operating
temperatures. These III-V nitrides occur in both zincblende and würtzite structures.
In this chapter, the würtzite structures will be considered rather than the zincblende
structures, since the treatment of the phonons in these würtzite structures is more
complicated than for the zincblendes. Throughout the remainder of this book,
phonon effects in nanostructures will be considered for both the zincblendes and
würtzites. This chapter focuses on the basic properties of phonons in bulk würtzite
structures as a foundation for subsequent discussions on phonons in würtzite
nanostructures.
The crystalline structure of a würtzite material is depicted in Figure 3.1. As in
the zincblendes, the bonding is tetrahedral. The würtzite structure may be generated
from the zincblende structure by rotating adjacent tetrahedra about their common
bonding axis by an angle of 60 degrees with respect to each other. As illustrated in
Figure 3.1, würtzite structures have four atoms per unit cell.
The total number of normal vibrational modes for a unit cell with s atoms in
the basis is 3s. As for cubic materials, in the long-wavelength limit there are three
acoustic modes, one longitudinal and two transverse. Thus, the total number of
optical modes in the long-wavelength limit is 3s − 3. These optical modes must,
of course, appear with a ratio of transverse to longitudinal optical modes of two.
16

3.1 Basic properties of phonons in würtzite structure

The numbers of the various long-wavelength modes are summarized in Table 3.1.
For the zincblende case, s = 2 and there are six modes: one LA, two TA, one
LO and two TO. For the würtzite case, s = 4 and there are 12 modes: one LA,
two TA, three LO and six TO. In the long-wavelength limit the acoustic modes are
simple translational modes. The optical modes for a würtzite structure are depicted
in Figure 3.2.
From Figure 3.2 it is clear that the A1 and E 1 modes will produce large electric
polarization fields when the bonding is ionic. Such large polarization fields result
in strong carrier–optical-phonon scattering. These phonon modes are known as
infrared active. As we shall see in Chapter 5, the fields associated with these infrared
modes may be derived from a potential describing the carrier–phonon interaction of
Figure 3.1. Unit cell of the
hexagonal würtzite crystal.

Table 3.1. Phonon modes associated with a unit cell
having s atoms in the basis.

Type of mode
Longitudinal acoustic (LA)
Transverse acoustic (TA)
All acoustic modes
Longitudinal optical (LO)
Transverse optical (TO)
All optical modes
All modes

Number of modes
1
2
3
s−1
2s − 2
3s − 3
3s

17

3 Phonons in bulk würtzite crystals

18

such modes. In Chapter 5, this carrier–phonon interaction potential will be identified
as the Fröhlich interaction. The dispersion relations for the 12 phonon modes of the
würtzite structure are depicted in Figure 3.3.
The low-frequency behavior of these modes near the  point makes it apparent
that three of these 12 modes are acoustic modes. This behavior is, of course,
consistent with the number of acoustic modes identified in Table 3.1.

3.2

Loudon model of uniaxial crystals

As discussed in subsection 2.3.4, Loudon (1964) advanced a model for uniaxial
crystals that provides a useful description of the longitudinal optical phonons in
würtzite crystals. In Loudon’s model of uniaxial crystals such as GaN or AlN, the
angle between the c-axis and q is denoted by θ, and the isotropic dielectric constant
of the cubic case is replaced by dielectric constants for the directions parallel and
perpendicular to the c-axis,  (ω) and ⊥ (ω) respectively. That is,

Figure 3.2. Optical phonons in würtzite structure. From Gorczyca et al. (1995),
American Physical Society, with permission.

3.2 Loudon model of uniaxial crystals



⊥ (ω)

(ω) = 

0
0

0
⊥ (ω)
0

19


0
0 
 (ω)

(3.1)

with
⊥ (ω)

=

⊥ (∞)

2
ω2 − ωLO,⊥

 (ω)

and

2
ω2 − ωTO,⊥

=

 (∞)

2
ω2 − ωLO,

,
2
ω2 − ωTO,
(3.2)

as required by the Lyddane–Sachs–Teller relation. The c-axis is frequently taken to
be in the z-direction and the dielectric constant is then sometimes labeled by the
z-coordinate; that is,  (ω) = z (ω). Figure 3.4 depicts the two dielectric constants
for GaN as well as those for AlN.
In such a uniaxial crystal, there are two types of phonon wave: (a) ordinary waves
where for any θ both the electric field E and the polarization P are perpendicular
to the c-axis and q simultaneously, and (b) extraordinary waves, for which the
orientation of E and P with respect to q and the c-axis is more complicated. As
discussed in subsection 2.3.4, the ordinary wave has E 1 symmetry, is transverse,
and is polarized in the ⊥-plane. There are two extraordinary waves, one associated
with the ⊥-polarized vibrations and having A1 symmetry and the other associated
with -polarized vibrations and having E 1 symmetry. For θ = 0, one of these modes
is the A1 (LO) mode and the other is the E 1 (TO) mode. As θ varies between 0 and
π/2, these modes evolve to the A1 (TO) and E 1 (TO) modes respectively. For values
of θ intermediate between 0 and π/2 they are mixed and do not have purely LO or

100



[110]

K

M

[100]



[001]

A

Energy (meV)

80
60
40
20
0
Reduced wavevector
Figure 3.3. Phonon dispersion curves for GaN crystal of würtzite structure. From
Nipko et al. (1998), American Institute of Physics, with permission.

20

3 Phonons in bulk würtzite crystals

TO character or A1 or E 1 symmetry (Loudon, 1964). For würtzite structures at the
 point, it will be obvious in Chapter 7 that only three of the nine optical phonon
modes, the A1 (Z ) and E 1 (X, Y ) modes, produce significant carrier–optical-phonon
scattering rates. These are the so-called infrared-active modes. For the case of
würtzite structures, Loudon’s model of uniaxial crystals is based upon generalizing
Huang’s equations, equations (A.8) and (A.9) of Appendix A, and the relationship
of subsection 2.3.4, equation (2.43). Specifically, for each of these equations there
is a set of two more equations, one in terms of quantities along the c-axis and the
other in terms of quantities perpendicular to the c-axis:
1/2 
V
⊥ (0) − ⊥ (∞) ωTO,⊥ E⊥ ,
4π µN
1/2


V
2
− ω2 )u =
(ωTO,
 (0) −  (∞) ωTO, E ,
4π µN


2
(ωTO,⊥
− ω2 )u⊥ =


P⊥ =

P =

µN
4π V

µN
4π V

1/2 
1/2




⊥ (0) − ⊥ (∞) ωTO,⊥ u⊥

+


 (0) −  (∞) ωTO, u

+

⊥ (∞) − 1

4π
 (∞) − 1

4π

(3.3)

(3.4)


E⊥ ,

(3.5)


E ,

(3.6)

Figure 3.4. Dielectric constants for GaN, 1⊥ (GaN) and 1z (GaN), and for AlN,
2⊥ (AlN) and 2z (AlN). From Lee et al. (1998), American Physical Society, with
permission.

3.2 Loudon model of uniaxial crystals

21

−4π [q⊥ (q · P) − ω2 P⊥ /c2 ]
,
q 2 − ω2 /c2
−4π [q (q · P) − ω2 P /c2 ]
.
E =
q 2 − ω2 /c2
E⊥ =

(3.7)
(3.8)

Eliminating u⊥ and u in the first four of these equations yields


2
1 [ ⊥ (0) − ⊥(∞)]ωTO,⊥
P⊥ =
+ [ ⊥ (∞) − 1] E⊥
2
4π
ωTO,⊥
− ω2
1
=
A⊥ E⊥
4π

(3.9)

and
1
P =
4π
=



[  (0) −

2
 (∞)]ωTO,

2
ωTO,
− ω2


+ [  (∞) − 1] E

1
A E ,
4π

(3.10)

where A⊥ and A may be written as
A⊥ =
A =

2
ωLO,⊥
− ω2

⊥ (∞) − 1,

(3.11)

 (∞) − 1,

(3.12)

2
ωTO,⊥
− ω2
2
ωLO,
− ω2
2
ωTO,
− ω2

upon using the Lyddane–Sachs–Teller relations

ωTO,⊥

⊥ (0)
⊥ (∞)

1/2


= ωLO,⊥

and

ωTO,

 (0)

1/2

 (∞)

= ωLO, .
(3.13)

For the ordinary wave, E = 0, P = 0, and q · P = 0, so that the derivation of
subsection 2.3.4 now gives
2
2
ωTO,⊥
⊥ (0) − ω ⊥ (∞)
q 2 c2
=
.
2
ω2
ωTO,⊥
− ω2

(3.14)

For the ordinary mode it also follows that u = u⊥ = 0.
For the extraordinary wave, q⊥ = q sin θ and q = q cos θ, where θ is the angle
between q and the c-axis. Then, it follows that
q · P = (q sin θ, q cos θ) · (P⊥ , P ) = q P⊥ sin θ + q P cos θ.
Thus,

(3.15)

3 Phonons in bulk würtzite crystals

22

q⊥ (q · P) = q 2 (P⊥ sin2 θ + P sin θ cos θ ),
(3.16)
q (q · P) = q 2 (P⊥ sin θ cos θ + P cos2 θ ).

In the limit where retardation effects are neglected, c → ∞ and it follows that
E⊥ =

−4π [q⊥ (q · P) − ω2 P⊥ /c2 ]
→ −4π(P⊥ sin2 θ + P sin θ cos θ)
q 2 − ω2 /c2

= − sin2 θ A⊥ E ⊥ − sin θ cos θ A E  ,

(3.17)

and
E =

−4π [q (q · P) − ω2 P /c2 ]
→ −4π(P⊥ sin θ cos θ + P cos2 θ)
q 2 − ω2 /c2

= − sin θ cos θ A⊥ E ⊥ − cos2 θ A E  .

(3.18)

These equations may be written as



E⊥
1 + sin2 θ A⊥ sin θ cos θ A
= 0,
sin θ cos θ A⊥ 1 + cos2 θ A
E

(3.19)

and it follows that the condition for non-trivial solutions to exist is
1 + sin2 θ A⊥ + cos2 θ A =

2
ωLO,⊥
− ω2

+
=

2
⊥ (∞) sin θ

2
ωTO,⊥
− ω2
2
− ω2
ωLO,
2
ωTO,
− ω2

2
⊥ (ω) sin θ

+

2
 (∞) cos θ
2
 (ω) cos θ

=0

(3.20)

or equivalently
2
⊥ (ω)q⊥

+

2
 (ω)q

= 0.

(3.21)

Since the high-frequency electronic response of a medium should not depend
strongly on the crystalline structure, it is usually assumed (Loudon, 1964) that
⊥ (∞) ≈  (∞). Thus
2
− ω2
ωLO,⊥
2
ωTO,⊥
− ω2

sin2 θ +

2
ωLO,
− ω2
2
ωTO,
− ω2

cos2 θ = 0

(3.22)

or equivalently
2
2
2
2
ωLO,
cos2 θ + ωLO,⊥
ωTO,
sin2 θ = 0,
ω4 − (ω12 + ω22 )ω2 + ωTO,⊥

(3.23)

3.3 Application of Loudon model to III-V nitrides

23

where
2
2
sin2 θ + ωTO,⊥
cos2 θ,
ω12 = ωTO,

(3.24)
2
2
cos2 θ + ωLO,⊥
sin2 θ.
ω22 = ωLO,


When ωTO, − ωTO,⊥  is very much less than ωLO, − ωTO, and ωLO,⊥ − ωTO,⊥
this equation has roots
ω2 =

1
2


(ω12 + ω22 ) ± [(ω12 − ω22 ) + 2ω2 (θ)] ,

(3.25)

where
ω2 (θ) = 2

2
2
2
2
(ωLO,
− ωLO,⊥
)(ωTO,
− ωTO,⊥
)

ω22 − ω12

sin2 θ cos2 θ;

(3.26)

thus
2
2
ω2 = ωTO,
sin2 θ + ωTO,⊥
cos2 θ

−

2
2
2
2
− ωLO,⊥
)(ωTO,
− ωTO,⊥
)
(ωLO,

ω22 − ω12

sin2 θ cos2 θ

2
2
sin2 θ + ωTO,⊥
cos2 θ
≈ ωTO,

(3.27)

and
2
2
ω2 = ωLO,
cos2 θ + ωLO,⊥
sin2 θ

+

2
2
2
2
− ωLO,⊥
)(ωTO,
− ωTO,⊥
)
(ωLO,

ω22 − ω12

sin2 θ cos2 θ

2
2
cos2 θ + ωLO,⊥
sin2 θ.
≈ ωLO,

3.3

(3.28)

Application of Loudon model to III-V nitrides





The conditions ωTO, − ωTO,⊥   ωLO, − ωTO, and ωTO, − ωTO,⊥  
ωLO,⊥ − ωTO,⊥ are satisfied reasonably well for a number of würtzite materials
including the III-V nitrides. Indeed, for GaN, (∞) = 5.26, ωLO,⊥ = 743 cm−1 ,
ωLO, = 735 cm−1 , ωTO,⊥ = 561 cm−1 , and ωTO, = 533 cm−1 (Azuhata
et al., 1995). For AlN, (∞) = 5.26, ωLO,⊥ = 916 cm−1 , ωLO, = 893 cm−1 ,
ωTO,⊥ = 673 cm−1 , and ωTO, = 660 cm−1 (Perlin et al., 1993). For these and
other würtzite crystals (Hayes and Loudon, 1978), Table 3.2 summarizes the various
frequency differences appearing in the previously stated frequency conditions.
As is clear from Table 3.2, the inequalities assumed in Section 3.2 are reasonably
well satisfied for both GaN and AlN as well as for the other materials listed. The

3 Phonons in bulk würtzite crystals

24

infrared-active modes in these III-V nitrides are the A1 (LO), A1 (TO), E 1 (LO), and
E 1 (TO) modes and the frequencies associated with these modes, ω A1 (LO) , ω A1 (TO) ,
ω E 1 (LO) , and ω E 1 (TO) are given by ωLO, , ωTO, , ωLO,⊥ , and ωTO,⊥ , respectively.
Let us consider the case of GaN in more detail. From the results of Section 3.2, it
follows immediately that
sin θ cos θ A
sin θ cos θ A
sin θ
E⊥
,
=−
=
=
E
cos θ
cos2 θ A
1 + sin2 θ A⊥
and
2
ωTO,
− ω2
u⊥
= 2
u
ωTO,⊥ − ω2

=

2
ωTO,
− ω2
2
ωTO,⊥




− ω2

⊥ (0) − ⊥ (∞)
 (0) −

 (∞)

⊥ (0) − ⊥ (∞)

1/2 
1/2 

 (0) −  (∞)

ωTO,⊥
ωTO,
ωTO,⊥
ωTO,




(3.29)

E⊥
E
sin θ
.
cos θ

(3.30)

Since q = (q⊥ , q ) = (q sin θ, q cos θ), the first of these relations illustrates the
fact that E  q, as expected from q 2 E = −4πq(q · P); this last equality follows from
∇ ·(E+4πP) = 0. The ratio u ⊥ /u  may be estimated for GaN for the transverse-like
2
2
sin2 θ + ωTO,⊥
cos2 θ, as
modes, with ω2 = ωTO,

1/2 

cos θ
ωTO,⊥ cos θ
u⊥
⊥ (0) − ⊥ (∞)
≈ −0.95
,
(3.31)
=−
u
(0)
−
(∞)
ω
sin
θ
sin θ


TO,
2
2
cos2 θ + ωLO,⊥
sin2 θ, as
and for the longitudinal-like modes, with ω2 = ωLO,
1/2 


2
ωTO,
− ω2
u⊥
ωTO,⊥ sin θ
⊥ (0) − ⊥ (∞)
= 2
u
ωTO, cos θ
ωTO,⊥ − ω2
 (0) −  (∞)




2
2
1/2
− ωLO
ωTO,
ωTO,⊥ sin θ
⊥ (0) − ⊥ (∞)
= 2
2
ωTO, cos θ
ωTO,⊥ − ωLO
 (0) −  (∞)

= 1.07

sin θ
,
cos θ

(3.32)

Table 3.2. Difference frequencies in cm−1 for GaN and AlN as well
as for other würtzite crystals.

Würtzite
GaN
AlN
AgI
BeO
CdS
ZnO
ZnS



ωTO, − ωTO,⊥ 

ωLO, − ωTO,

ωLO,⊥ − ωTO,⊥

27
59
0
44
9
33
0

211
279
18
403
71
199
76

186
243
18
375
64
178
76

3.3 Application of Loudon model to III-V nitrides

2 is taken to be equal to both ω2
2
2
2
where ωLO
LO, and ωLO,⊥ since ωLO, ≈ ωLO,⊥ .
The properties of uniaxial crystals derived in this section and in Section 3.2 will
be used extensively in Chapter 7 to determine the Fröhlich potentials in würtzite
nanostructures.

25

Chapter 4

Raman properties of bulk phonons
When you measure what you are speaking about and express it
in numbers, you know something about it; but when you
cannot measure it, when you cannot express it in numbers, your
knowledge is of a meagre and unsatisfactory kind; it may be the
beginning of knowledge but you have scarcely in your thoughts
advanced to the stage of science, whatever the matter may be.
Lord Kelvin, 1889

4.1

Measurements of dispersion relations for bulk
samples

This chapter deals with the application of Raman scattering techniques to measure
basic properties of phonons in dimensionally confined systems. It is, however,
appropriate at this point to emphasize that non-Raman techniques such as neutron
scattering (Waugh and Dolling, 1963) have been used for many years to determine
the phonon dispersion relations for bulk semiconductors. Indeed, for thermal
neutrons the de Broglie wavelengths are comparable to the phonon wavelengths.
For bulk samples, neutron scattering cross sections are large enough to facilitate
the measurement of phonon dispersion relations. This is generally not the case for
quantum wells, quantum wires, and quantum dots, where Raman and micro-Raman
techniques are needed to make accurate measurements of dispersion relations
in structures of such small volume. Further comparisons of neutron and Raman
scattering measurements of phonon dispersion relations are found in Section 7.5.

4.2

Raman scattering for bulk zincblende and würtzite
structures

Raman scattering has been a very effective experimental technique for observing
phonons; it involves measuring the frequency shift between the incident and
26

4.2 Raman scattering for bulk zincblende and würtzite structures

27

scattered photons. It is a three-step process: the incident photon of frequency ωi
is absorbed; the intermediate electronic state which is thus formed interacts with
phonons or other elementary excitations of energy via several mechanisms, creating
or annihilating them; finally, the scattered photon, of different energy ωs , is emitted.
Energy and momentum are conserved and are given by the following equations:
h̄ωi = h̄ωs ± h̄,

(4.1)

ki = ks ± q.

(4.2)

Since the momenta of the incident and scattered photons are small compared with
the reciprocal lattice vectors, only excitations with q  0 take part in the Raman
process illustrated in Figure 4.1. In the case of Raman scattering in semiconductors,
the absorption of photons gives rise to electron–hole pairs; hence the intensity of the
Raman scattering and the resonances reflect the underlying electronic structure of
the material. The Raman intensity, I (ωi ), is given by
I (ωi ) ∝ ωs4 |εε s Tεε i |2


α,β

1
(E α − h̄ωi )(E β − h̄ωs )

(4.3)

where the ωi and ωs are the frequencies of the incoming photon and of the
scattered photon respectively, E α and E β are the energies of the intermediate states,
T the Raman tensor, and ε i and ε s are the incident and scattered polarization
vectors. The summation is over all possible intermediate states. In general, for
semiconductors there may be the following real intermediate states: Bloch states,
which form the conduction or valence bands, exciton states and in-gap impurity
states. In equation (4.3), the second factor gives the Raman selection rules, which
come about from symmetry considerations of the interactions involved in a Raman
process. The selection rules are conveniently summarized in the form of Raman
tensors. These selection rules are essential tools for determining crystal orientation
and quality.
Details of the theoretical description of Raman scattering and these effects in
the vicinity of the critical points of the semiconductor are given in excellent books
and reviews elsewhere (Loudon, 1964; Hayes and Loudon, 1978; Cardona, 1975;
Cardona and Güntherodt, 1982a, b, 1984, 1989, 1991) and will not be repeated here.
Figure 4.1. Diagrammatic
representation of the Raman
process. The broken line
represents the phonon, the
wavy lines represent the
photons, and the dotted line
represents the electronic
state.

4 Raman properties of bulk phonons

28

Instead we will summarize key results in zincblende and würtzite crystals both for
the bulk case and, in Chapter 7, for quantum wells and superlattices. While first-rate
articles and book chapters exist for the results of the zincblende structures, the work
on the nitrides, with their würtzite structure, is more recent and hence in this book
we will cover the latter results in more detail.

4.2.1

Zincblende structures

The features that can be observed in a Raman experiment for particular values of
incident and scattered polarization can be determined from the symmetry properties
of the second-order susceptibility for the excitation concerned as well as from the
spatial symmetry of the scattering medium. The cubic zincblende structure has
a space-group symmetry Td2 , and there is one three-fold Raman active mode of
the T2 representation. The optic mode is polar so that the macroscopic field lifts
the degeneracy, producing a non-degenerate longitudinal mode that is at a higher
frequency than the two transverse modes. The allowed light-scattering symmetries,
as indicated by the second-order susceptibilities for the zincblende structure are
given below by appropriate matrices for the tensor T in the T2 representation:



0
d 
0

R(x) mode,


0 d
0 0 
0 0

R(y) mode,


0
0 
0

R(z) mode.

0 0
 0 0
0 d


0
 0
d


0
 d
0

d
0
0

(4.4)

Raman scattering has been used now for several decades as a characterization
tool in understanding, for example, crystal structure and quality, impurity content,
strain, interface disorder, and the effects of alloying and sample preparation. Much
work has been done in this class of cubic zincblende crystals since the first laser
measurements of Hobden and Russell (1964) in zincblende GaP. The prototypical
system that has been studied extensively is GaAs, and comprehensive reviews
are available (Loudon and Hayes, 1978; Cardona, 1975; Cardona and Güntherodt,
1982a, b, 1984, 1989, 1991). Frequencies of the LO and TO modes, ωLO and ωTO
respectively, for some of these systems are listed in Table 4.1.

4.2 Raman scattering for bulk zincblende and würtzite structures

4.2.2

Würtzite structures

In the last several years Raman scattering has also contributed a great deal to the
advances in understanding of the III-V nitride materials. The wealth of experiments
and information collected over the past 25 years on the GaAs-based material systems
is now starting to be duplicated in the nitride system, albeit somewhat slowly, as the
growth techniques and material systems continue to improve.
GaN-, AlN- and InN-based materials are highly stable in the hexagonal würtzite
structure although they can be grown in the zincblende phase and unintentional
phase separation and coexistence may occur. The würtzite crystal structure belongs
4 and group theory predicts zone-center optical modes are
to the space group C6v
A1 , 2B1 , E 1 and 2E 2 . The A1 and E 1 modes and the two E 2 modes are Raman
active while the B modes are silent. The A and E modes are polar, resulting in
a splitting of the LO and the TO modes (Hayes and Loudon, 1978). The Raman
tensors for the würtzite structure are as follows:


a 0 0
 0 a 0 
A1 (z) mode
0 0 b


0 0 c
 0 0 0 
E 1 (x) mode
c 0 0
(4.5)


0 0 0
 0 0 c 
E 1 (y) mode
0 c 0



f
0 0
0 −f 0
 0 −f 0   −f
E 2 mode.
0 0 
0
0 0
0
0 0
The vibrational modes in würtzite structures are given in Figure 3.3. Details of the
frequencies are given in Table 4.2.
Table 4.1. Frequencies in cm−1 of the
LO and TO modes for zincblende crystals.

AlN
GaAs
GaN
GaP
InP
ZnS

ωLO (cm−1 )

ωTO (cm−1 )

902
292
740
403
345
352

655
269
554
367
304
271

29

4 Raman properties of bulk phonons

30

Following some early work (Manchon et al., 1970; Lemos et al., 1972; Burns
et al., 1973) there has been a number of more recent experiments (Murugkar et al.,
1995; Cingolani et al., 1986; Azuhata et al., 1995) identifying the Raman modes
in these nitride materials. The early work was mainly on crystals in the form of
needles and platelets and the more recent work has been on epitaxial layers grown
on sapphire, on 6H-SiC, and on ZnO as well as some more unusual substrates. Table
4.2 gives the Raman modes as well as the scattering geometry in which they were
observed in the experiments of Azuhata et al. (1995). Experiments on AlN and InN
crystallites and films, particularly for the latter material, are more scarce, reflecting
the difficulties in achieving good growth qualities for these materials. In uniaxial
materials, when the long-range electrostatic field interactions of the polar phonons
dominate the short-range field of the vibrational force constants, phonons of mixed
symmetry can be observed (Loudon, 1964) under specific conditions of propagation
direction and polarization. They have been seen in the case of AlN (Bergman et al.,
1999).

4.3

Lifetimes in zincblende and würtzite crystals

Phonon–carrier interactions have an impact on semiconductor device performance
and, hence, a knowledge of the phonon lifetimes is important. Phonon lifetimes
demonstrate the effects of anharmonic interactions as well as scattering via point
defects and impurities. Anharmonic interactions (Klemens, 1958; Klemens, 1966;
Borer et al., 1971; Debernardi, 1998; Menéndez and Cardona, 1984; Ridley, 1996)
include the decay of phonons into other normal modes with the conservation of
energy and momentum. For a three-phonon decay process, a phonon of frequency
ω1 and wavevector q1 decays into two phonons of frequencies ω2 and ω3 , with
wavevectors q2 and q3 respectively, such that ω1 = ω2 + ω3 and q1 = q2 + q3 .
The investigation of the dynamical behavior of the vibrational modes provides a
direct measure of the electron–phonon interaction. The measurement of the decay
Table 4.2. Frequencies in cm−1 of the vibrational
modes in some würtzite structures.

E 21
E 22
A1 (TO)
A1 (LO)
E 1 (TO)
E 1 (LO)

ωAlN

ωCdS

ωGaN

252
660
614
893
673
916

44
252
228
305
235
305

144
569
533
735
561
743

ωInN
495
596

ωZnO
101
437
380
574
407
583

4.3 Lifetimes in zincblende and würtzite crystals

of the optical modes, which involves the anharmonic effects mentioned previously,
will be discussed here. Other processes that give experimental information on the
electron–phonon interaction include the generation of optical phonons by highenergy carriers, intervalley scattering between different minima in the conduction
band, and carrier–carrier scattering; these are reported by Kash and Tsang (1991)
for the prototypical system of GaAs.
Measurements of phonon linewidths for Raman and infrared measurements in
GaAs, ZnSe, and GaP give phonon lifetimes of 2–10 ps (von der Linde, 1980;
Menéndez and Cardona, 1984). For systems that are not far from equilibrium, the
lifetimes of the phonons can be described by anharmonic processes. The decay of
an optical phonon is frequently via pairs of acoustic phonons or via one acoustic
phonon and one optical phonon of appropriate energies and momenta (Cowley,
1963; Klemens, 1966). The first measurements with continuous-wave pumping
(Shah et al., 1970) of highly non-equilibrium LO phonons in GaAs yielded estimates
of LO-phonon lifetimes of approximately 5 ps at room temperature. This was
consistent with values obtained from linewidth studies. von der Linde (1980) used
time-resolved Raman scattering to obtain directly the time decay of non-equilibrium
LO phonons. They obtained a value of 7 ps for GaAs LO phonons at 77 K.
Subsequent experiments by Kash et al. (1985) led to the conclusion that the LO
phonon lifetime in GaAs was limited by its anharmonic decay into two acoustic
phonons.
Kash et al. (1987, 1988) and Tsen and Morkoç (1988a, b) used time-resolved
Raman scattering for the alloy system AlGaAs. The results for the lifetimes are
similar to those for pure GaAs; here, though, the phonon linewidths are broadened
owing to the disorder of the alloys and these inhomogeneous broadening effects need
to be considered. Secondly, although the dispersion relations of AlAs are different
from those of GaAs there is a similarity in decay times that is interesting and
unexpected. Tsen (1992) and Tsen et al. (1989) reported on the use of time-resolved
Raman studies of non-equilibrium LO phonons in GaAs-based structures.
Tsen et al. (1996, 1997, 1998) have studied the electron–phonon interactions in
GaN of würtzite structure via picosecond and sub-picosecond Raman spectroscopy.
Results on undoped GaN with an electron density of n = 5 × 1016 cm−3 showed
that the relaxation mechanism of the hot electrons is via the emission of LO phonons
and that the Fröhlich interaction is much stronger than the deformation-potential
interaction in that material. The measured lifetime was found to be 3 ps at 300 K and
5 ps at 5–25 K (Tsen et al., 1996, 1997, 1998). The electron–LO-phonon scattering
rate was seen to be an order of magnitude larger than that for GaAs and was
attributed to the much larger ionicity in GaN. These experiments also indicated that
the longitudinal phonons decay into a TO and an LO phonon or two TO phonons.
Raman investigations of phonon lifetimes have been reported by Bergman et al.
(1999) in GaN, AlN, and ZnO würtzite crystals. These lifetimes were obtained
from measured Raman linewidths using the uncertainty relation, after correcting

31

4 Raman properties of bulk phonons

32

for instrument broadening (Di Bartolo, 1969). These results demonstrate that the E 21
mode has a lifetime of 10 ps, an order of magnitude greater than that of the E 22 ,
E 1 (TO), A1 (TO) and A1 (LO) modes. This result was found to be true for samples
of high-quality GaN, AlN, ZnO as well as for AlN with a high level of impurities.
An explanation of the relative long lifetime of the E 22 phonons was given in terms
of factors including energy conservation constraints, density of final states, and
anharmonic interaction coefficients. The E 22 mode lies at the lowest energy of the
optical phonon modes in the würtzite dispersion curves (Nipko et al., 1998; Nipko
and Loong, 1998; Hewat, 1970) and only the acoustic phonons provide channels of
decay. At the zone edges, the acoustic phonons are equal to or larger than those of
the E 22 mode. Thus, for energy conservation to hold, the E 22 phonons have to decay
to acoustic phonons at the zone center, where their density is low.

4.4

Ternary alloys

The phonons of the ternary alloys ABx C1−x formed from the binaries AB and AC
crystals in the III-V as well as the II-VI semiconductors have been studied for some
time (Chang and Mitra, 1968). The III-nitrides have been studied more recently and
the alloys of the würtzite materials show some interesting features (Hayashi et al.,
1991; Behr et al., 1997; Cros et al., 1997; Demangeot et al., 1998; Wisniewski
et al., 1998). The ABx C1−x mixed crystals of the zincblende materials fall into
two main groups when classified according to the characteristics of the phonons.
These two classes are generally referred to as one-mode or two-mode behavior,
where ‘one-mode’ refers to the situation where the frequency of the AB phonons
gradually approaches the frequency of the AC phonons as the x-value of the alloy
increases. In the two-mode situation, the phonon frequencies are distinct and in
the limit of x = 0 (1) the AC (AB) phonon frequency is a local mode in the
AB (AC) crystal. Intermediate behavior has also been observed for certain crystals
(Lucovski and Chen, 1970). While there is no general agreement, several criteria
for phonon-mode behavior based on the mass differences of the atoms have been
proposed (Chang and Mitra, 1968). Typically, when the frequencies of the phonons
in the AB and the AC binary crystals are very different a two-mode behavior is
expected; otherwise, a one-mode behavior is seen. There is more uncertainty as well
as a smaller number of reports in the case of the würtzite nitrides. Hayashi et al.
(1991) reported studies on AlGaN würtzite films in the range 0 < x < 0.15. The
E 2 , E 1 (TO), E 1 (LO) and A1 (TO) modes were investigated and, in the composition
range studied, one-mode behavior was observed. Similar results were obtained by
Behr et al. (1997) in a narrow composition range. The E 2 mode was seen to be
unaffected by a change in composition. Cros et al. (1997) studied the AlGaN alloys
over the whole concentration range. They concluded that the E 2 mode exhibits
two-mode character, while the A1 (LO) mode is one-mode; the results for the

4.5 Coupled plasmon–phonon modes

A2 (TO) mode were inconclusive. Studies by Demangeot et al. (1998) concluded that
the A1 (LO), A1 (TO), and E 1 (TO) modes all exhibit one-mode behavior. However,
infrared reflectance experiments indicate that the E 1 (TO) mode displays two-mode
behavior (Wisniewski et al., 1998).
Yu et al. (1998) extended the modified random-element isodisplacement model
developed for zincblende structures by including the additional phonon modes and
the anisotropy of the würtzite structure. According to this model, the E 1 and the
A1 phonon modes should show one-mode behavior. In zincblende AlGaN crystals,
the results of Harima et al. (1999) indicate that the LO-phonon shows one-mode
behavior while the TO mode shows two-mode behavior.
Raman experiments using two ultraviolet wavelengths were performed by Alexson et al. (2000) on InGaN in the range 0 < x < 0.5. They investigated the A1 and
the E 2 phonons. These studies show a one-mode behavior of the A1 (LO) phonon
while the E 2 phonon demonstrates a two-mode characteristic.
The fact that the E 2 mode behaves differently from the E 1 and A1 modes is
not surprising, when one considers the specific atoms giving rise to the vibrations.
This in fact emerges from experiments on GaN würtzite films from natural GaN as
well as from GaN containing the isotope 15 N reported by Zhang et al. (1997). All
the A1 and the E 1 modes observed in the 15 N isotope were seen to shift to lower
frequencies. Niether the E 21 nor E 22 mode showed a similar shift; the E 21 mode was
essentially unaffected by the different isotopic mass. The authors thus concluded
that the E 21 vibration is due to the motion of the Ga atoms, which are heavy, alone
and, thus, that there is no frequency response to an isotropic change in the nitrogen
mass, which is considerably lighter.

4.5

Coupled plasmon–phonon modes

A coupling of the LO phonons to the plasma oscillations of the free carriers –
these oscillations are known as plasmons – occurs when an appreciable free-carrier
concentration is present in a polar semiconductor. These coupled phonon–plasmon
modes may be observed by Raman scattering, so providing information about the
free-carrier density of a given sample.
The free electrons scatter light weakly, although in solids the effect is enhanced
by band structure effects. Detailed accounts are provided by Platzman and Wolff
(1973), Yafet (1966), and Klein (1975). Mooradian and Wright (1966) made the
first observation of plasmons in n-GaAs using a Nd:YAG laser. These results
demonstrated the coupling between the LO phonon and the plasmon via the
interaction of the longitudinal electric fields produced by each of these excitations.
Other systems for which plasmon–phonon scattering has been observed include
zincblende structures such as InSb, GaP, and InAs (Hon and Faust, 1973; Patel and
Slusher, 1968). Various würtzite crystals have been investigated, for example CdS

33

34

4 Raman properties of bulk phonons

(Scott et al., 1969), SiC (Klein et al., 1972) and, more recently, the nitrides discussed
in the next paragraph.
Here, we summarize the experimental Raman scattering results of phonon–
plasmon coupled modes in GaN, which as grown tends to be an n-type material
(Edgar, 1994); a higher carrier concentration is achieved via intentionally doping
the material. Due to this fairly high carrier concentration, the plasmons in GaN
are considered to be overdamped, similarly to those in SiC (Klein et al., 1972).
Experiments by Kozawa et al. (1998) in GaN films with a relatively low concentration of carriers show a broadening and a weakening of the intensity of the Raman
features as well as a shift to higher frequencies. The carrier concentrations obtained
by fitting the Raman lineshape as in Klein et al. (1972), Hon and Faust (1973), and
Irmer et al. (1983) give values similar to those obtained from Hall measurements
(Kozawa et al., 1998). Other experiments on GaN to study these effects at lower
free-carrier concentrations have been carried out by Wetzel et al. (1996) and Ponce
et al. (1996) and show similar results. Kirillov et al. (1996) studied the effect on
the Raman modes in the high-carrier-concentration limit. This study found that with
these higher carrier concentrations the Raman features corresponding to the upper
branch of the phonon–plasmon coupled modes are too broad to extract meaningful
information. Demangeot et al. (1997) carried out a study of the lower branch of
the coupled phonon–plasmon mode for GaN for higher carrier concentrations. The
broad Raman peak observed could be fitted by the models referenced above (Klein
et al., 1972; Hon and Faust, 1973).

Chapter 5

Occupation number representation
Oh mighty-mouthed inventor of harmonies.
Alfred, Lord Tennyson, 1863

5.1

Phonon mode amplitudes and occupation numbers

In the study of carrier–phonon interactions in nanostructures it is convenient to
use the so-called phonon-number-occupation basis. In this basis the phonon system
is modeled by the Hamiltonian for a simple harmonic oscillator. Specifically, the
familiar conjugate variables of position and momentum are replaced by creation
and annihilation operators. These creation and annihilation operators act on states
each having a given number of phonons. In particular, the creation operator acting
on a state of n q phonons of wavevector q increases the number of phonons to n q + 1
and the phonon annihilation operator acting on a state of n q phonons of wavevector
q decreases the number of phonons to n q − 1.
In some applications – such as those involving the ground state of a Bose–
Einstein condensation – the creation and annihilation operators are essential to
describing the physical properties of the system. However, in the applications
considered in this book, these operators are used merely as a convenient way of
keeping track of the number of phonons before and after a carrier–phonon scattering
event; they do not introduce new physics. Nevertheless, it is important to understand
these operators since they are used widely by the semiconductor community in
modeling the electronic and optical properties of bulk semiconductors as well as
nanostructures.
The Hamiltonian describing the harmonic oscillator associated with a phonon
mode of wavevector q is
35

5 Occupation number representation

36

Hq =

pq2

1
+ mωq2 u q2 ,
2m
2

(5.1)

where m is the mass of the oscillator, ωq is the frequency of the phonon, u q is the
displacement associated with it, and pq is its momentum. Introducing the operators,
aq and aq† ,


mωq
1
uq + i
pq
(5.2)
aq =
2h̄
2h̄mωq
and

aq†

=

mωq
uq − i
2h̄



1
pq ,
2h̄mωq

(5.3)

it is straightforward to show that





mωq
mωq
1
1
†
aq aq =
pq
pq
uq − i
uq + i
2h̄
2h̄mωq
2h̄
2h̄mωq
=

1 1 2
i
1 mωq2 2
u +
p +
[u q , pq ].
2 h̄ωq q 2m h̄ωq q 2h̄

(5.4)

Here the commutator [u q , pq ] ≡ u q pq − pq u q = i h̄, from the properties of the
quantum mechanical operators u q and pq . Thus


1
1
2 2
†
+ mωq u q = h̄ωq aq aq +
.
2m
2
2
pq2

(5.5)

Since the energy of a quantum-mechanical harmonic oscillator is h̄ωq (n q + 12 ),
where n q is the number of phonons having wavevector q, it is clear that Nq =
aq† aq operating on an eigenstate of Nq phonons |Nq  has eigenvalue Nq . That
is, Nq |Nq  = n q |Nq . Moreover, by calculating aq aq† in the same manner used
to derive an expression for aq† aq , it follows that aq aq† − aq† aq = [aq , aq† ] = 1,
[aq , Nq ] = aq aq† aq − aq† aq aq = [aq , aq† ]aq = aq and [aq† , Nq ] = aq† aq† aq −
aq† aq aq† = aq† [aq† , aq ] = −aq† .
Accordingly,
Nq (aq† |Nq ) = (aq† Nq + aq† )|Nq  = (n q + 1)aq† |Nq 

(5.6)

Nq (aq |Nq ) = (aq Nq − aq )|Nq  = (n q − 1)aq |Nq .

(5.7)

and

Thus, aq† acting on |Nq  gives a new eigenstate with eigenvalue increased by 1
and aq acting on |Nq  gives a new eigenstate with eigenvalue decreased by 1; that
is,

5.1 Phonon mode amplitudes and occupation numbers

37

aq† |Nq  =


n q + 1|Nq + 1

(5.8)

aq |Nq  =

√
n q |Nq − 1,

(5.9)

and

where the eigenvalues are consistent with the relation Nq |Nq  = n q |Nq  with Nq =
aq† aq . The eigenstates |Nq  are orthonormal, so that
Nq ||Nq  ≡ Nq |Nq  = δ Nq ,Nq

(5.10)

where δ Nq ,Nq is the Kronecker delta function.
Thus, the only non-zero matrix elements of aq† are those that couple states Nq |
and |Nq  for which Nq = Nq + 1:
Nq |aq† |Nq  =


n q + 1 δ Nq ,Nq +1 .

(5.11)

√
n q δ Nq ,Nq −1 .

(5.12)

Likewise,
Nq |aq |Nq  =

The phonon occupation number n q may be determined at a temperature T from
the relation   = h̄ωq (n q + 12 ), where the average energy   is calculated for the
case where the eigenenergies are those of a harmonic oscillator, E n = h̄ωq (n + 12 ).
Then taking the weighting factor to be the Boltzmann factor f (E n ) = e−E n /k B T ,
we find that


f (E n ),
(5.13)
 =
E n f (E n )
where the sums are over all n from 0 to ∞. Then



h̄ωq
 =
h̄ωq ne−h̄ωq n/k B T
e−h̄ωq n/k B T +
2

 
h̄ω
q
,
=
h̄ωq nx n
xn +
2
where x = e−h̄ωq /k B T . Then

 
h̄ωq
d  n
  = h̄ωq x
xn +
x
dx
2
h̄ωq
x/(1 − x)2
+
= h̄ωq
1/(1 − x)
2
h̄ωq
h̄ωq x
+
=
1−x
2
h̄ωq
h̄ωq
,
+
= −h̄ω /k T
q
B
2
e
−1

(5.14)

(5.15)

38

5 Occupation number representation

and it follows that
nq =

1
e−h̄ωq /k B T

−1

,

(5.16)

which is known as the Bose–Einstein distribution. Equations (5.10)–(5.12) and
(5.16) are used frequently in calculating carrier–phonon scattering rates in nanostructures. Such calculations will be the subject of following chapters. In these
calculations the carrier wavefunctions and the phonon eigenstates are written as
products to express the total wavefunctions for the system of carriers and phonons.
When matrix elements are evaluated between the final and initial states of the
system, the phonon eigenstates and the phonon operators are grouped together since
they commute with the carrier wavefunctions and operations. Thus, carrier and
phonon matrix elements always appear as products and are evaluated separately.
This procedure will be illustrated in Chapters 8, 9, and 10. In Sections 5.2–5.4
we will relate the phonon displacement amplitudes to the interaction Hamiltonians
describing the dominant carrier–phonon interaction processes. In these calculations
and those of Chapters 8, 9, and 10 it will be convenient to express the normal-mode
phonon displacement u q in terms of the phonon creation and annihilation operators,
aq† and aq respectively. By adding equations (5.11) and (5.12) it will be seen readily
that

h̄
(aq + aq† ).
(5.17)
uq =
2mωq
In calculations of carrier–phonon scattering probabilities, the normal-mode
phonon displacement u(r) will appear in various linear forms in the Hamiltonians
entering the matrix elements being evaluated. In evaluating these matrix elements,
it will be important to keep track of not only the phonon occupation numbers n q
emerging from the phonon matrix elements but also the necessary conservation of
momentum and energy for each scattering process. In these matrix elements either
phonon creation or phonon absorption will take place but not both; therefore, as is
manifest from equations (5.11) and (5.12), only one of the two terms appearing in
equation (5.17) will contribute to the process under consideration.
u(r) is, of course, a Fourier series over the modes u q . In phonon absorption processes the phonon appears as an incoming wave and the factor ei(q·r−ωt) multiplies
the amplitudes associated with the phonon fields. Likewise, in phonon emission
processes the phonon appears as an outgoing wave and the factor ei(−q·r−ωt) multiplies the amplitudes associated with the phonon fields. These factors are essential
in ensuring proper conservation of momentum and energy and it is convenient to
include them along with the associated creation or annihilation operator. Indeed,
they appear naturally in the Fourier decomposition of u(r).
Moreover, each incoming or outgoing phonon will be associated with a unit
polarization vector; these unit polarization vectors will be denoted by êq, j for

5.1 Phonon mode amplitudes and occupation numbers

39

incoming waves and by ê∗q, j for outgoing waves. The factor e−iωt is common to
both incoming and outgoing phonons and it is generally included – along with
factors of E/h̄ associated with the carrier phases – in the integral over time that
appears in the Fermi golden rule. As a means of including the phase factors to
ensure proper accounting of energy and momentum as well as the appropriate unit
polarization vectors, (5.17) will now be written as a sum over all wavevectors q; the
appropriate non-temporal phase factors appear as multipliers of the corresponding
phonon operators:


1  
h̄  iq·r
u(r) = √
aq e êq, j + aq† e−iq·r ê∗q, j
2mω
N q j=1,2,3
q


h̄
1  
†
) eiq·r ≡
u(q) eiq·r .
êq, j (aq + a−q
=√
N q j=1,2,3 2mωq
q
(5.18)
where q is summed over all wavevectors in the Brillouin zone and N is the
number of unit cells in the sample. Chapter 7 will treat the role of dimensional
confinement in modifying the optical and acoustic phonon modes in nanostructures.
Two major modifications result: the phase space is restricted and the plane-wave
nature of the phonon is modified. Both of these effects will be treated by appropriate
modifications of equation (5.18): the changes in the phase space due to dimensional
confinement will be addressed by modifying the sum over q and the dimensional
confinement of the plane wave will be described by introducing suitable envelope
functions. Equations (5.8)–(5.18) will find many applications here and in Chapters
8, 9 and 10.

The factor h̄/2mωq in the final expression for u(r) ensures that the desired
Hamiltonian is consistent with the phonon mode amplitude. It is convenient in
practical calculations of u(r) to cast the amplitude constraints implied by [u q , pq ] =
i h̄ in the form of an integral:


  h̄
1

∗
3

êq · êq
u (r)u(r)d r =
e−i(q −q)·r d 3 r
√

2N
m
ω
ω
q q
q q
=

1  h̄
,
nm q 2ωq

(5.19)

where n = N /V and the integral is performed through use of the identity
 −i(q −q)·r 3
d r = V δq,q  . For a single mode q, the so-called phonon normalization
e
condition then becomes

√
√
h̄
.
(5.20)
[ nmu∗ (r)] · [ nmu(r)]d 3 r =
2ωq
In terms of u(q), the normalization condition is then
√
√
h̄ 1
.
[ nmu∗ (q)] · [ nmu(q)] =
2ωq V

(5.21)

5 Occupation number representation

40

In the literature, quantized fields – such as the displacement field u(r) – are
expressed in terms of aq and aq† in a number of different ways. Indeed, it is possible
to perform canonical transformations that essentially exchange the roles of u q and
pq . Specifically, from the definitions of aq and aq† in terms of u q and pq , it follows
that


pq
h̄
h̄


†
(aq + aq )
and
= −i
(a  − aq† ),
uq =
2mωq
mωq
2mωq q
(5.22)
where the primes have been added in preparation for the following substitutions:
aq → −iaq and aq† → +iaq† . With these substitutions it follows that
u q →

pq
mωq

and

pq
mωq

→ −u q .

(5.23)

Thus, the canonical transformation aq → −iaq has the result that the roles of
u q and pq /mωq are interchanged. Clearly, this canonical transformation leaves
the harmonic-oscillator Hamiltonian unchanged. This is, of course, true for this
Hamiltonian whether expressed in terms of u q and pq or in terms of aq and aq† .
As a result of this canonical transformation, the quantized fields – such as the
displacement field u(r) – may be expressed in terms of aq and aq† in a number
of different ways. Indeed, the literature testifies to the fact that all the different
forms are used widely. Evidently, factors of ±i as multipliers of aq and aq† do
not change the matrix element of any such field times its complex conjugate, since
(±i)(±i)∗ = 1; thus, quantum-mechanical transition rates are not affected by this
transformation, as should clearly be the case. There is one other invariance that
is used frequently to rewrite fields expressed in terms of aq and aq† in the most
convenient form for a particular application. Namely, in expressions where a sum
or integral over positive and negative values of q is present, substitutions of the
type aq eiq·r → a−q e−iq·r leave the expression unchanged if all other multiplicative
factors contain even powers of q. Again, the quantum-mechanical transition rates
are not affected by such a change of variable.

5.2

Polar-optical phonons: Fröhlich interaction

One of the most important carrier–phonon scattering mechanisms in semiconductors
occurs when charge carriers interact with the electric polarization, P(r), produced
by the relative displacement of positive and negative ions. In low-defect polar semiconductors such as GaAs, InP, and GaN, carrier scattering in polar semiconductors
at room temperature is dominated by this polar–optical-phonon (POP) scattering
mechanism. The POP–carrier interaction is referred to as the Fröhlich interaction,

5.2 Polar-optical phonons: Fröhlich interaction

41

after H. Fröhlich, who formulated the first qualitatively correct formal description.
In this book, the potential energy associated with the Fröhlich interaction will be
denoted by φFr (r). Clearly the polarization P associated with polar-optical phonons
and the potential energy associated with the Fröhlich interaction, φFr (r), are related
by
∇ 2 φFr (r) = 4π e∇ · P(r).

(5.24)

In terms of the phonon creation and annihilation operators of Section 5.1, P(r)
may be written as
P(r) = ζ

 
j=1,2,3


d 3 q  iq·r
aq e eq, j + aq† e−iq·r e∗q, j
3
(2π )

(5.25)

where eq, j represents the polarization vector associated with P(r) and q is the
phonon wavevector; then, it follows that
 

4π∇ · P(r) = 4πiζζ

j=1,2,3


d 3 q  iq·r
aq e q · eq, j − aq† e−iq·r q · e∗q, j .
3
(2π )
(5.26)

Consider the case of a polar crystal with two atoms per unit cell, such as GaAs.
Clearly, the dominant contribution to P(r) results from the phonon modes in which
the normal distance between the planes of positive and negative charge varies. Such
modes are obviously the LO modes since in the case of LO modes eq, j is parallel
to q. However, TO phonons produce displacements of the planes of charge such
that they remain at fixed distances from each other; that is, the charge planes ‘slide’
by each other but the normal distance between planes of opposite charge does not
change. So, TO modes make negligible contributions to P(r). For TO phonons,
eq, j · q = 0. Accordingly,

4π ∇ · P(r) = 4πiζζ


d 3 q  iq·r
aq e q − aq† e−iq·r q ,
3
(2π )

(5.27)

and the potential energy associated with the Fröhlich interaction, φFr (r), is given by

HFr = φFr (r) = −4πieζζ


d 3 q 1  iq·r
aq e
− aq† e−iq·r ,
3
q
(2π )

(5.28)

where φFr (r), has been denoted by HFr , the Fröhlich interaction Hamiltonian since
φFr (r) is the only term contributing to it.
The dependence of φFr (r) on q −1 is familiar from the Coulomb interaction; the
coupling constant ζ remains to be determined.

5 Occupation number representation

42

From subsection 2.3.3, the electric polarization P(r) may be written as
N e∗
uq (r)
(∞)

h̄
N e∗ 1   
 

=
√

mM
(∞) N q j=1,2,3 
ωLO
2
m+M

× aq eiq·r eq, j + aq† e−iq·r e∗q, j ),

P(r) =

(5.29)

where the division by (∞) accounts for screening, and the normal-mode expression
(5.18) has been used for u(r). By noticing that

d 3q
1 
↔
√
(2π )3
V q

(5.30)

and by comparing expressions (5.25) and (5.29), it follows that

h̄
N e∗ 1 


.
ζ =
√  
mM
(∞) N 
ωLO
2
m+M

(5.31)

However, from (2.26) evaluated for ω = ω0 , it follows that


2 − ω2
ωLO
1
N e∗2 1
TO
+
=
M
4π (∞)
(∞)2 m

2 
ω2
1
1 ωTO
−
= LO
2
4π
(∞)
(∞) ωLO


ω2
1
1
−
,
= LO
4π
(∞)
(0)

(5.32)

so that



2 
1
1
h̄
h̄ ωLO
N e∗ 1 


,
−
=
ζ =
√  
mM
(∞) N 
2ωLO 4π
(∞)
(0)
2
ωLO
m+M
(5.33)
and



d 3 q 1  iq·r
aq e
− aq† e−iq·r
3
(2π ) q




2π e2 h̄ωLO
1
1  1  iq·r
− aq† e−iq·r .
−
aq e
= −i
V
(∞)
(0) q q

HFr = φFr (r) = −4πieζζ

(5.34)

5.3 Acoustic phonons and deformation-potential interaction

5.3

43

Acoustic phonons and deformation-potential
interaction

The deformation-potential interaction arises from local changes in the crystal’s energy bands arising from the lattice distortion created by a phonon. The deformationpotential interaction, introduced by Bardeen and Shockley, is one of the most
important interactions in modern semiconductor devices and it has its origin in
the displacements caused by phonons. Indeed, the displacements associated with
a phonon set up a strain field in the crystal. In the simple case of a one-dimensional
lattice, the energy of the conduction band, E c , or the energy of the valence band,
E v , will change by an amount
E c,v = E c,v (a) − E c,v (a + u),

(5.35)

where a is the lattice constant and u is the displacement produced by the phonon
mode. Since a  u, it follows that
E c,v (a) = (d E c,v (a)/da)u.

(5.36)

Thus the phonon displacement field u produces a local change in the band energy;
the energy associated with the change is known as the deformation potential and
it represents one of the major scattering mechanisms in non-polar semiconductors.
Indeed, the deformation-potential interaction is a dominant source of electron energy
loss in silicon-based electronic devices. The three-dimensional generalization of
E c,v is
E c,v (a) = (d E c,v (a)/d V )V,

(5.37)

where V is a volume element and V is the change in the volume element due to
the phonon field. For an isotropic medium V /V = ∇ · u and the last expression
becomes,
E c,v (a) = V (d E c,v (a)/d V )∇ · u,

(5.38)

which is usually written as
c,v
= E c,v (a) = E 1c,v ∇ · u.
Hdef
c,v
Hdef

(5.39)

E 1c,v

and
are necessary since the deformation potential
The superscripts on
for electrons is different from that for holes. Chapter 9 provides a discussion of the
case where the medium is not assumed to be isotropic.

5.4

Piezoelectric interaction

The piezoelectric interaction occurs in all polar crystals lacking an inversion
symmetry. In the general case, the application of an external strain to a piezoelectric

44

5 Occupation number representation

crystal will produce a macroscopic polarization as a result of the displacements
of ions. Thus an acoustic phonon mode will drive a macroscopic polarization in
a piezoelectric crystal. In rectangular coordinates, the polarization created by the
piezoelectric interaction in cubic crystals, including zincblende crystals, may be
written as


P = 12 ex4 (∂w/∂ y + ∂v/∂z), 12 ex4 (∂u/∂z + ∂w/∂ x), 12 ex4 (∂u/∂ y + ∂v/∂ x) ,
(5.40)
where ex4 is the piezoelectric coupling constant and, as will be described in
Section 7.2, the factors multiplying ex4 are the components of the strain tensor
that contribute to the piezoelectric polarization in a zincblende crystal. As with
the Fröhlich and deformation-potential interactions, phonons play an essential role
in producing piezoelectric interactions. Piezoelectric interactions will be discussed
further in Chapter 9.

Chapter 6

Anharmonic coupling of phonons
With a name like yours, you might be any shape almost.
Lewis Carroll, Through the Looking Glass, 1872

6.1

Non-parabolic terms in the crystal potential for
ionically bonded atoms

The crystal potential may be expanded in powers of the displacements of the ions
from their equilibrium positions to yield a sum over quadratic and higher-order
terms. The quadratic terms, of course, represent the harmonic modes considered
at length in Chapter 5. The cubic and higher-order terms, containing products of
three or more displacements, are generally known as the anharmonic terms. These
anharmonic terms lead to modifica