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Introduction to Semiconductor Devices
For Computing and Telecommunications Applications
From semiconductor fundamentals to state-of-the-art semiconductor devices
used in the telecommunications and computing industries, this book provides
a solid grounding in the most important devices used in the hottest areas of
electronic engineering today. The book includes coverage of future approaches
to computing hardware and RF power amplifiers, and explains how emerging
trends and system demands of computing and telecommunications systems
influence the choice, design, and operation of semiconductor devices.
The book begins with a discussion of the fundamental properties of semi-
conductors. Next, state-of-the-art field effect devices are described, including
MODFETs and MOSFETs. Short channel effects and the challenges faced by
continuing miniaturization are then addressed. The rest of the book discusses
the structure, behavior, and operating requirements of semiconductor devices
used in lightwave and wireless telecommunications systems.
This is both an excellent senior/graduate text, and a valuable reference for
engineers and researchers in the field.
Kevin Brennan (1956–2003) was the recipient of a National Science Foun-
dation Presidential Young Investigator Award. He was named School of ECE
Distinguished Professor at Georgia Tech in 2002, and awarded a special com-
mendationfromtheViceProvostforResearchinrecognitionofhiscontributions
to graduate-level education in 2002. In 2003, he received the highest honor that
a Georgia Tech faculty member can attain: the Class of 1934 Distinguished Pro-
fessor Award. He also served as an IEEE Electron Device Society Distinguished
Lecturer.

Introduction to
Semiconductor Devices
For Computing and
Telecommunications Applications
KEVIN F. BRENNAN

CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo, Mexico City
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521153614
© Cambridge University Press 2005
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2005
Reprinted 2006
First paperback printing 2010
A catalogue record for this publication is available from the British Library
ISBN 978-0-521-83150-5 Hardback
ISBN 978-0-521-15361-4 Paperback
Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party Internet Web sites referred to in
this publication, and does not guarantee that any content on such Web sites is,
or will remain, accurate or appropriate.

To my family, Lea, Casper, and Jack

Contents
Preface page xi
List of physical constants xv
List of materials parameters for important semiconductors,
Si and GaAs xvi
1 Semiconductor fundamentals 1
1.1 Deﬁnition of a semiconductor 2
1.2 Equilibrium carrier concentrations and intrinsic material 7
1.3 Extrinsic material 16
Problems 21
2 Carrier action 23
2.1 Drift and diffusion 23
2.2 Generation–recombination 28
2.3 Continuity equation and its solution 33
Problems 36
3 Junctions 38
3.1 p–n homojunction in equilibrium 38
3.2 p–n homojunctions under bias 47
3.3 Deviations from ideal diode behavior 57
3.4 Carrier injection, extraction, charge control analysis,
and capacitance 61
3.5 Schottky barriers 68
Problems 75
4 Bipolar junction transistors 78
4.1 BJT operation 78
4.2 Secondary effects in BJTs 92
4.2.1 Drift in the base region 92
4.2.2 Base narrowing or the Early Effect 94
4.2.3 Avalanche breakdown 95
4.3 High frequency operation of a BJT 97
Problems 99

viii Contents
5 JFETs and MESFETs 101
5.1 JFET operation 101
5.2 MESFET and MODFET operation 104
5.3 Quantitative description of JFETs and MESFETs 112
5.4 Small signal model for a JFET 121
Problems 124
6 Metal–insulator–semiconductor structures
and MOSFETs 127
6.1 MIS systems in equilibrium 127
6.2 MIS systems under bias 133
6.3 Basic theory of MOSFET operation 144
6.4 Small signal operation of MESFETs and MOSFETs 155
6.5 CMOS circuits 160
Problems 165
7 Short-channel effects and challenges to CMOS 169
7.1 Short-channel effects 169
7.2 Scaling theory 176
7.3 Processing challenges to further CMOS miniaturization 183
Problems 186
8 Beyond CMOS 188
8.1 Evolutionary advances beyond CMOS 188
8.2 Carbon nanotubes 195
8.3 Conventional vs. tactile computing, molecular and
biological computing 197
8.4 Moletronics – molecular diodes and diode–diode logic 201
8.5 Defect tolerant computing 206
8.6 Quantum dot cellular automata 210
Problems 219
9 Telecommunications systems – an overview 220
9.1 Fiber transmission 220
9.2 Ampliﬁers and repeaters 223
9.3 Mobile cellular telecommunications systems 225
9.4 Device types for cellular systems 228
10 Optoelectronic devices – emitters, light ampliﬁers, and
detectors 230
10.1 LEDs 230
10.2 Stimulated emission 238
10.3 Laser operation 244

Contents ix
10.4 Types of semiconductor lasers 248
10.5 EDFAs 255
10.6 SOAs 258
10.7 p–i–n photodetectors 260
10.8 Avalanche photodiodes 265
Problems 273
11 Transistors for high frequency, high power
ampliﬁers for wireless systems 275
11.1 Transistor ﬁgures of merit for wireless systems 275
11.2 Heterostructures 281
11.3 MODFET devices 286
11.4 HBTs 290
11.5 Wide band gap semiconductors 294
Problems 298
References 300
Index 303

Preface
At the time of this writing the microelectronics industry is poised at the threshold of
a major turning point. For nearly fifty years, the industry has grown from the initial
invention of the integrated circuit through the continued refinement and miniaturization
of silicon based transistors. Along with the development of complementary metal
oxide semiconductor circuitry, miniaturization of semiconductor devices created what
has been called the information revolution. Each new generation of devices leads to
improved performance of memory and microprocessor chips at ever reduced cost, thus
fueling the expansion and development of computing technology. The growth rate in
integrated circuit technology, a doubling in chip complexity every eighteen months or
so, is known as Moore’s First Law. Interestingly, the semiconductor industry has been
able to keep pace with Moore’s First Law and at times exceed it over the past forty
years. However, now at the beginning of the twenty-first century doubts are being raised
as to just how much longer the industry can follow Moore’s First Law. There are many
difficult challenges that confront CMOS technology as device dimensions scale down
below 0.1 ␮m. Many people have predicted that several of these challenges will be so
difficult and expensive to overcome that continued growth in CMOS development will
be threatened. Further improvement in device technology will then require a disruptive,
revolutionary technology.
One might first wonder why is it important to continue to improve microprocessor
speed and memory storage much beyond current levels? Part of the answer to this
question comes from the simultaneous development of the telecommunications indus-
try. Both lightwave communications and cellular communications systems have grown
rapidly. Over just the past ten years, the cellular telephone industry has increased expo-
nentially, making it one of the fastest growing industries in the world. The expansion
of cellular telephony to the transmission of data, internet connections, and video infor-
mation is already beginning. Cellular transmission of video information will require
much higher bandwidth operation and greater sophistication than is currently available
in cellular systems. Lightwave systems already handle video and internet communica-
tions and are pressed to improve bandwidth for faster operation. Though improvement
in software and algorithms has been highly instrumental in improving telecommuni-
cations system capacity, hardware improvements are equally as important to maintain
growth in these systems. Therefore, there is an acute need for faster electronics with a
concurrent memory enhancement to improve telecommunications systems, thus fur-
ther fueling the information revolution.
It is my opinion that the microelectronics industry will necessarily continue to
grow to meet the demands of future computing and telecommunications systems.

xii Preface
However, this growth may not be confined to silicon CMOS but may extend into
several other technologies as well. The goal of this book is to present an introductory
discussion to undergraduate students of the basic workings of current semiconductor
devices used in computing and telecommunications systems and to present some of
the emerging revolutionary approaches that microelectronics could take in the near
future. Throughout the book, the applications and operating requirements imposed
on semiconductor hardware by computing and telecommunications applications are
used to describe the important figures of merit of each device. In this way, the student
can clearly see what fundamental properties a particular device must have to meet the
system application requirements for which it is designed.
One might wonder why yet another book is needed on semiconductor devices for
undergraduate education. This question is particularly relevant in that several univer-
sities have recently decided to abandon requiring an undergraduate course in semi-
conductor devices, making it solely an elective instead. Given that there are several
excellent texts, such as Streetman and Banerjee Solid State Electronic Devices (2000)
or Pierret Semiconductor Device Fundamentals (1996), one might wonder why another
undergraduate book is needed especially in light of the fact that the need for undergrad-
uate books is apparently decreasing. Though the above mentioned books are unques-
tionably excellent, they do not provide a discussion of the future of microelectronics
and how it relates to the greatest existing growth industries of computing and telecom-
munications. It is the primary purpose of this book to provide the context, namely
computing and telecommunications, in which semiconductor devices play their most
important and ubiquitous role. Further, the present book provides a look at not only the
state-of-the-art devices but also future approaches that go beyond current technology.
In this way, a new, refreshing, up-to-date approach to teaching semiconductor devices
and exciting the students about the future of the field is provided. It is my opinion that
through an enlightened approach the negative trend of the removal of microelectron-
ics courses from undergraduate curriculums can be reversed. Ironically, I believe that
microelectronics is poised for its greatest surge. Thus rather than abandoning teach-
ing microelectronics, it should be more widely presented and the approach should
be more interdisciplinary at least addressing possibilities in molecular and biologi-
cal systems for future computing hardware. This book presents a first cut at such an
interdisciplinary approach.
This book has grown out of notes used for an undergraduate course I teach in the
School of Electrical and Computer Engineering at Georgia Tech. The course is one
semester long and follows a required course in circuit theory that includes some of the
basics of semiconductor devices. However, the book does not draw on the student’s
knowledge of circuits and can thus be used as a first course in semiconductor devices.
Given that the presentation is a bit briefer than most semiconductor device texts on
the fundamentals, the book is probably better suited for either a second level course,
as is done at Georgia Tech, or a first level course for more advanced students. As
for scientific and mathematical background, the book requires knowledge of calculus
and differential equations. However, no knowledge of quantum mechanics, solid state
physics or statistical mechanics is required. Computer based assignments have not been

Preface xiii
included in the text. The main reasons for their exclusion is that we are preparing a
computer based exercise book for use in all of our undergraduate level microelectronics
courses. The proposed book will have computer exercises that follow the present book
providing another path for learning.
Thepresentbookisorganizedasfollows.Itbeginswithapresentationoftheessential
fundamentals of semiconductors. The second chapter discusses carrier action. The
third chapter focuses on junctions including p–n homojunctions, Schottky barriers,
and ohmic contacts. In the fourth chapter, bipolar junction transistors are presented.
JFETs and MESFETs are discussed in Chapter 5, including ac models. Chapter 6
presents a discussion of metal insulator semiconductor systems particularly MOS
devices, long channel MOSFETs, and CMOS circuits. Short channel devices, scaling
and challenges to further improvement of CMOS devices are discussed in Chapter 7.
Chapter 8 presents a discussion of several different technical approaches that go beyond
CMOS. The topics in Chapter 8 are limited to those that do not require knowledge of
quantum mechanics. These topics are included in the graduate level textbook Theory
of Modern Electronic Semiconductor Devices (2002) by Kevin F. Brennan and April
S. Brown. The balance of the book focuses on device use in lightwave and cellular
telecommunications systems. Chapter 9 gives an overview of telecommunications
systems, both wired and wireless. In Chapter 10 a discussion of optoelectronic devices
used in lightwave communications systems such as LEDs, lasers, erbium doped fiber
amplifiers, semiconductor optical amplifiers and photodetectors is presented. The book
concludes with a discussion of transistors used in high frequency, high power amplifiers
such as MODFETs and HBTs in Chapter 11.
This book is designed to be the first in a series of texts written by the current
author. It provides an introduction to semiconductor devices using only for the most
part classical physics. Some limited discussion about spatial quantization is included,
however. Thus the present book is well suited to the typical junior or senior level
undergraduate student. After completing a course that utilizes the present book, the
student is prepared for graduate level study. At Georgia Tech graduate students in
microelectronics begin their study, following an undergraduate course at the level of
the present book, with the basic science of quantum mechanics, statistical mechanics,
and solid state physics covered in The Physics of Semiconductors with Applications to
Optoelectronic Devices (1999), by Kevin F. Brennan. This material is covered in a first
semester graduate level course that is followed by a second semester graduate level
course on modern electronic devices. The textbook for the second semester graduate
level course at Georgia Tech is Theory of Modern Electronic Semiconductor Devices
(2002) by Kevin F. Brennan and April S. Brown.
Pedagogically, the undergraduate course this book has been developed from is taught
three times a year at Georgia Tech. This course is a second level course in semicon-
ductor devices that follows a required course that contains both circuit theory and
elementary semiconductor material. Since the book is used at Georgia Tech for a sec-
ond level course, we typically quickly cover the topics in Chapters 1–3 in about 2–3
weeks. Depending upon the student’s preparation, the fourth chapter can be skipped,
substituting a brief review instead. The course gets “down to business” beginning with

xiv Preface
Chapter 5 and goes through the remaining chapters for the balance of the semester.
Often we skip the section on CMOS (this is covered in the circuits level course) as
well as Chapter 9 which is generally just assigned reading. Homework problems are
typically selected from those at the back of the chapters. Two in-class quizzes and a
final examination are given. Instructors can obtain a solutions manual for the prob-
lems on-line at www.ece.gatech.edu/research/labs/comp elec. The solutions manual
can be downloaded and is password protected. Instructors only are given access to the
solutions. Please follow the directions at the web site to obtain the necessary password.
The author would like to thank his many colleagues and students at Georgia Tech
that have provided constructive criticism in the writing of this book. Specifically, the
author is thankful to Mike Weber for his help on some of the figures and for assisting
in creating the book web site. Thanks go to Professor Wolfgang Porod of Notre Dame
University and to Dr. Phaedron Avouris at IBM for granting permission to reproduce
some of their work.
Finally, I would like to thank my family and friends for their enduring support and
patience.
Postscript
Professor Kevin Brennan, my colleague at Georgia Tech, and one of my best friends,
passed away on August 2, 2003. After he became ill, he continued to work on this
text during the last year of his life, and had essentially completed it at the time of
his death. I became involved at the copy-editing stage, and would like to express my
appreciation to his wife, Lea McLees, for allowing me to assist in bringing this text
to conclusion. I would like to acknowledge the effort of Ms. Maureen Storey, whose
meticulous attention to detail was essential to the completion of the project. Most of my
corrections and additions were reactive to her questions and comments. Eric Willner
at Cambridge University Press showed considerable patience in both coaxing us and
allowing us time to polish the text. The Chair of the School of Electrical and Computer
Engineering at Georgia Tech, Dr. Roger Webb, provided both emotional and tangible
support during this difficult period. Professor Christiana Honsberg and Professor Tom
Gaylord at Georgia Tech provided answers to questions from me at critical junctures.
Kevin Brennan was a superb teacher, accomplished researcher, and prolific author. I
am appreciative of the fact that he is able to teach us one last time.
Atlanta, GA W. Russell Callen
March, 2004

Physical constants
Avogadro’s constant NAVO 6.022 × 1023
Mol−1
Boltzmann’s constant kB 1.38 × 10−23
J/K
8.62 × 10−5
eV/K
Electron charge q 1.6 × 10−19
C
Electron rest mass m0 0.511 × 106
eV/C2
9.11 × 10−31
kg
Permeability – free space µ0 1.2566 × 10−8
H/cm
Permittivity – free space ε0 8.85 × 10−14
F/cm
Planck’s constant h 4.14 × 10−15
eV s
6.63 × 10−34
J s
Reduced Planck’s constant h̄ 6.58 × 10−16
eV s
1.055 × 10−34
J s
Speed of light c 3.0 × 1010
cm/s
Thermal voltage – 0300 K kBT/q 0.0259 V

Material parameters for important
semiconductors, Si and GaAs
Bulk material parameters for silicon
Lattice constant (Å) a = 5.43
Dielectric constant 11.9
Intrinsic carrier concentration (cm−3
) 1.0 × 1010
Energy band gap (eV) 1.12
Sound velocity (cm/s) 9.04 × 105
Density (g cm−3
) 2.33
Effective mass along X (m∗
/m0) – transverse 0.19
/m0) – longitudinal 0.916
Effective mass along L (m∗
/m0) – transverse 0.12
/m0) – longitudinal 1.59
Heavy hole mass (m∗
/m0) 0.537
Electron mobility at 300 K (cm2
/(V s)) 1450
Hole mobility at 300 K (cm2
/(V s)) 500
Thermal conductivity at 300 K (W/(cm ◦
C)) 1.5
Effective density of states in conduction band (cm−3
) 2.8 × 1019
Effective density of states in valence band (cm−3
) 1.04 × 1019
Nonparabolicity along X (eV−1
) 0.5
Intravalley acoustic deformation potential (eV) 9.5
Optical phonon energy at (eV) 0.062
Intervalley separation energy, X–L (eV) 1.17
Bulk material parameters for GaAs
Lattice constant (Å) a = 5.65
Low frequency dielectric constant 12.90
High frequency dielectric constant 10.92
Energy band gap at 300 K (eV) 1.425
Intrinsic carrier concentration (cm−3
) 2.1 × 106
Electron mobility at 300 K (cm2
/(V s)) 8500
Hole mobility at 300 K (cm2
/(V s)) 400
Longitudinal sound velocity (cm/s) along (100)
direction 4.73 × 105
Density (g/cm3
) 5.36

Materials parameters for Si and GaAs xvii
Effective mass at (m∗
/m0) 0.067
/m0) 0.56
/m0) 0.85
Heavy hole mass (m∗
/m0) 0.62
Effective density of states conduction band (cm−3
) 4.7 × 1017
Effective density of states valence band (cm−3
) 7.0 × 1018
Thermal conductivity at 300 K (W/(cm ◦
C)) 0.46
Nonparabolicity at (eV−1
) 0.690
Intravalley acoustic deformation potential (eV) 8.0
Optical phonon energy at (eV) 0.035
Intervalley separation energy, –L (eV) 0.284
Intervalley separation energy, –X (eV) 0.476
Note: designates a point in k-space; X and L designate directions in k-space. refers to the
k = 0 point at the center of the Brillouin zone. X refers to the {100} directions and L to the {111}
directions.

1
Semiconductor fundamentals
In this chapter, we review the basic fundamentals of semiconductors that will be used
throughout the text. Only the fundamental issues that we will need to begin our study
of semiconductor devices utilized in computing and telecommunications systems are
discussed.
Before we begin our study it is useful to point out how semiconductor devices are
instrumental in many applications. In this book we will mainly examine the application
of semiconductor devices to computing and telecommunications systems. Specifically,
we will examine the primary device used in integrated circuits for digital systems, the
metal oxide semiconductor field effect transistor, MOSFET. The discussion will focus
on state-of-the-art MOSFET devices and future approaches that extend conventional
MOSFETs and revolutionary approaches that go well beyond MOSFETs. It is expected
that computing hardware will continue to improve, providing faster and more powerful
computers in the future using either some or all of the techniques discussed here or
perhaps using completely new technologies. In any event, there is almost certainly
going to be a large growth in computing hardware in order to maintain the pace of
computer development and this book will help introduce the student to emerging
technologies that may play a role in future computing platforms.
The second major topic of this book involves discussion of semiconductor devices
for telecommunications applications. We will examine devices of use in lightwave
communications as well as wireless communications networks. Among these devices
are emitters, detectors, amplifiers, and repeaters.
Some mention should be made of the various commercial products that are and
will be greatly impacted by semiconductor devices. The development of blue and
blue-green light emitting diodes (LEDs) and lasers foments the evolution of new,
highly efficient, rugged, ultra-long-life illumination elements. White light emitters
using LEDs are now becoming commercially available. These emitters are far more
efficient than incandescent bulbs, cost about the same or less, have lifetimes measured
in years rather than months, are rugged and durable. It is expected that replacing
incandescent lighting by LEDs worldwide can result in a substantial energy savings
and potentially reduce consumption of fossil fuels. Perhaps this will lead to a reduction
in greenhouse gas emission and help combat global warming and environmental decay
in general. Blue lasers enable the development of very small compact discs for data
storage, video and audio systems thus greatly expanding the storage capacity of CDs.
New semiconductor materials, such as gallium nitride (GaN) and silicon carbide
(SiC), are emerging that are far more tolerant of high temperatures, and operate at
significantly higher current densities and frequencies than existing devices. Devices

2 Semiconductor fundamentals
made from these materials are highly attractive for high power, high frequency, and
high temperature operation. Specific applications are as power amplifiers for base
stations in wireless telecommunications systems, hybrid electric vehicles, switching
elements for electric power grids, and high power amplification for radar and satel-
lite communications. Thus GaN and SiC may emerge as important semiconductor
materials for many important applications.
1.1 Definition of a semiconductor
The first question one might raise is why are semiconductor materials important in
electrical engineering? To answer this question let us first consider a useful character-
ization scheme for solids based on their electrical properties, specifically their elec-
trical conductivity. Generally, all crystalline solids can be classified into one of four
categories. These categories, arranged from highest electrical conductivity to lowest,
are metals, semimetals, semiconductors, and insulators. The distinction among these
four categories is of course, somewhat vague. For instance, some materials can be
either metallic or semimetallic depending upon the form into which they crystallize.
Additionally, the distinction between semiconductors and insulators can often become
blurred for the wide band gap materials. Nevertheless, we will find it convenient to
classify solids into one of these four categories.
Of the four classes of materials, semiconductors are arguably the most important in
electrical engineering. The principal reason underlying the importance of semiconduc-
tors is the fact that their electrical properties can be readily engineered. Semiconductors
are unique in that their conductivity can be significantly altered in several different
ways. For the other three types of solids, metals, semimetals, and insulators, their
conductivity cannot be readily and significantly altered making them far less attractive
for electrical engineering.
There are numerous ways in which the conductivity of a semiconductor can be
altered. In this book, we will address most of these approaches and how they can be
utilized to make useful semiconductor devices. Before we outline the approaches to
manipulating the electrical conductivity of a semiconductor, we should first review
what a semiconductor is.
The most commonly used semiconductors are the elemental semiconductors silicon
and germanium, and the compound semiconductors, consisting of compound mate-
rials. There are numerous compound semiconductors but they are generally formed
from two, three, or four different elements and are referred to as binary, ternary, and
quaternary compounds respectively. The most important compound semiconductors
are based on Column IIIA and Column VA elements in the Periodic Table. For this
reason, these compounds are called the III–V compound semiconductors or III–Vs.
Examples of the III–V compounds are gallium arsenide (GaAs), indium phosphide
(InP), aluminum arsenide (AlAs), indium arsenide (InAs), etc. Notice that in each case
the cation is a Column III element while the anion is a Column V element. Ternary
compounds can be formed using three elements such as AlxGa1−xAs, where the sub-
script x represents the mole fraction of aluminum present in the compound. Similarly,

quaternary compounds can be formed in which four elements are combined. An exam-
ple of a quaternary compound semiconductor is InxGa1−xAsyP1−y.
How though can we identify which materials are semiconductors? To answer this
question we must first consider a fundamental result in the physics of solids. Every
crystalline solid has translational symmetry. A system is said to have translational
symmetry if it can be broken into a set of identical basic unit cells such that when
the system is translated by a distance equal to the length of one unit cell it remains
invariant. An obvious example is that of a uniform brick wall. If one translates a row
of bricks by a length equal to that of a single brick, the wall looks precisely the same
as before. The wall is said to be invariant under a linear translation. A similar situation
holds for a crystalline solid. The arrangements of atoms forming a crystalline solid
are like the bricks of a uniform wall. The atoms, much like the bricks, are arranged in
periodic intervals. Therefore, when the system is translated by a distance equal to the
separation between two adjacent atom centers, called the lattice constant, the system
remains the same and is said to be invariant. Since the arrangement of the positions
of the atoms in a crystalline solid is periodic, the electrostatic potential corresponding
to the atoms is also periodic. The potential of the solid is thus also translationally
symmetric. The fact that all crystalline solids have a periodic potential is extremely
important. There is a fundamental result from quantum mechanics that applies to any
system with a periodic potential. This result (Brennan, 1999, Chapter 8) states that
for a system with a translationally symmetric potential, the electron energy levels are
arranged in bands. These bands can either be conducting or forbidden. As the name
implies a conduction band is one in which the electrons can propagate or conduct.
Conversely, a forbidden band is one in which no conducting states exist. Electrons
cannot be placed into a forbidden band.
In addition to the formation of energy bands, the presence of a periodic potential
introducesenergygapsintheallowedenergyspectrum.Thesegapsarecalledforbidden
gaps. Forbidden gaps correspond to energy ranges wherein no allowed electronic
states exist. A typical diagram showing a valence band, forbidden energy band and
conduction band is shown in Fig. 1.1. As can be seen from the figure, allowed energy
states exist only within the conduction and valence energy bands. As mentioned above,
electrons within the conduction band can propagate through the crystal and thus carry
a current. Electrons cannot be located within the forbidden band. In the valence band,
electronic states exist but these states are not free. In other words, electrons within the
valence band are localized into bound states that are formed by the molecular bonds
between the constituent host atoms of the crystal.
A completely empty band cannot conduct a current. This should be obvious since
an empty band has no carriers within it and thus there is nothing to carry the current. A
less obvious fact is that a completely filled energy band also cannot conduct a current.
This follows from the fact that no two electrons can simultaneously occupy the same
quantum state. The general formal statement of this is the Pauli Principle, which applies
to the class of particles called fermions, and includes electrons, protons, and neutrons.
The Pauli Principle plays a strong role in the formation of atoms. As the reader is
aware from fundamental chemistry, each atom in the Periodic Table is formed by

conduction band
forbidden band
valence band
Figure 1.1 Sketch of the conduction, forbidden, and valence bands within a semiconductor.
Electrons within the conduction band can freely propagate through the crystal and thus can
carry a current. Electrons within the valence band are localized into bound electronic states
formed by the molecular bonding of the constituent atoms of the crystal. In the forbidden
band, no electronic states exist and thus electrons cannot exist within the forbidden band.
The forbidden band is also called the energy gap.
progressively adding an electron and proton (and possibly neutrons) to each previous
atom starting with hydrogen. In the case of hydrogen the only charged particles present
are one electron and one proton. The electron is placed into the lowest lying energy
state of the atom. The next element is helium which comprises two electrons and
two protons as well as two neutrons. The additional electron cannot be added to the
same quantum state as the first electron and is placed into the first level, 1s, but with a
different spin state. The 1s level is completely filled by two electrons. Thus for the next
element, lithium with three electrons and three protons plus neutrons, the third electron
in lithium must go into a higher energy state than that of the first two electrons, the 2s
orbital. Thus ever larger atoms containing more electrons and protons are configured
such that the additional electrons enter higher energy states. If electrons did not obey
the Pauli Principle, then all of the electrons in an atom, no matter how many electrons
are present, would be put into the lowest energy, 1s state. As a result, chemistry would
be very different from what is observed.
According to the Pauli Principle, an electron cannot move into an already occupied
state. This situation is similar to that of parking automobiles in a parking lot. No
two cars can be put into the same parking spot simultaneously. Obviously, a parking
spot must initially be unoccupied in order to place a car into it. Electrons behave in
much the same way. In the case of electrons, quantum states assume the same role as
parking spaces do for cars. It is important to further recognize that a filled parking lot
cannot accept any more cars without removing one and similarly a filled energy band
cannot accept any more electrons without removing one. Now we can understand why
a filled energy band does not conduct a current. For a current to flow, electrons must
move from one state to another. In a filled band there are no vacancies into which the
electrons can move since all possible states are filled. Hence, no current can flow.

The distinction among each of the four categories of solids can now be made based
on the energy bands in the material. An insulator is a material in which the highest
occupied band, called the valence band, is completely filled and the next available
energy band, called the conduction band, is completely empty. The energy separation
between the filled and empty bands is called the energy gap. In order for a material to
be insulating, it is also necessary that the energy gap be very high such that carriers
cannot be readily promoted from the valence band into the conduction band. Therefore
in an insulator, the valence band is completely filled and the conduction band is com-
pletely empty and no current can flow in the material. Conversely, a metal is a highly
conductive material. Metals are solids in which the conduction band is only partially
filled. The conduction band consists then of many electrons and many empty states.
A large current can be supported within a metal since most of the electrons within
the conduction band can contribute to the current conduction since there exist many
vacancies into which the electrons can move under the action of a driving field. Con-
sequently, metals have a very high electrical conductivity. The other two categories of
materials, semimetals and semiconductors, are somewhat intermediate between metals
and insulators. Semimetals are materials like insulators in that the conduction band is
unoccupied and the valence band is fully occupied at zero temperature. However, in
semimetals the energy gap vanishes in part such that the conduction and valence bands
intersect. Electrons from the valence band can be readily accelerated into the conduc-
tion band at the point or points of intersection of the two bands and the material can
thus support a current. In this way, semimetals exhibit a relatively high conductivity but
not as high as that of a metal. Finally, a semiconductor is something like an insulator
but with a relatively small energy gap separating the conduction and valence bands.
At absolute zero temperature within a semiconductor the conduction band is com-
pletely empty and the valence band is completely filled. However, as the temperature
is raised to room temperature, the energy gap is sufficiently small that some measurable
population of the conduction band occurs. Therefore, a semiconductor will conduct
a current at room temperature but with a much higher resistance than that of a metal.
The electrical resistance of a crystal is a function of the electron concentration in
the conduction band. In a metal, the electron concentration within the conduction band
is extremely high, on the order of ∼1023
cm−3
. In a semiconductor the electron con-
centration within the conduction band is many orders of magnitude lower. Therefore,
the conductivity of a semiconductor is much less than that of a metal. To quantify the
conductivity it is essential to determine the electron concentration. In the next section
the technique used to determine the electron concentration within a semiconductor
will be discussed.
Before we end this section, it is useful to discuss the shape of the energy bands in a
crystal. One of the basic concepts of quantum mechanics is that fundamental particles
have a wave-particle duality. This implies that a fundamental particle like an electron
for example sometimes manifests itself as a wave and sometimes as a particle, but never
simultaneously. Therefore, an electron has a wavelength associated with it, called the
de Broglie wavelength, that accounts for its wavelike behavior. The momentum of an
electron can be described using its wavelike behavior as

energy
0,0 k
Figure 1.2 Sketch of the energy vs. k relationship for free electrons. Energy bands that obey
this relationship are called parabolic energy bands. To a good approximation the energy bands
within a semiconductor, at least near the band edge (bottom of the conduction band and top of
the valence band), are parabolic.
p = h̄k (1.1)
where h̄ is Planck’s constant divided by 2π and k is deﬁned as
k =
2π
λ
(1.2)
λ is the electron wavelength and k is called the electron wavevector. A free electron
has only kinetic energy given by
E =
p2
2m
(1.3)
Substituting into (1.3) for p the expression given by (1.1) obtains
E =
h̄2
k2
2m
(1.4)
The energy of the electron varies quadratically with the wavevector, k. The relationship
between E and k given by (1.4) is called a parabolic energy vs. k relationship and is
sketched in Fig. 1.2. Notice that the energy vs. k diagram shown in Fig. 1.2 is a parabola
with vertex at E = 0, k = 0. Since the electron energy varies with respect to the electron
wavevector, the E(k) relationship is very important in semiconductors. The behavior
of the electron as a function of k is referred to as the electron motion in k-space. In
general, the wavevector
k for an electron in a crystal is a three-dimensional vector. In
free space, we can replace the vector
k by its one-dimensional scalar magnitude, k. We
can also often use this scalar one-dimensional model to gain insight into the behaviour
of an actual semiconductor.
Typically, the mass that appears in the denominator of (1.4) is quite different from
the free space mass and is referred to as the effective mass, usually written as m∗
.
The effective mass is usually less than the free space mass and takes into account

the motion of the electron within the crystalline lattice. The electron effective mass is
defined as
1
m∗
=
1
h̄2
d2
E
dk2
(1.5)
Notice that (1.5) implies that the curvature of the E(k) relationship determines the
effective mass of the electron. If the curvature is high, meaning that E changes greatly
with a small change in k, then the effective mass of the electron is small. Conversely, if
the curvature is low, implying that the energy E changes slowly with change in k, then
the effective mass of the electron is large. In the limit of a horizontal line in the E(k)
relationship, the effective mass is infinite; the energy never changes for any change in
momentum or k.
The energy bands within most semiconductors deviate from the simple parabolic
energy relationship given by (1.4) at high energy, defined as several kT above the
conduction band minimum or edge or several kT below the valence band edge.†
The
valence band edge is the point of minimum hole energy within the valence band and
typically lies at k = 0 in k-space. The energy band structure in general is very compli-
cated in most semiconductors, yet can have a profound effect on device operation, as
will be seen in later chapters.
1.2 Equilibrium carrier concentrations and intrinsic material
It is important first to understand the concept of equilibrium. A full discussion of
equilibrium can be found in the book by Brennan (1999). The most exacting definition
of a system in equilibrium is that a closed system, isolated from the external environ-
ment, if left to itself over time will evolve towards equilibrium. Under equilibrium
conditions there are no external agents, i.e., external voltages, external fields, radiative
excitations, or any other external perturbation acting on the system. The system is
completely isolated from the external world and as such is unperturbed. There is an
important difference between equilibrium and steady-state. In steady-state the system
does not change with time, but it is not isolated from the external world. In equilibrium
the system is completely isolated from the external world and thus does not change
with time but also has no net current flow. A system in steady-state though it does not
change with time still has a net current flow. One simple way to view the difference
between equilibrium and steady-state is to imagine a partially filled sink. In equilib-
rium the water level does not change and remains constant. Additionally, there is no
net current flow. There is no input or output of water from the sink, the faucet is off
and the drain is closed. For a sink in steady-state the water level also doesn’t change.
However, there is a net current flow. The faucet is on and the drain is open such that
the input matches the output and thus the water level does not change. However, the
† Here, k = kB, Boltzmann’s constant. It is usually multiplied by T, the absolute temperature. The factor kT
appears in the Fermi–Dirac distribution function, discussed in Section 1.2.

system interacts with its external environment and thus is not in equilibrium but in
steady-state instead.
In order to calculate the electron concentration within the conduction band of a
semiconductor in equilibrium it is useful to again draw an analogy to parking spaces
and cars. In order to park one’s car two conditions must be met. First, there must be
a parking space. One cannot park one’s car, at least legally and safely, in the middle
of the road. There must be a parking space. However, the mere presence of a parking
space does not ensure that one can park one’s car. The second condition is that the
space must be unoccupied. The obvious statement that one must have a vacant parking
space available to park one’s car has an analogy for electrons. In order to put an
electron into an energy state, a similar set of two conditions must exist. These are that
there must exist a state matching the energy of the electron into which it can be put
and this state must be unoccupied. The total number of electrons in the conduction
band depends upon the number of available states at a given energy multiplied by the
probability that each state is occupied. Summing this product over all possible energies
will give the total number of electrons within the conduction band. Mathematically,
we can determine the electron concentration in the conduction band by integrating
the product of the function that describes the number of available states at a given
energy, called the density of states, D(E), and the function that gives the probability
that a state at that energy will be occupied, called the distribution function, f (E). The
electron concentration, n, is given then as
n =

D(E) f (E)dE (1.6)
where the integration is taken over the full range of energy values. In order to evaluate
this expression it is necessary to determine both D(E) and f (E). The density of states
function D(E)for a three-dimensional system is given as(Brennan,1999, Section 5.1),
D(E) =
1
2π2

2m
h̄2
3
2 √
E (1.7)
where h̄ is the reduced Planck constant, h/2π.
The probability distribution function, f (E), depends upon whether the system is in
equilibrium or not. What then is the form of the equilibrium probability distribution
function for electrons? To answer this question let us consider Fig. 1.3. Figure 1.3
shows a collection of bins, arranged in ascending energy into which one can place
an electron. Let each bin represent an allowed energy state. It is important to recall
that no two electrons can occupy the same quantum state simultaneously in accor-
dance with the Pauli Principle. Therefore, once an electron has been placed into a
bin, no additional electrons can be put into that bin. To attain the minimum energy
configuration of the system the first electron must be put into the first bin. The next
electron must then be placed into the second bin, the third electron into the third bin
and so forth. This process continues until all of the electrons are placed into a bin.
For example, in Fig. 1.3, if only six electrons are present they are placed into the first
six bins as shown in the diagram. The normalized probability of each of the first six

energy
f(E)
(a)
(b)
energy
1
Ef
Figure 1.3 (a) Collection of energy bins representing energy states arranged in ascending
energy. Into each bin only one electron can be placed in accordance with the Pauli Principle.
Each circle represents an electron. The figure shows the minimum energy configuration of an
arrangement of six electrons. (b) Corresponding probability distribution function, f (E).
bins being occupied is thus 1. All bins above the sixth bin are empty in the example
since no additional electrons are present. Hence the normalized probability of the bins
higher than six being occupied is zero. The resulting probability distribution function
is shown in Fig. 1.3(b). Note that the probability distribution shown in Fig. 1.3(b)
holds for T = 0 K. Clearly, the probability distribution function reflects the physical
situation, each of the first six states or bins is occupied, while those above six are
empty. Inspection of Fig. 1.3 shows that the distribution has the value of 1 until an
energy, Ef, is reached. This energy is called the Fermi level and is related to the number
of electrons present in the system. For the present example, the energy corresponding
to the Fermi level lies at the energy corresponding to the sixth bin.
What happens though at temperatures greater than absolute zero? Temperature is
a measure of the internal energy of the system. At temperatures greater than zero,
the total energy of the system must be greater than that corresponding to T = 0 K.
Let us again consider a system with only six electrons. For simplicity let us set the
energy of each bin to be an integer multiple of E. Thus for the system shown in
Fig. 1.3(a), the total energy is given as the sum of the occupied bins as E + 2E + 3E +
4E + 5E + 6E = 21E. The next highest energy configuration, or higher temperature
of the system is obtained by promoting the sixth electron into the seventh bin. The
corresponding energy of the resulting configuration is then equal to E + 2E + 3E +
4E + 5E + 7E = 22E, which is obviously higher than that of the T = 0 K configuration.
Higher temperature configurations are similarly achieved, i.e. by promoting electrons
from the lower energy states into higher energy states. An example system is shown

energy
energy
f(E)
T 0 K
T 0 K
(a)
(b)
Ef
1/2
1
Figure 1.4 (a) Distribution of a collection of six electrons in energy bins corresponding to a
collective energy or temperature greater than 0 K. (b) Corresponding probability distribution
function for a T 0 K distribution.
in Fig. 1.4 along with the corresponding probability distribution function. As can be
seen from the figure, at higher energies there exists a tail in the distribution function.
This implies that there is a nonzero probability of a state with energy greater than
Ef being occupied and correspondingly a nonzero probability that states below Ef are
unoccupied.
The equilibrium probability distribution function, f (E), called the Fermi–Dirac
distribution can be expressed mathematically. Its derivation is rather complicated and
will not be repeated here. The interested reader is referred to the books by Brennan
(1999, Chapter 5) or Kittel and Kroemer (1980). The Fermi–Dirac distribution is given
as
f (E) =
1
1 + e
(E−Ef)
kT
(1.8)
where k is Boltzmann’s constant.
It is instructive to examine how f (E) behaves and to show it replicates the distributions
shown in Figs. 1.3 and 1.4. Consider first its behavior at T = 0 K. There are two
conditions, E Ef and E Ef. For E Ef, the exponent in (1.8) is negative infinity
(due to the division by zero), and exp of negative infinity is zero. Thus f (E) for E
Ef, is 1/(1 + 0) or simply 1. This is of course exactly what is expected; for energies
less than the Fermi level, at T = 0 K, f (E) = 1. The second case, E Ef at T = 0 K
leads to the following. Notice that in this case, the exponent is now positive infinity,

and exp of positive infinity is infinity. Thus the denominator of (1.8) becomes infinity
and f (E) = 1/∞ = 0. Again this is consistent with Fig. 1.3; for energies greater than
Ef at T = 0 K, f (E) = 0. At energy E = Ef, the Fermi–Dirac function has value 1
/2 as
is readily seen from (1.8).
For temperatures greater than 0 K, f (E) is no longer a simple step function and
has a tail as shown in Fig. 1.4. If we consider the situation where E is large, such that
e(E−Ef)/kT
1, then f (E) can be approximated as
f (E) =
1
1 + e
(E−Ef)
kT
∼
1
e
(E−Ef)
kT
∼ e−
(E−Ef)
kT (1.9)
Under this condition, the Fermi–Dirac distribution behaves as the Maxwell–Boltzmann
distribution, and clearly the occupation probability of a state of energy E decreases
exponentially with increasing energy. This is as it should be since few electrons, if
any, will occupy very high energy states.
At this point, we can now determine the equilibrium electron concentration in a
semiconductor using (1.6) by substituting in for D(E) and f (E) the expressions given
by (1.7) and (1.8). The general expression, valid for all possibilities, involves using
the Fermi–Dirac distribution for f (E). However, this choice of f (E) necessitates that
the integral in (1.6) be solved numerically. Alternatively a closed form solution can be
obtained if the distribution function f (E) is approximated by the Maxwell–Boltzmann
distribution as
f (E) ∼ e−
(E−Ef)
kT (1.10)
Usage of the Maxwell–Boltzmann distribution for f (E) holds when the Fermi level
lies within the forbidden gap about 3kT below the conduction band edge or 3kT above
the valence band edge. When this condition is valid, the semiconductor is said to be
nondegenerate. If the Fermi level lies close to or within the conduction or valence bands
the material is said to be degenerate and the full Fermi–Dirac formulation for f (E)
should be used. A degenerate material is produced by heavily doping the semiconduc-
tor. The ranges in which a semiconductor is nondegenerate and degenerate are shown in
Fig. 1.5. Using the Maxwell–Boltzmann distribution the electron concentration within
the conduction band can be determined from
n =
∞

0
8πm
3
2
√
2E
h3
e−
(E−Ef)
kT dE (1.11)
In (1.11) the lower bound on the integral is set to zero since we assume that the
minimum energy is the conduction band edge. Of course, the upper limit on the
energy in a realistic energy band is not infinity but the upper bound can be extended
to infinity with little error since the probability distribution decreases exponentially
with increasing energy. Therefore, the error introduced by integrating n(E) to infinity
is exceedingly small. Equation (1.11) can be evaluated using
∞

0
√
xe−ax
dx =
√
π
2a
√
a
(1.12)

Ec
Ev
nondegenerate
degenerate if
Ef lies within
this range.
degenerate if
Ef lies within
this range.
Figure 1.5 Sketch of the energy bands of a semiconductor illustrating the conditions for
degenerate and nondegenerate doping. Notice that a degenerate material is highly doped such
that the Fermi level lies either near or above the conduction band in n-type material or near or
below the valence band in p-type material.
to be
n = 2

2πm∗
ekT
h2
3
2
e
Ef
kT (1.13)
where m∗
e is the electron effective mass. The mass of an electron within the semicon-
ductor is not the same as in free space. Instead, the electron behaves within the crystal
as if it has a different mass, called the reduced mass. The reduced mass arises from
the motion of the electrons in the periodic potential of the ions forming the crystalline
lattice. If we call the bottom of the conduction band Ec instead of zero, then (1.13)
becomes
n = 2

2πm∗
ekT
h2
3
2
e
−(Ec−Ef)
kT (1.14)
Deﬁning the effective density of states, Nc, as
Nc = 2

2πm∗
ekT
h2
3
2
(1.15)
the value of n can be written as
n = Nce−
(Ec−Ef)
kT (1.16)
A similar expression holds for the equilibrium hole concentration within the valence
band as
p = Nve−
(Ef−Ev)
kT (1.17)

(b)
(a)
increasing electron energy
increasing hole energy
k
k
Figure 1.6 Sketch of (a) the conduction band and (b) the valence band showing the direction
of increasing electron and hole energy.
where Nv is the same as (1.15) with the hole mass, m∗
h, used in place of the electron
mass, m∗
e, yielding
Nv = 2

2πm∗
hkT
h2
3
2
(1.18)
Equations (1.16) and (1.17) apply for the electron and hole concentrations for a semi-
conductor in equilibrium.
An intrinsic semiconductor has no intentionally added impurities. Conversely, an
extrinsic semiconductor has intentionally added impurities called dopants. In an intrin-
sic semiconductor promoting electrons from the valence band produces carriers within
the conduction band. The vacancies produced within the valence band from the pro-
motion of electrons are called holes. The two most salient features of holes necessary
for our discussion are that holes are positively charged and that hole energy increases
downwards in the energy band diagram as opposed to electron energy which increases
upwards as shown in Fig. 1.6. A hole is a vacancy within an otherwise filled band.
Thus when an electron is promoted from the valence band into the conduction band a
vacancy or hole is left behind in the valence band. As a result, the conduction band is
no longer empty and can now conduct a current. This is obvious since there is now an
electron within the band and it can move between different unoccupied energy states
under the action of an applied field. Similarly, the valence band is no longer com-
pletely filled and thus it too can conduct a current. Given that there are now vacancies
within the valence band, the electrons within the valence band can move from one

vacancy to the next. This movement of electrons within the nearly filled valence band
is equivalent to the movement of the corresponding number of holes within an empty
band, provided that the holes have opposite sign to that of the electrons. For example,
if there are N − 1 electrons within the valence band equivalently there is one hole.
The current carrying species within the conduction band is of course the free electron
that has been promoted from the valence band. The current carrying species within
the valence band is the hole.
What though is the current carried by the hole? If the band is completely filled, then
no current flows. The corresponding current density is given by summing over all of
the electron velocities as

j = −q
N

i=1

vi (1.19)
For a completely filled band, as mentioned above the net current is zero. This implies
that there are on average, an equal number of electrons crossing a Gaussian surface
moving to the left as are moving to the right. Therefore, the net flux of electrons
across the surface on average is zero. Consequently, the net average velocity of the
carriers must also be zero; for every electron motion there is another electron with
an opposite but equal motion. Thus the net velocity of the entire system vanishes as
well as the current density. The current density corresponding to the motions of the
electrons within a partially filled band can be related to the motion of the vacancies by
recognizing that

jfilled −
joccupied =
jvacancies (1.20)
But the current density due to the filled band is zero. Consequently, the current density
produced by the motion of the vacancies must be the exact negative of that produced
by the motion of the electrons. The current density due to the motion of the holes
(vacancies) is then

j = +q
N

i=1

vi (1.21)
A hole thus behaves like a positively charged particle. Hence, holes move in the
opposite direction from electrons under the action of an applied field.
Within an intrinsic material the electron concentration is equal to the hole concen-
tration, n = p. The intrinsic carrier concentration in equilibrium is called ni. The Fermi
level in intrinsic material is referred to as the intrinsic level, Ei. The position of the
intrinsic level can be determined as follows. Since n = p, the equilibrium electron and
hole concentrations in intrinsic material can be related as
Nce
(Ei−Ec)
kT = Nve
(Ev−Ei)
kT (1.22)

where Ei has been inserted in place of Ef in (1.16) and (1.17). Solving for Ei in (1.22)
obtains
Nv
Nc
= e
(Ei−Ec−Ev+Ei)
kT
Ei =
(Ec + Ev)
2
+
kT
2
ln

Nv
Nc

(1.23)
Substituting into (1.23) the relationships for Nv and Nc given by (1.15) and (1.18)
obtains
Ei =
(Ec + Ev)
2
+
3kT
4
ln

m∗
h
m∗
e

(1.24)
In some materials the effective masses of the electrons and holes are roughly equal.
In this case, the intrinsic level lies at midgap, halfway between the conduction and
valence bands. Even if the effective masses are substantially different it is a reasonable
assumption to set the intrinsic level equal to the midgap energy. For example, in GaAs
the electron and hole effective masses are 0.067 and 0.62 respectively. The last term
in (1.24) is equal to 0.043 eV. The midgap energy is 0.71, so we see that the correction
due to the difference in the effective masses, even when they are substantially different,
is small.
The intrinsic concentration, ni, can be obtained as follows. For an intrinsic semi-
conductor the electron and hole concentrations are equal:
n = p = ni (1.25)
As we discussed above, the Fermi level, Ef can be replaced by Ei the intrinsic level.
Using the above results, ni can be written as
ni = Nce
(Ei−Ec)
kT ; ni = Nve
(Ev−Ei)
kT (1.26)
Rearranging the terms in (1.26) ni can be expressed as
Nce− Ec
kT = nie−
Ei
kT (1.27)
The electron concentration, n, can now be expressed in terms of the intrinsic concen-
tration using (1.27). Starting with (1.16)
n = Nce− Ec
kT e
Ef
kT (1.28)
Substituting (1.27) into (1.28) obtains
n = nie
(Ef−Ei)
kT (1.29)
Similarly, the hole concentration can be written as
p = nie
(Ei−Ef)
kT (1.30)
Equations (1.29) and (1.30) hold for a nondegenerate semiconductor in equilibrium
in which approximating the Fermi–Dirac distribution by the Maxwell–Boltzmann
distribution is valid.

Consider the product of n and p for an intrinsic semiconductor. Using (1.29) and
(1.30) the np product is given as
np = n2
i e
(Ef−Ei)
kT e
(Ei−Ef)
kT (1.31)
which is simply
np = n2
i (1.32)
Equation (1.32) is called the Law of Mass Action. The Law of Mass Action applies
only in equilibrium but it is true for any semiconductor either intrinsic or extrinsic.
The np product can be written in an alternative manner using (1.16) and (1.17) as
n = Nce− Ec
kT e
Ef
kT p = Nve
Ev
kT e−
Ef
kT (1.33)
Taking the product of n and p given by (1.33) obtains,
np = Nc Nve− (Ec−Ev)
kT = Nc Nve−
(Eg)
kT = n2
i (1.34)
where Eg = Ec − Ev, the energy difference between the top of the valence band and the
bottom of the conduction band, the “gap energy” of the semiconductor. Substituting
in for Nc and Nv we ﬁnally obtain for ni
ni =
2
h3
(2πkT )
3
2 (m∗
em∗
h)
3
4 e−
Eg
2kT (1.35)
which is a constant.
1.3 Extrinsic material
A semiconductor into which impurities, called dopants, are intentionally added in order
to alter the conductivity of the sample is said to be extrinsic. There are two dopant
types. These are n-type dopants called donors and p-type dopants called acceptors. In
a semiconductor doped with donors the equilibrium electron concentration becomes
larger than the equilibrium hole concentration and the semiconductor is said to be
n-type. Similarly, if the semiconductor is doped with acceptors, the equilibrium hole
concentration is greater than the equilibrium electron concentration and the semicon-
ductor is said to be p-type.
An example donor atom in silicon is phosphorus. Phosphorus is a Column VA
element while silicon is a Column IVA element. Therefore, phosphorus has an outer
valence of ﬁve while silicon has an outer valence of four. Silicon crystallizes such that
each silicon atom forms bonds with four other silicon atoms fully accommodating all
four outer valence electrons. If a phosphorus atom is substituted for a silicon atom
in an otherwise silicon lattice, then one of the outer valence electrons within the
phosphorus atom is not bound to a neighboring silicon atom as shown in Fig. 1.7(a).
The phosphorus atom then only weakly holds the unbound electron. The other four
valence electrons in phosphorus are chemically bound to four neighboring silicon

(a)
(b)
Figure 1.7 (a) A two-dimensional representation of a donor atom, phosphorus, shown in
black, within a silicon matrix. Each silicon atom is represented by an open circle. Each line
represents an outer valence electron. Notice that the extra electron in the outer shell of the
phosphorus atom is unbound. (b) A two-dimensional representation of an acceptor atom,
boron, shown in gray, within a silicon matrix. Each silicon atom is again represented by an
open circle. Notice that one of the bonds is not ﬁlled yielding a hole.
atoms. The unbound electron can be readily ionized since it is not chemically bound.
Once ionized, the unbound electron can move freely through the crystal and thus lies
within the conduction band.
p-type doping can be achieved by adding an atom with fewer electrons in the
outer shell than silicon. An example of a p-type dopant is aluminum. Aluminum is a
Column IIIA element and as such has only three electrons in its outermost atomic
orbital. If an atom of aluminum substitutes for an atom of silicon within the silicon
matrix, one of the four bonds to the nearest neighbor atoms is unﬁlled as shown in
Fig. 1.7(b). The vacant bond is called a vacancy or a hole. The vacancy can propa-
gate through the lattice as a result of electrons jumping from one occupied state into
another, each time leaving a vacancy behind.
It is useful to picture the donor and acceptor states in an energy level diagram.
The key to understanding the energy level diagrams for donors and acceptors is to
recognize that the donor and acceptor states, being impurity states, lie somewhere
between the conduction and valence bands of the host semiconductor material. Donor
and acceptor atoms are special types of impurities in that they introduce levels near
the conduction and valence band edges respectively as shown in Fig. 1.8. Deep levels
formed by impurity atoms added to the host semiconductor cannot easily be ionized.
As a result these levels act as traps. The energy levels in this case lie near midgap and
thus require extensive energy in order to be ionized.

(a)
Ec
Ev
donor levels
(b)
Ec
Ev
acceptor levels
Figure 1.8 Sketch of the energy band diagrams of a semiconductor doped with (a) donors and
(b) acceptors. Notice that the donor and acceptor atoms lie near the conduction and valence
band edges respectively. Therefore, only a small amount of energy is needed to ionize either
dopant.
In Section 1.2 we found that the electron and hole concentrations can be written in
terms of the intrinsic level and Fermi level as
n = nie
(Ef−Ei)
kT p = nie
(Ei−Ef)
kT (1.36)
The position of the Fermi level with respect to the intrinsic level determines the doping
type in the semiconductor. Notice that the product of n and p once again yields the
Law of Mass Action.
To decide if a semiconductor is n- or p-type it is necessary to compare the electron
and hole concentrations with the intrinsic concentration. To do this the charge neutrality
condition must be invoked. The charge neutrality condition is given as
0 = p − n + N+
d − N−
a (1.37)
Equation (1.37) implies that the net charge within the semiconductor is zero. The net
positive charge contributed by the holes and ionized donors is balanced by the net
negative charge contributed by the electrons and ionized acceptors. In most situations,
only one ionized dopant atom dominates, either the donors or acceptors. Consider
ﬁrst an intrinsic semiconductor. In an intrinsic semiconductor the donor and acceptor
concentrations are zero. Thus (1.37) becomes
0 = p − n (1.38)
which of course is simply p = n. Alternatively for extrinsic material if Nd Na then the
material is n-type. Under this assumption, the acceptor concentration can be neglected
with respect to the donor concentration. The electron concentration can be determined
as follows. The charge neutrality condition becomes
0 = p − n + N+
d (1.39)

The hole concentration can be expressed in terms of n using the Law of Mass Action
(equilibrium conditions) as
p =
n2
i
n
(1.40)
Substituting (1.40) into (1.39) obtains
n2
i
n
− n + Nd = 0 (1.41)
Equation (1.41) is simply a quadratic equation in n given as
n2
− Ndn − n2
i = 0 (1.42)
It can be solved as
n =
Nd ±

N2
d + 4n2
i
2
(1.43)
Notice that if ni Nd then (1.43) simpliﬁes to
n = Nd (1.44)
Similarly, if Na Nd and ni Na then we obtain
p = Na (1.45)
and of course the material is p-type.
If the condition given by (1.44) holds then the Fermi level can be calculated as
follows. The electron concentration in n-type material is given as
n = Nd = nie
(Ef−Ei)
kT (1.46)
Solving for Ef obtains
Ef = Ei + kT ln

Nd
ni

(1.47)
for n-type material. Notice that the Fermi level in n-type material is greater than the
intrinsic level. Similarly, for p-type material the Fermi level is given as
Ef = Ei − kT ln

Na
ni

(1.48)
In this case the Fermi level lies below the intrinsic level. Thus for n-type material the
Fermi level lies above the intrinsic level and in p-type material the Fermi level lies
below the intrinsic level as shown in Fig. 1.9.
Example Problem 1.1
Consider a silicon sample doped with donors at 1 × 1017
cm−3
. If the intrinsic con-
centration within silicon is 1.0 × 1010
cm−3
, determine the location of the Fermi level
relative to the valence band.

Ec
Ei
Ev
Ef
(a)
(b)
Ec
Ei
Ev
Ef
Figure 1.9 Sketch of the band structure showing the intrinsic level. In (a) the material is
n-type since the Fermi level lies above the intrinsic level. In (b) the material is p-type since the
Fermi level lies below the intrinsic level.
Since the donor doping concentration is very much larger than the intrinsic concen-
tration, using the approximation that the electron concentration is equal to the donor
concentration is valid. Therefore, the Fermi level relative to the intrinsic level is given
as
Ef = Ei + kT ln

Nd
ni

Substituting in for each term, kT = 0.0259 eV at 300 K, the Fermi level becomes
Ef = Ei + 0.417 eV
By evaluating Ei, the position of the intrinsic level relative to the valence band can be
determined. Ei is given in the text as
Ei =
(Ec + Ev)
2
+
3kT
4
ln

m∗
h
m∗
e

The effective masses in silicon for holes and electrons are
m∗
e = 0.328 m∗
h = 0.55
With these values for the effective masses the intrinsic level relative to the valence
band edge is
Ei =
1.12
2
eV + 0.01 eV = 0.57 eV

Problems 21
Ec
Ev
X = 0
X = L
Ef
X = L/2
Figure 1.10 Energy band diagram as a function of position. L is the total length of the sample.
Ef corresponds to the flat dashed line in the figure.
Thus the Fermi level is located at 0.57 eV + 0.417 eV or 0.987 eV above the valence
band edge.
Problems
1.1 Show that the probability that a state E above the Fermi level, Ef, is filled is
equal to the probability that a state E below the Fermi level, Ef, is empty.
1.2 Determine the position of the Fermi level relative to the intrinsic level in silicon at
300 K if the electron concentration is 1 × 1016
cm−3
. Use ni = 1.0 × 1010
cm−3
.
1.3 (a) Show that the minimum conductivity of a semiconductor sample occurs when
n0 = ni

µp
µn
(b) What is the expression for the minimum conductivity, σmin?
(c) Calculate σmin for Si at 300 K and compare with the intrinsic conductivity.
µn = 1350 cm2
/(V s) µp = 480 cm2
/(V s) ni = 1.0 × 1010
cm−3
.
1.4 What are the equilibrium concentrations of electrons and holes at T = 300 K in:
(a) Si doped with Nd = 3 × 1014
cm−3
;
(b) Ge doped with Na = 3 × 1014
cm−3
;
ni(Si) = 1.0 × 1010
cm−3
ni(Ge) = 2.4 × 1013
cm−3
1.5 An intrinsic semiconductor sample has a resistance of 5 at 360 K and a resistance
of 50 at 330 K. Assume that the only factor that changes the resistance between
the two cases is the change in the intrinsic carrier concentration. Determine an
expression relating the energy gaps at 360 K and 330 K.
1.6 A silicon sample has a length of 1 cm, a height of 0.01 cm and a width of
0.1 cm. The temperature is assumed to be 300 K. The electron mobility is given as

1450 cm2
/(V s) and the hole mobility is given as 500 cm2
/(V s). Determine the
resistance of the sample if the doping concentration is given as
(a) intrinsic
(b) donor doping of 1016
cm−3
(c) acceptor doping of 1015
cm−3
1.7 A semiconductor is characterized by the energy band diagram sketched in
Fig. 1.10. The system is doped such that there is a band bending as shown in the
ﬁgure. Given Eg = 1.12 eV, ni = 1010
cm−3
, and kT = 0.0259 eV determine the
following.
(a) Determine n and p at X = L/2
(b) Determine n at X = L/4
(c) If L = 1 cm, what is the magnitude of the electric ﬁeld in the semiconductor?
(d) In the region, L/2 X L is the material n- or p-type? Explain why.
1.8 Consider a semiconductor sample doped n-type with 5.0 × 1012
cm−3
donors. If
the intrinsic concentration of the semiconductor is 1013
cm−3
what is the electron
concentration in the semiconductor? Neglect the acceptor concentration.
1.9 Consider a collection of electrons at a temperature of 300 K. If the Fermi energy
is equal to 1.2 eV, determine the probability that a state is NOT occupied at an
energy of 1.25 eV. Assume kT is 0.0259 eV.
1.10 Determine the energy band gap of a semiconductor at 300 K if the electron and
hole masses are 0.067 and 0.50 times the free electron mass respectively. The
intrinsic carrier concentration of the semiconductor is 1.5 × 107
cm−3
.

2
Carrier action
In this chapter we examine the dynamics of carriers in semiconductors. We consider
three general types of dynamics: drift, diffusion, and generation–recombination. In
the first section, we discuss both drift and diffusion, which govern electron transport
dynamics in semiconductors. The next section is devoted to the study of generation–
recombination mechanisms active in semiconductors. Finally, we conclude with a
discussion of the carrier continuity equation and its solution.
2.1 Drift and diffusion
The two major mechanisms that govern current flow in a semiconductor are drift and
diffusion. Drift is charged particle motion in response to an applied electric field. The
carrier drifts under the action of the applied electric field E as
F = qE (2.1)
where F is the force acting on the electron.†
For an electron q is negative while for a
hole q is positive. Notice that for an electron the force acts in the opposite direction
from the field. For a hole the force and field point in the same direction. The work
done on the carrier from a constant electric field is given as
E = qEx (2.2)
where E is the change in energy of the carrier. The carrier cannot be accelerated
continuously; otherwise its energy would “run away” and approach infinity, which is
of course not observed. The electron suffers an occasional scattering with the lattice
leading to energy transfer from the electron to the lattice. In this way, the electron
energy is reduced. Ultimately a balance is achieved between the gain of energy from
the field and the loss of energy via lattice collisions. Once this balance is achieved, the
electron has no net energy gain; the time rate of change of the energy of the carrier is
zero. This condition is steady-state. Once steady-state is reached the system no longer
changes with time. Under steady-state conditions the energy gained from the field must
be equal to the energy lost to the lattice through scatterings which can be expressed as
E(gain from the field) = E(lost to the lattice) (2.3)
† In this chapter, although electric field and current density are vector quantities, we consider only a one-
dimentional approach.

24 Carrier action
The electron motion under the application of an applied field is directed. If the
field is shut off the carriers relax through lattice scattering back to their equilibrium
distribution. As a result there is no net current flow in any direction.
The drift current density can be expressed as
J = qnvd (2.4)
where vd is the drift velocity. The current density is simply I/A, where I is the current
and A the area. Thus the electron drift current can be written as
I = −qnvd A (2.5)
while the hole drift current is given as
I = qnvd A (2.6)
There exists a linear relationship between the drift velocity and the field. This rela-
tionship is
vd = µE (2.7)
where µ is the mobility. The mobility is a measure of how readily a carrier can move
through the crystal. The drift velocity is then
vd = −µmE (2.8)
for electrons and
vd = µpE (2.9)
for holes. Therefore, the hole and electron drift current densities become
Jp(drift) = qµp pE Jn(drift) = qµnnE (2.10)
The electrical conductivity in general is equal to the ratio of the current density to the
electric field. Thus the conductivity for n-type material is
σn = qµnn (2.11)
The corresponding electrical conductivity for p-type material is
σp = qµp p (2.12)
The mobility itself can be expressed (for electrons) as
µn =
qτ
m∗
e
(2.13)
where τ is the mean free time between collisions. Equation (2.13) shows how the
mobility varies with both the effective mass and mean time between scatterings, τ.
Notice that as the mean time between scatterings increases, implying that the scattering
rate (which is inversely proportional to τ) decreases, the mobility increases. This is
as expected since if the scattering rate is decreased then there are fewer impeding
collisions. As a result, the carrier more easily moves through the crystal and hence
has a higher mobility. The mobility is also a function of the effective mass. As the

effective mass increases, the mobility decreases. An increased effective mass produces
a higher inertia for the carrier and thus it is less easily moved. At high doping levels
the mobility is reduced due to the enhanced scattering rate caused by ionized impurity
scattering. The mobility varies also with temperature. For a low doped material, the
mobility decreases with increasing temperature. This is because the lattice scattering
increases with increasing temperature reducing the mobility (the mean time between
collisions is lowered).
When both electrons and holes are present in the material the total current density
due to drift, Jtotal(drift), is equal to the sum of the electron and hole drift current
densities:
Jtotal(drift) = Jn + Jp (2.14)
Substituting in the expressions for Jn and Jp given by (2.10) Jtotal(drift) becomes
Jtotal(drift) = qnµnε + qpµpε (2.15)
which simplifies to
Jtotal(drift) = q(nµn + pµp)ε (2.16)
The electrical conductivity is defined as the ratio of the current density to the field so
when both carriers are present the electrical conductivity is given as
σ =
Jtotal(drift)
E
= q(nµn + pµp) (2.17)
Let us next consider the form of the conductivity for three different cases. Case 1 is
intrinsic material. In this situation we have
n = p = ni
σ = qni(µn + µp) (2.18)
ρ =
1
σ
=
1
qni(µn + µp)
where ρ is the resistivity. Case 2 is for n-type material. In this case in equilibrium,
n = Nd p =
n2
i
Nd
p n (2.19)
Under these conditions the electrical conductivity and resistivity are given as
σ = qNdµn
ρ =
1
σ
=
1
qNdµn
(2.20)
Finally, in the third case the material is p-type. In this case in equilibrium the electron
and hole concentrations are given as
p = Na n =
n2
i
Na
n p (2.21)

26 Carrier action
Ec
Ev
Ei
positive potential
negative potential
e
Figure 2.1 Energy bands under the application of an applied field.
The resulting conductivity and resistivity are
σ = qNaµp
ρ =
1
σ
=
1
qNaµp
(2.22)
Consider next band bending and its effect on electron and hole motion. Energy
band diagrams depict electron energy not potential. Recall that the electron energy is
lowered near a positive potential and raised near a negative potential. The band bending
is shown in Fig. 2.1. Notice that the potential energy is positive at the lowest point
and negative for the highest potential. An electron will be attracted by the positive
potential and repelled by the negative potential. Thus an electron will “roll downhill”
in the diagram towards the positive potential. This is a general result. Electrons “roll
downhill” in potential energy diagrams. Let us examine the electric field. The electric
field can be found from the potential as the negative gradient of the potential. This is
given as
E = −
dV
dx
(2.23)
But V (x) is given as Ei/(−q). With this definition the electric field becomes
E = −
dV
dx
= −
d
dx

Ei
−q

=
1
q
dEi
dx
(2.24)
Aside from drift motion a carrier can move via diffusion. Carriers will diffuse from
regions of high concentration to regions of low concentration until the concentration
gradient is zero. A vivid example of diffusion is an opened perfume bottle in a closed

room. Initially all of the perfume is in the bottle and no vapor has spread. However,
after some time the perfume molecules will diffuse throughout the room until the
concentration gradient vanishes. Under these conditions the concentration of perfume
molecules is everywhere the same. Diffusion is governed by Fick’s Law. The particle
density current is proportional to the concentration gradient:
J
= −D
dn
dx
(2.25)
where D is the diffusion constant. Notice that the particle density current flows from
the region of high concentration to the region of low concentration which produces
the negative sign. For electrons the diffusion current is given as
Jn(diff) = (−q)

−Dn
dn
dx

= q Dn
dn
dx
(2.26)
Similarly, for holes the diffusion current is given as
Jp(diff) = (q)

−Dp
dp
dx

= −q Dp
dp
dx
(2.27)
Using (2.26) and (2.27) the total current densities can be determined. The total current
densities may be expressed as the sum of the drift and diffusion current densities:

Jp = qµp p
ε − q Dp

∇ p (2.28)
for holes and

Jn = qµnn
ε + q Dn

∇n (2.29)
for electrons.
In equilibrium no net current flows by drift and diffusion. The current due to the
drift of carriers in the applied electric field must exactly balance on average the current
due to diffusion. When the current vanishes the gradient of the Fermi level is zero
dEf
dx
= 0 (2.30)
Hence in equilibrium, the Fermi level is flat. The electron carrier concentration in
equilibrium can be written as
n = nie
(Ef−Ei)
kT (2.31)
The derivative of n with respect to x is
dn
dx
= nie
(Ef−Ei)
kT
1
kT

dEf
dx
−
dEi
dx

(2.32)
But dEf/dx is zero. With this substitution dn/dx becomes
dn
dx
= −
1
kT
n
dEi
dx
(2.33)

28 Carrier action
but
E =
1
q
dEi
dx
(2.34)
Combining (2.33) and (2.34) obtains
dn
dx
= −
q
kT
nE (2.35)
In equilibrium the electron current density vanishes. Thus,
J = 0 = qµnnE + q Dn
dn
dx
(2.36)
Substituting (2.35) into (2.36) yields
qµnnE + q Dn

−
q
kT
nE

= 0 (2.37)
Simplifying, (2.37) becomes
µn =
q
kT
Dn (2.38)
which is known as the Einstein relation. The Einstein relation relates the diffusion
constanttothemobilitybutitholdsrigorouslyonlyinequilibrium.Asimilarexpression
exists for holes.
2.2 Generation–recombination
Generation and recombination events change the electron and hole concentrations. A
generation event creates free electrons and holes while a recombination event removes
free electrons and holes. There are two general types of generation–recombination
events. These are band-to-band and band-to-bound transitions. Band-to-band transi-
tions are between the valence and conduction bands. A band-to-band generation event
occurs when an electron within the valence band is promoted into the conduction band
as shown in Fig. 2.2(a). A band-to-band recombination event occurs when an electron
within the conduction band recombines with a vacancy in the valence band as shown
in Fig. 2.2(b).
Conversely, a band-to-bound event occurs between a band state, either the conduc-
tion or valence band, and an impurity state located within the band gap. Examples of
band-to-bound events are shown in Fig. 2.3. In Fig. 2.3(a) an electron is generated from
an impurity state located within the energy gap. As can be seen from the ﬁgure, the
impurity state lies near midgap. Similarly, an electron can be captured by an impurity
state and thus be removed from the conduction band as shown in Fig. 2.3(b). This is a
band-to-bound recombination event.
A net electron-hole pair can be either generated or recombined through band-to-
bound transitions. First the electron is trapped by the bound impurity state which lies
near midgap. Then after some time the electron can be emitted from the impurity state
and recombine with a vacancy in the valence band. As a result an electron–hole pair
has recombined. Alternatively, an electron–hole pair can be generated by the action of

(a)
Ec
Ev
(b)
Ec
Ev
Figure 2.2 Band-to-band generation–recombination processes: (a) band-to-band generation
event; (b) band-to-band recombination event. The solid circle represents the ﬁnal state of the
electron.
(a)
Ec
Ev
impurity states
Ec
Ev
impurity states
(b)
Figure 2.3 Band-to-bound generation–recombination events. In (a) an electron is generated
from an impurity state within the energy gap. In (b) an electron is captured by a impurity state
leading to an electron recombination event. The solid circle represents the ﬁnal state of the
electron in the process.

30 Carrier action
the impurity state. An electron from the valence band jumps up to a trap state leaving
behind a hole within the valence band. After some time, the electron can be emitted
from the trap state into the conduction band creating an electron–hole pair.
There are three different mechanisms by which either a band-to-band or band-to-
bound generation–recombination event can proceed. These are: thermal, radiative, and
Auger.Athermalgeneration–recombinationeventoccursbytheemissionorabsorption
ofthequantumoflatticevibrations,aphonon.Phononsaresimilartophotons.Aphoton
is the quantum of an electromagnetic vibration (light). A phonon is the quantum of a
lattice vibration. In a phonon emission event a phonon is given up by the electron to
the lattice. Thus the electron energy is lowered by the energy of a phonon. In a phonon
absorption event a phonon is absorbed by the electron thus increasing the electron’s
energy by the amount given by the phonon.
A radiative generation–recombination event occurs with either the absorption or the
emission of a photon. A photon absorption event leads to generation while a photon
emission event leads to recombination. In order for a band-to-band radiative generation
event to occur, the incident photon must have energy equal to or greater than the energy
band gap. Conversely, a band-to-band radiative recombination event emits a photon
with an energy equal to or greater then the energy band gap.
Finally, the last mechanism is Auger generation or recombination. An Auger
generation–recombination event occurs via energy transfer between two carriers, either
two electrons or two holes. A band-to-band Auger generation event occurs when a
high energy carrier, either an electron or a hole, makes a collision with the lattice and
transfers its excess kinetic energy to the lattice to produce an electron–hole pair. A
band-to-band Auger generation event is a threshold process since the initial carrier
must have kinetic energy equal to or greater than the energy gap in order to produce
an electron–hole pair. Band-to-band Auger generation is often referred to as impact
ionization. Band-to-band Auger recombination occurs when an electron–hole pair
recombines and transfers its excess kinetic energy to either a free electron or hole.
The free carrier is then promoted to a high energy within either the conduction band
(electron) or the valence band (hole).
Let us next consider generation–recombination quantitatively. Deﬁne n0 and p0 as
the equilibrium electron and hole concentrations respectively; n and p are the general
carrier concentrations within the material. The excess concentrations are given as
δn = n − n0 δp = p − p0 (2.39)
The time rate of change of the carrier concentration due to generation–recombination
is written as
∂n
∂t

R−G
∂p
∂t

R−G
(2.40)
Radiative transitions depend to some extent upon the energy band structure of
the semiconductor. The simplest energy relationship for a semiconductor assumes
completely free carriers. Under this condition the energy is completely kinetic and is

Γ Γ
X U L K
energy
(eV)
10
8
6
4
2
0
−2
−4
−6
−8
Wavevector k
Figure 2.4 Electronic energy band structure of bulk Si. Notice that the conduction band
minimum occurs at the X point, the (100) point, in k-space. Since the maximum hole energy
occurs at the point, the material is indirect.
given by (1.4) as
E =
h̄2
k2
2m
(2.41)
The energy is a quadratic function of the wavevector, k; E(k) is parabolic. Under this
condition, the energy bands assume their simplest form. Generally, the energy band
structure of a realistic semiconductor departs from this simple relationship. The band
structures of Si and GaAs are shown in Figs. 2.4 and 2.5 respectively. Notice that in
both diagrams there exist multiple bands for the valence bands. These are the heavy
and light hole bands. Only a single conduction band is shown for both materials in
the diagram. Hole energy increases downwards in the diagram while electron energy
increases upwards. At k = (000), the valence bands reach their minimum energy value
in both materials. However, the minimum electron energy occurs at different points in
k-space for GaAs and Si. As can be seen from the ﬁgures, the minimum electron energy
within the conduction band occurs at k = (000) for GaAs but at k = (100) for Si. Since
the minimum electron and hole energies occur at the same location in k-space in GaAs,
it is called a direct gap semiconductor. Alternatively, since the minimum electron and
hole energies occur at different points in k-space in Si the material is called an indirect
semiconductor. A direct gap semiconductor like GaAs is a far more efﬁcient absorber
and emitter of radiation than an indirect gap semiconductor like Si. As we will see in
Chapter 10, most optoelectronic devices are made using direct gap semiconductors.
For a constant radiative generation rate, GL, the time rate of change of the carrier
distributions is given as
∂n
∂t

R−G
=
∂p
∂t

R−G
= GL (2.42)

32 Carrier action
4
3
2
1
0
−1
−2
−3
−4
L X
Wavevector k
K
Γ Γ
energy
(eV)
Figure 2.5 Electronic energy band structure of bulk GaAs. Notice that the conduction and
valence band minimums occur at the same point in k-space, the point. For this reason the
material is said to be direct.
Radiative absorption requires that the incident photon have energy equal to or greater
than the energy band gap. Let hν Eg and I(x) be the intensity of light incident onto
the semiconductor sample. The rate of change of the intensity is proportional to the
intensity:
dI(x)
dx
= −αI(x) (2.43)
which can be readily solved for I(x) as
I(x) = I0e−αx
(2.44)
where α is called the absorption coefﬁcient. The intensity of the light a distance l into
the sample is given then as
I(l) = I0e−αl
(2.45)
If an external agent, such as illumination, acts on a semiconductor to generate
carriers the semiconductor is driven out of equilibrium. As a result, the excess carrier
concentration is increased above the equilibrium concentration. To relax back to
equilibrium once the illumination is removed, there must be a net recombination rate

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The various preparations are met with as fine thin threads
(vermicelli), thin sticks and pipes (spaghetti, macaroni), small
lozenges, stars, disks, ellipses, c. (pastes). These various forms are
prepared in a uniform manner from a granular product of hard
wheat, which, under the name of semolina or middlings, is a
commercial article. The semolina is thoroughly mixed with boiling
water and incorporated in a kneading machine, such as is used in
bakeries, into a stiff paste or dough. It is then further kneaded by
passing frequently between rollers or under edge runners, till a
homogeneous mass has been produced which is placed in a strong
steam-jacketed cylinder, the lower end of which is closed with a thick
disk pierced with openings corresponding with the diameter or

section of the article to be made. Into this cylinder an accurately
fitting plunger or piston is introduced and subjected to very great
pressure, which causes the stiff dough to squeeze out through the
openings in the disk in continuous threads, sticks or pipes, as the
case may be. Vermicelli is cut off in short bundles and laid on trays
to dry, while macaroni is dried by hanging it in longer lengths over
wooden rods in stoves or heated apartments through which currents
of air are driven. It is only genuine macaroni, rich in gluten, which
can be dried in this manner; spurious fabrications will not bear their
own weight, and must, therefore, be laid out flat to be dried. In
making pastes the cylinder is closed with a disk pierced with holes
having the sectional form of the pastes, and a set of knives revolving
close against the external surface of the disk cut off the paste in thin
sections as it exudes from each opening. True macaroni can be
distinguished by observing the flattened mark of the rod over which
it has been dried within the bend of the tubes; it has a soft yellowish
colour, is rough in texture, elastic and hard, and breaks with a
smooth glassy fracture. In boiling it swells up to double its original
size without becoming pasty or adhesive. It can be kept any length
of time without alteration or deterioration; and it is on that account,
in many circumstances, a most convenient as well as a highly
nutritious and healthful article of food.

MACARONICS, a species of burlesque poetry, in which words
from a modern vernacular, with Latin endings, are introduced into
Latin verse, so as to produce a ridiculous effect. Sometimes Greek is
used instead of Latin. Tisi degli Odassi issued a Carmen
macaronicum de Patavinis in 1490. The real founder of the practice,
however, was Teofilo Folengo (1491-1544), whose mock-heroic Liber
Macaronices appeared in 1517. Folengo (q.v.) was a Benedictine
monk, who escaped from his monastery and wandered through Italy,
living a dissolute life, and supporting himself by his absurd verses,
which he described as an attempt to produce in literature something
like macaroni, a gross, rude and rustic mixture of flour, cheese and
butter. He wrote under the pseudonym of Merlinus Coccaius, and his
poem is an elaborate burlesque epic, in twenty-five books, or
macaronea; it is an extraordinary medley of chivalrous feats,
ridiculous and squalid adventures, and satirical allegory. Its effect
upon the mind of Rabelais was so extraordinary that no examination
of Pantagruel can be complete without a reference to it (cf.
Gargantua, i. 19). It was immediately imitated in Italy by a number
of minor poets; and in France a writer whose real name was Antoine
de la Sablé, but who called himself Antonius de Arena (d. 1544),
published at Avignon in 1573 a Meygra entrepriza, which was a
burlesque account of Charles V.’s disastrous campaign in Provence.
Folengo in Italy and Arena in France are considered as the
macaronic classics. In the 17th century, Joannes Caecilius Frey
(1580-1631) published a Recitus veritabilis, on a skirmish between
the vine-growers of Rueil and the bowmen of Paris. Great popularity
was achieved later still by an anonymous macaronic, entitled
Funestissimus trepassus Micheli Morini, who died by falling off the
branch of an elm-tree:—

De branche in brancham degringolat, et faciens pouf
Ex ormo cadit, et clunes obvertit Olympo.
Molière employed macaronic verse in the ceremonial scene with the
doctors in Le Malade imaginaire. Works in macaronic prose are rarer.
An Anti-Clopinus by Antony Hotman may be mentioned and the
amusing Epistolae obscurorum virorum (1515). Macaronic prose was
not unknown as an artifice of serious oratory, and abounds (e.g.) in
the sermons of Michel Menot (1440-1518), who says of the prodigal
son, Emit sibi pulcheras caligas d’écarlate, bien tirées.
The use of true macaronics has never been frequent in Great
Britain, where the only prominent example of it is the Polemo-
Middinia ascribed to William Drummond of Hawthornden. This short
epic was probably composed early in the 17th century, but was not
published until 1684. The Polemo-Middinia follows the example set
by Arena, and describes with burlesque solemnity a quarrel between
two villages on the Firth of Forth. Drummond shows great ingenuity
in the tacking on of Latin terminations to his Lowland Scots
vernacular:—
Lifeguardamque sibi saevas vocat improba lassas,
Maggaeam, magis doctam milkare cowaeas,
Et doctam sweepare flooras, et sternere beddas,
Quaeque novit spinnare, et longas ducere threedas.
There is a certain macaronic character about many poems of
Skelton and Dunbar, as well as the famous Barnabae itinerarium
(1638) of Richard Brathwait (1588-1673), but these cannot be
considered legitimate specimens of the type as laid down by
Folengo.

See Ch. Nodier, Du Langage factice appelé macaronique
(1834); Genthe, Histoire de la poésie macaronique (1831).
(E. G.)
MACARSCA (Serbo-Croatian, Makarska), the chief town of an
administrative district in Dalmatia, Austria; situated opposite to the
island of Brazza, about 32 m. S.E. of Spalato. Pop. (1900), of town
1805; of commune, 11,016, chiefly Serbo-Croatian. Macarsca is a
port of call for the Austrian Lloyd steamers, and has a brisk trade in
wine, grain and fruit. Under the name of Mocrum, Macarsca was a
thriving Roman city, and a bishopric until 639, when it was destroyed
by the Avars. In the 10th century it is mentioned by Constantine
Porphyrogenitus as a city of the pagan Narentines. Its bishopric was
revived in 1320, but the bishops resided at Almissa. In 1481 the city
was purchased from the duke of Herzegovina by Venice; in 1499 it
was conquered by the Turks; and in 1646, after a successful revolt,
it again welcomed the sovereignty of Venice. The see of Macarsca
was merged in that of Spalato in 1830.

MACARTNEY, GEORGE MACARTNEY, Earl (1737-
1806), was descended from an old Scottish family, the Macartneys of
Auchinleck, who had settled in 1649 at Lissanoure, Antrim, Ireland,
where he was born on the 14th of May 1737. After graduating at
Trinity College, Dublin, in 1759, he became a student of the Temple,
London. Through Stephen Fox, elder brother of C. J. Fox, he was
taken up by Lord Holland. Appointed envoy extraordinary to Russia
in 1764, he succeeded in negotiating an alliance between England
and that country. After occupying a seat in the English parliament,
he was in 1769 returned for Antrim in the Irish parliament, in order
to discharge the duties of chief secretary for Ireland. On resigning
this office he was knighted. In 1775 he became governor of the
Caribbee Islands (being created an Irish baron in 1776), and in 1780
governor of Madras, but he declined the governor-generalship of
India, and returned to England in 1786. After being created Earl
Macartney in the Irish peerage (1792), he was appointed the first
envoy of Britain to China. On his return from a confidential mission
to Italy (1795) he was raised to the English peerage as a baron in
1796, and in the end of the same year was appointed governor of
the newly acquired territory of the Cape of Good Hope, where he
remained till ill health compelled him to resign in November 1798.
He died at Chiswick, Middlesex, on the 31st of May 1806, the title
becoming extinct, and his property, after the death of his widow
(daughter of the 3rd earl of Bute), going to his niece, whose son
took the name.
An account of Macartney’s embassy to China, by Sir George
Staunton, was published in 1797, and has been frequently
reprinted. The Life and Writings of Lord Macartney, by Sir John
Barrow, appeared in 1807. See Mrs Helen Macartney Robbins’s

biography, The First English Ambassador to China (1908), based
on previously unpublished materials in possession of the family.
MACASSAR (Makassar, Mangkasar), the capital of a district of
the same name in the island of Celebes, Dutch East Indies, and the
chief town of the Dutch government of Celebes. Pop. 17,925 (940
Europeans, 2618 Chinese, 168 Arabs). It stands on the west coast of
the southern peninsula of the island, near the southern extremity of
the Macassar Strait, which separates Celebes from Borneo. Macassar
consists of the Dutch town and port, known as Vlaardingen, and the
Malay town which lies inland. Macassar’s trade amounts to about
£1,250,000 annually, and consists mainly of coffee, trepang, copra,
gums, spices and valuable timber.
For the Macassar people and for the Strait, see Celebes.
“Macassar oil” is a trade name, not geographical: see
Antimacassar.

MACAULAY, THOMAS BABINGTON MACAULAY,
Baron (1800-1859), English historian, essayist and politician, was
born at Rothley Temple, Leicestershire, on the 25th of October 1800.
His father, Zachary Macaulay (1768-1838), had been governor of
Sierra Leone, and was in 1800 secretary to the chartered company
which had founded that colony; an ardent philanthropist, he did
much to secure the abolition of the slave trade, and he edited the
abolitionist organ, the Christian Observer, for many years. Happy in
his home, the son at a very early age gave proof of a determined
bent towards literature. Before he was eight years of age he had
written a Compendium of Universal History, which gave a tolerably
connected view of the leading events from the creation to 1800, and
a romance in the style of Scott, in three cantos, called The Battle of
Cheviot. A little later he composed a long poem on the history of
Olaus Magnus, and a vast pile of blank verse entitled Fingal, a Poem
in Twelve Books. After being at a private school, in October 1818
young Macaulay went to Trinity College, Cambridge, where he
afterwards became a fellow. He gained in 1824 a college prize for an
essay on the character of William III. He also won a prize for Latin
declamation and a Craven scholarship, and wrote the prize poems of
1819 and 1821.
In 1826 Macaulay was called to the bar and joined the northern
circuit. But he soon gave up even the pretence of reading law, and
spent many more hours under the gallery of the house of commons
than in the court. His first attempt at a public speech, made at an
anti-slavery meeting in 1824, was described by the Edinburgh
Review as “a display of eloquence of rare and matured excellence.”
His first considerable appearance in print was in No. 1 of Knight’s
Quarterly Magazine, a periodical which enjoyed a short but brilliant
existence, and which was largely supported by Eton and Cambridge.

In August 1825 began Macaulay’s connexion with the periodical
which was to prove the field of his literary reputation. The Edinburgh
Review was at this time at the height of its power, not only as an
organ of the growing opinion which, leant towards reform, but as a
literary tribunal from which there was no appeal. His essay on Milton
(Aug. 1825), so crude that the author afterwards said that “it
contained scarcely a paragraph such as his matured judgment
approved,” created for him at once a literary reputation which
suffered no diminution to the last, a reputation which he established
and confirmed, but which it would have been hardly possible to
make more conspicuous. The publisher John Murray declared that it
would be worth the copyright of Childe Harold to have Macaulay on
the staff of the Quarterly Review, and Robert Hall, the orator,
writhing with pain, and well-nigh worn out with disease, was
discovered lying on the floor employed in learning by aid of grammar
and dictionary enough Italian to enable him to verify the parallel
between Milton and Dante.
This sudden blaze of popularity, kindled by a single essay, is partly
to be explained by the dearth of literary criticism in England at that
epoch. For, though a higher note had already been sounded by
Hazlitt and Coleridge, it had not yet taken hold of the public mind,
which was still satisfied with the feeble appreciations of the
Retrospective Review, or the dashing and damnatory improvisation
of Wilson in Blackwood or Jeffrey in the Edinburgh. Still, allowance
being made for the barbarous partisanship of the established critical
tribunals of the period, it seems surprising that a social success so
signal should have been the consequence of a single article. The
explanation is that the writer of the article on Milton was, unlike
most authors, also a brilliant conversationalist. There has never been
a period when an amusing talker has not been in great demand at

London tables; but when Macaulay made his debut witty
conversation was studied and cultivated as it has ceased to be in the
more busy age which has succeeded. At the university Macaulay had
been recognized as pre-eminent for inexhaustible talk and genial
companionship among a circle of such brilliant young men as Charles
Austin, Romilly, Praed and Villiers. He now displayed these gifts on a
wider theatre. Launched on the best that London had to give in the
way of society, Macaulay accepted and enjoyed with all the zest of
youth and a vigorous nature the opportunities opened for him. He
was courted and admired by the most distinguished personages of
the day. He was admitted at Holland House, where Lady Holland
listened to him with deference, and scolded him with a
circumspection which was in itself a compliment. Samuel Rogers
spoke of him with friendliness and to him with affection. He was
treated with almost fatherly kindness by “Conversation” Sharp.
Thus distinguished, and justifiably conscious of his great powers,
Macaulay began to aspire to a political career. But the shadow of
pecuniary trouble early began to fall upon his path. When he went to
college his father believed himself to be worth £100,000. But
commercial disaster overtook the house of Babington Macaulay,
and the son now saw himself compelled to work for his livelihood.
His Trinity fellowship of £300 a year became of great consequence to
him, but it expired in 1831; he could make at most £200 a year by
writing; and a commissionership of bankruptcy, which was given him
by Lord Lyndhurst in 1828, and which brought him in about £400 a
year, was swept away, without compensation, by the ministry which
came into power in 1830. Macaulay was reduced to such straits that
he had to sell his Cambridge gold medal.

In February 1830 the doors of the House of Commons were
opened to him through what was then called a “pocket borough.”
Lord Lansdowne, who had been struck by two articles on James Mill
and the Utilitarians, which appeared in the Edinburgh Review in
1829, offered the author the seat at Calne. The offer was
accompanied by the express assurance that the patron had no wish
to interfere with Macaulay’s freedom of voting. He thus entered
parliament at one of the most exciting moments of English domestic
history, when the compact phalanx of reactionary administration
which for nearly fifty years had commanded a crushing majority in
the Commons was on the point of being broken by the growing
strength of the party of reform. Macaulay made his maiden speech
on the 5th of April 1830, on the second reading of the Bill for the
Removal of Jewish Disabilities. In July the king died and parliament
was dissolved; the revolution took place in Paris. Macaulay, who was
again returned for Calne, visited Paris, eagerly enjoying a first taste
of foreign travel. On the 1st of March 1831 the Reform Bill was
introduced, and on the second night of the debate Macaulay made
the first of his reform speeches. It was, like all his speeches, a
success. Sir Robert Peel said of it that “portions were as beautiful as
anything I have ever heard or read.”
Encouraged by this first success, Macaulay now threw himself with
ardour into the life of the House of Commons, while at the same
time he continued to enjoy to the full the social opportunities which
his literary and political celebrity had placed within his reach. He
dined out almost nightly, and spent many of his Sundays at the
suburban villas of the Whig leaders, while he continued to supply the
Edinburgh Review with articles. On the triumph of Earl Grey’s
cabinet, and the passing of the Reform Act in June 1832, Macaulay,
whose eloquence had signalized every stage of the conflict, became

one of the commissioners of the board of control, and applied
himself to the study of Indian affairs. Giving his days to India and his
nights to the House of Commons, he could only devote a few hours
to literary composition by rising at five when the business of the
house had allowed of his getting to bed in time on the previous
evening. Between September 1831 and December 1833 he furnished
the Review with eight important articles, besides writing his ballad
on the Armada.
In the first Reform Parliament, January 1833, Macaulay took his
seat as one of the two members for Leeds, which up to that date
had been unrepresented in the House of Commons. He replied to
O’Connell in the debate on the address, meeting the great agitator
face to face, with high, but not intemperate, defiance. In July he
defended the Government of India Bill in a speech of great power,
and he was instrumental in getting the bill through committee
without unnecessary friction. When the abolition of slavery came
before the house as a practical question, Macaulay had the prospect
of having to surrender office or to vote for a modified abolition, viz.
twelve years’ apprenticeship, which was proposed by the ministry,
but condemned by the abolitionists. He was prepared to make the
sacrifice of place rather than be unfaithful to the cause to which his
father had devoted his life. He placed his resignation in Lord
Althorp’s hands, and spoke against the ministerial proposal. But the
sense of the house was so strongly expressed as unfavourable that,
finding they would be beaten if they persisted, the ministry gave
way, and reduced apprenticeship to seven years, a compromise
which the abolition party accepted; and Macaulay remained at the
board of control.

While he was thus growing in reputation, and advancing his public
credit, the fortunes of the family were sinking, and it became
evident that his sisters would have no provision except such as their
brother might be enabled to make for them. Macaulay had but two
sources of income, both of them precarious—office and his pen. As
to office, the Whigs could not have expected at that time to retain
power for a whole generation; and, even while they did so,
Macaulay’s resolution that he would always give an independent vote
made it possible that he might at any moment find himself in
disagreement with his colleagues, and have to quit his place. As to
literature, he wrote to Lord Lansdowne (1833), “it has been hitherto
merely my relaxation; I have never considered it as the means of
support. I have chosen my own topics, taken my own time, and
dictated my own terms. The thought of becoming a bookseller’s
hack, of spurring a jaded fancy to reluctant exertion, of filling sheets
with trash merely that sheets may be filled, of bearing from
publishers and editors what Dryden bore from Tonson and what
Mackintosh bore from Lardner, is horrible to me.” Macaulay was thus
prepared to accept the offer of a seat in the supreme council of
India, created by the new India Act. The salary of the office was
fixed at £10,000, out of which he calculated to be able to save
£30,000 in five years. His sister Hannah accepted his proposal to
accompany him, and in February 1834 the brother and sister sailed
for Calcutta.
Macaulay’s appointment to India occurred at the critical moment
when the government of the company was being superseded by
government by the Crown. His knowledge of India was, when he
landed, but superficial. But at this juncture there was more need of
statesmanship directed by general liberal principles than of a
practical knowledge of the details of Indian administration.

Macaulay’s presence in the council was of great value; his minutes
are models of good judgment and practical sagacity. The part he
took in India has been described as “the application of sound liberal
principles to a government which had till then been jealous, close
and repressive.” He vindicated the liberty of the press; he
maintained the equality of Europeans and natives before the law;
and as president of the committee of public instruction he
inaugurated the system of national education.
A clause in the India Act 1833 occasioned the appointment of a
commission to inquire into the jurisprudence of the Eastern
dependency. Macaulay was appointed president of that commission.
The draft of a penal code which he submitted became, after a
revision of many years, and by the labour of many experienced
lawyers, the Indian criminal code. Of this code Sir James Stephen
said that “it reproduces in a concise and even beautiful form the
spirit of the law of England, in a compass which by comparison with
the original may be regarded as almost absurdly small. The Indian
penal code is to the English criminal law what a manufactured article
ready for use is to the materials out of which it is made. It is to the
French code pénal, and to the German code of 1871, what a finished
picture is to a sketch. It is simpler and better expressed than
Livingston’s code for Louisiana; and its practical success has been
complete.”
Macaulay’s enlightened views and measures drew down on him,
however, the abuse and ill-will of Anglo-Indian society. Fortunately
for himself he was enabled to maintain a tranquil indifference to
political detraction by withdrawing his thoughts into a sphere remote
from the opposition and enmity by which he was surrounded. Even
amid the excitement of his early parliamentary successes literature

had balanced politics in his thoughts and interests. Now in his exile
he began to feel more strongly each year the attraction of European
letters and European history. He wrote to his friend Ellis: “I have
gone back to Greek literature with a passion astonishing to myself. I
have never felt anything like it. I was enraptured with Italian during
the six months which I gave up to it; and I was little less pleased
with Spanish. But when I went back to the Greek I felt as if I had
never known before what intellectual enjoyment was.” In thirteen
months he read through, some of them twice, a large part of the
Greek and Latin classics. The fascination of these studies produced
their inevitable effect upon his view of political life. He began to
wonder what strange infatuation leads men who can do something
better to squander their intellect, their health and energy, on such
subjects as those which most statesmen are engaged in pursuing.
He was already, he says, “more than half determined to abandon
politics and give myself wholly to letters, to undertake some great
historical work, which may be at once the business and the
amusement of my life, and to leave the pleasures of pestiferous
rooms, sleepless nights, and diseased stomachs to Roebuck and to
Praed.”
In 1838 Macaulay and his sister Hannah, who had married Charles
Trevelyan in 1834, returned to England. He at once entered
parliament as member for Edinburgh. In 1839 he became secretary
at war, with a seat in the cabinet in Lord Melbourne’s ministry. His
acceptance of office diverted him for a time from prosecuting the
plan he had already formed of a great historical work. But in less
than two years the Melbourne ministry fell. In 1842 appeared his
Lays of Ancient Rome, and in the next year he collected and
published his Essays. He returned to office in 1846, in Lord John
Russell’s administration, as paymaster-general. His duties were very

light, and the contact with official life and the obligations of
parliamentary attendance were even of benefit to him while he was
engaged upon his History. In the sessions of 1846-1847 he spoke
only five times, and at the general election of July 1847 he lost his
seat for Edinburgh. The balance of Macaulay’s faculties had now
passed to the side of literature. At an earlier date he had relished
crowds and the excitement of ever new faces; as years went
forward, and absorption in the work of composition took off the
edge of his spirits, he recoiled from publicity. He began to regard the
prospect of business as worry, and had no longer the nerve to brace
himself to the social efforts required of one who represents a large
constituency.
Macaulay retired into private life, not only without regret, but with
a sense of relief. He gradually withdrew from general society, feeling
the bore of big dinners and country-house visits, but he still enjoyed
close and constant intercourse with a circle of the most eminent men
that London then contained. At that time social breakfasts were in
vogue. Macaulay himself preferred this to any other form of
entertainment. Of these brilliant reunions nothing has been
preserved beyond the names of the men who formed them—Rogers,
Hallam, Sydney Smith, Lord Carlisle, Lord Stanhope, Nassau Senior,
Charles Greville, Milman, Panizzi, G. C. Lewis, Van de Weyer. His
biographer thus describes Macaulay’s appearance and bearing in
conversation: “Sitting bolt upright, his hands resting on the arms of
his chair, or folded over the handle of his walking-stick, knitting his
eyebrows if the subject was one which had to be thought out as he
went along, or brightening from the forehead downwards when a
burst of humour was coming, his massive features and honest
glance suited well with the manly sagacious sentiments which he set
forth in his sonorous voice and in his racy and intelligible language.

To get at his meaning people had never the need to think twice, and
they certainly had seldom the time.”
But, great as was his enjoyment of literary society and books, they
only formed his recreation. In these years he was working with
unflagging industry at the composition of his History. His
composition was slow, his corrections both of matter and style
endless; he spared no pains to ascertain the facts. He sacrificed to
the prosecution of his task a political career, House of Commons
fame, the allurements of society. The first two volumes of the
History of England appeared in December 1848. The success was in
every way complete beyond expectation. The sale of edition after
edition, both in England and the United States, was enormous.
In 1852, when his party returned to office, he refused a seat in
the cabinet, but he could not bring himself to decline the
compliment of a voluntary amende which the city of Edinburgh paid
him in returning him at the head of the poll at the general election in
July of that year. He had hardly accepted the summons to return to
parliamentary life before fatal weakness betrayed itself in deranged
action of the heart; from this time forward till his death his strength
continued steadily to sink. The process carried with it dejection of
spirits as its inevitable attendant. The thought oppressed him that
the great work to which he had devoted himself would remain a
fragment. Once again, in June 1853, he spoke in parliament, and
with effect, against the exclusion of the master of the rolls from the
House of Commons, and at a later date in defence of competition for
the Indian civil service. But he was aware that it was a grievous
waste of his small stock of force, and that he made these efforts at
the cost of more valuable work.

In November 1855 vols. iii. and iv. of the History appeared and
obtained a vast circulation. Within a generation of its first
appearance upwards of 140,000 copies of the History were printed
and sold in the United Kingdom alone; and in the United States the
sales were on a correspondingly large scale. The History was
translated into German, Polish, Danish, Swedish, Hungarian,
Russian, Bohemian, Italian, French, Dutch and Spanish. Flattering
marks of respect were heaped upon the author by foreign
academies. His pecuniary profits were (for that time) on a scale
commensurate with the reputation of the book: the cheque he
received for £20,000 has become a landmark in literary history.
In May 1856 he quitted the Albany, in which he had passed fifteen
happy years, and went to live at Holly Lodge, Campden Hill, then,
before it was enlarged, a tiny bachelor’s dwelling, but with a lawn
whose unbroken slope of verdure gave it the air of a considerable
country house. In the following year (1857) he was raised to the
peerage by the title of Baron Macaulay of Rothley. “It was,” says
Lady Trevelyan, “one of the few things that everybody approved; he
enjoyed it himself, as he did everything, simply and cordially.” It was
a novelty in English life to see eminence which was neither that of
territorial opulence nor of political or military services recognized and
rewarded by elevation to the peerage.
But Macaulay’s health, which had begun to give way in 1852, was
every year visibly failing. In May 1858 he went to Cambridge for the
purpose of being sworn in as high steward of the borough, to which
office he had been elected on the death of Earl Fitzwilliam. When his
health was given at a public breakfast in the town-hall he was
obliged to excuse himself from speaking. In the upper house he
never spoke. Absorbed in the prosecution of his historical work, he

had grown indifferent to the party politics of his own day. Gradually
he had to acquiesce in the conviction that, though his intellectual
powers remained unimpaired, his physical energies would not carry
him through the reign of Anne; and, though he brought down the
narrative to the death of William III., the last half-volume wants the
finish and completeness of the earlier portions. The winter of 1859
told on him, and he died on the 28th of December. On the 9th of
January 1860 he was buried in Westminster Abbey, in Poets’ Corner,
near the statue of Addison.
Lord Macaulay never married. A man of warm domestic affections,
he found their satisfaction in the attachment and close sympathy of
his sister Hannah, the wife of Sir Charles Trevelyan. Her children
were to him as his own. Macaulay was a steadfast friend, and no act
inconsistent with the strictest honour and integrity was ever imputed
to him. When a poor man, and when salary was of consequence to
him, he twice resigned office rather than make compliances for
which he would not have been severely blamed. In 1847, when his
seat in parliament was at stake, he would not be persuaded to
humour, to temporize, even to conciliate. He had a keen relish for
the good things of life, and desired fortune as the means of
obtaining them; but there was nothing mercenary or selfish in his
nature. When he had raised himself to opulence, he gave away with
an open hand, not seldom rashly. His very last act was to write a
letter to a poor curate enclosing a cheque for £25. The purity of his
morals was not associated with any tendency to cant.
The lives of men of letters are often records of sorrow or
suffering. The life of Macaulay was eminently happy. Till the closing
years (1857-1859), he enjoyed life with the full zest of healthy
faculty, happy in social intercourse, happy in the solitude of his

study, and equally divided between the two. For the last fifteen
years of his life he lived for literature. His writings were remunerative
to him far beyond the ordinary measure, yet he never wrote for
money. He lived in his historical researches; his whole heart and
interest were unreservedly given to the men and the times of whom
he read and wrote. His command of literature was imperial.
Beginning with a good classical foundation, be made himself familiar
with the imaginative, and then with the historical, remains of Greece
and Rome. He went on to add the literature of his own country, of
France, of Italy, of Spain. He learnt Dutch enough for the purposes
of his history. He read German, but for the literature of the northern
nations he had no taste, and of the erudite labours of the Germans
he had little knowledge and formed an inadequate estimate. The
range of his survey of human things had other limitations more
considerable still. All philosophical speculation was alien to his mind;
nor did he seem aware of the degree in which such speculation had
influenced the progress of humanity. A large—the largest—part of
ecclesiastical history lay outside his historical view. Of art he
confessed himself ignorant, and even refused a request to furnish a
critique on Swift’s poetry to the Edinburgh Review. Lessing’s
Laocoon, or Goethe’s criticism on Hamlet, “filled” him “with wonder
and despair.”
Of the marvellous discoveries of science which were succeeding
each other day by day he took no note; his pages contain no
reference to them. It has been told already how he recoiled from the
mathematical studies of his university. These deductions made, the
circuit of his knowledge still remains very wide—as extensive
perhaps as any human brain is competent to embrace. His literary
outfit was as complete as has ever been possessed by any English
writer; and, if it wants the illumination of philosophy, it has an

equivalent resource in a practical acquaintance with affairs, with
administration, with the interior of cabinets, and the humour of
popular assemblies. Nor was the knowledge merely stored in his
memory; it was always at his command. Whatever his subject, he
pours over it his stream of illustration, drawn from the records of all
ages and countries. His Essays are not merely instructive as history;
they are, like Milton’s blank verse, freighted with the spoils of all the
ages. As an historian Macaulay has not escaped the charge of
partisanship. He was a Whig; and in writing the history of the rise
and triumph of Whig principles in the latter half of the 17th century
he identified himself with the cause. But the charge of partiality, as
urged against Macaulay, means more than that he wrote the history
of the Whig revolution from the point of view of those who made it.
When he is describing the merits of friends and the faults of enemies
his pen knows no moderation. He has a constant tendency to glaring
colours, to strong effects, and will always be striking violent blows.
He is not merely exuberant but excessive. There is an overweening
confidence about his tone; he expresses himself in trenchant
phrases, which are like challenges to an opponent to stand up and
deny them. His propositions have no qualifications. Uninstructed
readers like this assurance, as they like a physician who has no
doubt about their case. But a sense of distrust grows upon the more
circumspect reader as he follows page after page of Macaulay’s
categorical affirmations about matters which our own experience of
life teaches us to be of a contingent nature. We inevitably think of a
saying attributed to Lord Melbourne: “I wish I were as cocksure of
any one thing as Macaulay is of everything.” Macaulay’s was the
mind of the advocate, not of the philosopher; it was the mind of
Bossuet, which admits no doubts or reserves itself and tolerates

none in others, and as such was disqualified from that equitable
balancing of evidence which is the primary function of the historian.
Macaulay, the historian no less than the politician, is, however,
always on the side of justice, fairness for the weak against the
strong, the oppressed against the oppressor. But though a Liberal in
practical politics, he had not the reformer’s temperament. The world
as it is was good enough for him. The glories of wealth, rank,
honours, literary fame, the elements of vulgar happiness, made up
his ideal of life. A successful man himself, every personage and
every cause is judged by its success. “The brilliant Macaulay,” says
Emerson, “who expresses the tone of the English governing classes
of the day, explicitly teaches that ‘good’ means good to eat, good to
wear, material commodity.” Macaulay is in accord with the average
sentiment of orthodox and stereotyped humanity on the relative
values of the objects and motives of human endeavour. And this
commonplace materialism is one of the secrets of his popularity, and
one of the qualities which guarantee that that popularity will be
enduring.
(M. P.)
Macaulay’s whole works were collected in 1866 by his sister,
Lady Trevelyan, in 8 vols. The first four volumes are occupied by
the History; the next three contain the Essays, and the Lives
which he contributed to the Encyclopaedia Britannica. In vol. viii.
are collected his Speeches, the Lays of Ancient Rome, and some
miscellaneous pieces. The “life” by Dean Milman, printed in vol.
viii. of the edition of 1858-1862, is prefixed to the “People’s
Edition” (4 vols., 1863-1864). Messrs. Longmans, Green Co.
published a complete edition, the “Albany,” in 12 vols., in 1898.
There are numerous editions of the Critical and Historical

Essays, separately and collectively; they were edited in 1903 by
F. C. Montagu.
The Life and Letters of Lord Macaulay (2 vols., 1876), by his
nephew, Sir George Otto Trevelyan, is one of the best
biographies in the English language. The life (1882) in the
“English Men of Letters” series was written by J. Cotter Morison.
Far further criticism, see Hepworth Dixon, in his Life of Penn
(1841); John Paget, The New Examen: Inquiry into Macaulay’s
History (1861) and Paradoxes and Puzzles (1874); Walter
Bagehot, in the National Review (Jan. 1856), reprinted in his
Literary Studies (1879); James Spedding, Evenings with a
Reviewer (1881), discussing his essay on Bacon; Sir L. Stephen,
Hours in a Library, vol. ii. (1892); Lord Morley, Critical
Miscellanies (1877), vol. ii.; Lord Avebury, Essays and Addresses
(1903); Thum, Anmerkungen zu Macaulay’s History of England
(Heilbronn, 1882). A bibliography of German criticism of
Macaulay is given in G. Körting’s Grd. der engl. Literatur (4th
ed., Münster, 1905).
MACAW, or, as formerly spelt, Maccaw, the name given to some
fifteen or more species of large, long-tailed birds of the parrot-
family, natives of the neotropical region, and forming a very well-
known and easily recognized genus Ara, and to the four species of
Brazilian Hyacinthine macaws of the genera Anodorhynchus and

Cyanopsittacus. Most of the macaws are remarkable for their gaudy
plumage, which exhibits the brightest scarlet, yellow, blue and green
in varying proportion and often in violent contrast, while a white
visage often adds a very peculiar and expressive character.1 With
one exception the known species of Ara inhabit the mainland of
America from Paraguay to Mexico, being especially abundant in
Bolivia, where no fewer than seven of them (or nearly one half)
have been found (Proc. Zool. Soc., 1879, p. 634). The single extra-
continental species, A. tricolor, is one of the most brilliantly coloured,
and is peculiar to Cuba, where, according to Gundlach (Ornitologia
Cubana, p. 126), its numbers are rapidly decreasing so that there is
every chance of its becoming extinct.2
Of the best known species of the group, the blue-and-yellow
macaw, A. ararauna, has an extensive range in South America from
Guiana in the east to Colombia in the west, and southwards to
Paraguay. Of large size, it is to be seen in almost every zoological
garden, and it is very frequently kept alive in private houses, for its
temper is pretty good, and it will become strongly attached to those
who tend it. Its richly coloured plumage, sufficiently indicated by its
common English name, supplies feathers eagerly sought by salmon-
fishers for the making of artificial flies. The red-and-blue macaw, A.
macao, is even larger and more gorgeously clothed, for, besides the
colours expressed in its ordinary appellation, yellow and green enter
into its adornment. It inhabits Central as well as South America as
far as Bolivia, and is also a common bird in captivity, though perhaps
less often seen than the foregoing. The red-and-yellow species, A.
chloroptera, ranging from Panama to Brazil, is smaller, or at least has
a shorter tail, and is not quite so usually met with in menageries.
The red-and-green, A. militaris, smaller again than the last, is not
unfrequent in confinement, and presents the colours of the name it

bears. This has the most northerly extension of habitat, occurring in
Mexico and thence southwards to Bolivia. In A. manilata and A.
nobilis the prevailing colour is green and blue. The Hyacinthine
macaws A. hyacinthinus, A. leari, A. glaucus and Cyanopsittacus spixi
are almost entirely blue.
The macaws live well in captivity, either chained to a perch or kept
in large aviaries in which their strong flight is noticeable. The note of
these birds is harsh and screaming. The sexes are alike; the
lustreless white eggs are laid in hollow trees, usually two at a time.
The birds are gregarious but apparently monogamous.
(A. N.)
1 This serves to separate the macaws from the long-tailed parakeets
of the New World (Conurus), to which they are very nearly allied.
2 There is some reason to think that Jamaica may have formerly
possessed a macaw (though no example is known to exist), and if so it
was most likely a peculiar species. Sloane (Voyage, ii. 297), after
describing what he calls the “great maccaw” (A. ararauna), which he had
seen in captivity in that island, mentions the “small maccaw” as being
very common in the woods there, and P. H. Gosse (Birds of Jamaica, p.
260) gives, on the authority of Robinson, a local naturalist of the last
century, the description of a bird which cannot be reconciled with any
species now known, though it must have evidently been allied to the
Cuban A. tricolor.

MACBETH, king of Scotland (d. 1058), was the son of
Findlaech, mormaer or hereditary ruler of Moreb (Moray and Ross),
who had been murdered by his nephews in 1020. He probably
became mormaer on the death of Malcolm, one of the murderers, in
1029, and he may have been one of the chiefs (the Maclbaethe of
the Saxon Chronicle) who submitted to Canute in 1031. Marianus
records that in 1040 Duncan, the grandson and successor of
Malcolm king of Scotland, was slain by Macbeth. Duncan had shortly
before suffered a severe defeat at the hands of Thorfinn, the
Norwegian earl of Orkney and Caithness, and it was perhaps this
event which tempted Macbeth to seize the throne. As far as is
known he had no claim to the crown except through his wife Gruach,
who appears to have been a member of the royal family. Macbeth
was apparently a generous benefactor to the Church, and is said to
have made a pilgrimage to Rome in 1050. According to S. Berchan
his reign was a time of prosperity for Scotland. The records of the
period, however, are extremely meagre, and much obscurity prevails,
especially as to his relations with the powerful earl Thorfinn. More
than one attempt was made by members of the Scottish royal family
to recover the throne; in 1045 by Crinan, the lay abbot of Dunkeld,
son-in-law of Malcolm II., and in 1054 by Duncan’s son Malcolm with
the assistance of Siward the powerful earl of Northumbria, himself a
connexion of the ousted dynasty. Three years later in 1057 Malcolm
and Siward again invaded Scotland and the campaign ended with
the defeat and death of Macbeth, who was slain at Lumphanan.
Macbeth is, of course, chiefly famous as the central figure of
Shakespeare’s great tragedy.
See W. F. Skene, Chronicles of the Picts and Scots (1867) and
Celtic Scotland (1876); Sir John Rhys, Celtic Britain (1904).

MACCABEES, the name (in the plural) of a distinguished
Jewish family dominant in Jerusalem in the 2nd century b.c.
According to 1 Macc. ii. 4, the name Maccabaeus (Gr. Μακκαβῖος-?
Heb. ‫)מקבי‬ was originally the distinctive surname of Judas, third son
of the Jewish priest Mattathias, who struck the first blow for religious
liberty during the persecution under Antiochus IV. (Epiphanes).
Subsequently, however, it obtained a wider significance, having been
applied first to the kinsmen of Judas, then to his adherents, and
ultimately to all champions of religion in the Greek period. Thus the
mother of the seven brethren, whose martyrdom is related in 2
Macc. vi., vii., is called by early Christian writers “the mother of the
Maccabees.” The name is used still more loosely in the titles of the
so-called Third, Fourth and Fifth Books of Maccabees. It is now
customary to apply it only to the sons and descendants of
Mattathias. As, however, according to Josephus (Ant. xii. 6. 1), this
brave priest’s great-great-grandfather was called Ḥasmon (i.e. “rich”
= magnate; cf. Ps. lxviii. 31 [32]), the family is more correctly
designated by the name of Hasmonaeans or Asmoneans (q.v.). This
name Jewish authors naturally prefer to that of Maccabees; they
also style 1 and 2 Macc. “Books of the Hasmonaeans.”
If Maccabee (maqqābi) is the original form of the name, the most
probable derivation is from the Aramaic maqqābā (Heb. ‫מקבת‬,
Judg. iv. 21, c.) = “hammer.” The surname “hammerer” might have

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