To the many students I＊ve had the privilege of teaching over the years who have contributed in many
ways to the broad field of electrical engineering, and to future students who will contribute in ways
we cannot now imagine.
ABOUT THE AUTHOR
Donald A. Neamen is a professor emeritus in the Department of Electrical and
Computer Engineering at the University of New Mexico where he taught for more than
25 years. He received his Ph.D. from the University of New Mexico and then became
an electronics engineer at the Solid State Sciences Laboratory at Hanscom Air Force
Base. In 1976, he joined the faculty in the ECE department at the University of New
Mexico, where he specialized in teaching semiconductor physics and devices courses
and electronic circuits courses. He is still a part-time instructor in the department.
In 1980, Professor Neamen received the Outstanding Teacher Award for the
University of New Mexico. In 1983 and 1985, he was recognized as Outstanding
Teacher in the College of Engineering by Tau Beta Pi. In 1990, and each year from
1994 through 2001, he received the Faculty RecognitionAward, presented by graduating
ECE students. He was also honored with the Teaching ExcellenceAward in the
College of Engineering in 1994.
In addition to his teaching, Professor Neamen served as Associate Chair of the
ECE department for several years and has also worked in industry with Martin
Marietta, Sandia National Laboratories, and Raytheon Company. He has published
many papers and is the author of Electronic Circuit Analysis and Design, Second Edition
and Semiconductor Physics and Devices: Basic Principles, Third Edition.
PREFACE
PHILOSOPHY AND GOALS
The purpose of this text is to provide a basis for understanding the characteristics,
operation, and limitations of semiconductor devices. In order to gain this understanding,
it is essential to have a thorough knowledge of the physics of the semiconductor
material. The goal of this book is to bring together the fundamental physics of
the semiconductor material and the semiconductor device physics.
Since the objective of this text is to provide an introduction to the theory of
semiconductor devices, there is a great deal of advanced theory that is not considered.
This material is found in more advanced texts. There are occasions in the text
where equations and relationships are simply stated with no or very little derivation.
Again, the details are found in more advanced texts. However, the author feels that
there is enough mathematics included to provide a good foundation for the basic understanding
of semiconductor devices in this first course.
PREREQUISITES
This text is intended for junior and senior undergraduates in electrical engineering.
The prerequisites for understanding the material are college mathematics, up to and
including differential equations, and college physics, including an introduction to
modern physics and electrostatics. Prior completion of an introductory course in
electronic circuits is helpful, but not essential.
ORDER OF PRESENTATION
Each instructor has a personal preference for the order in which the course material
is presented. The order of presentation of topics in this text is somewhat different
compared to many semiconductor textbooks. Chapters 1每4 cover the basic physics of
the semiconductor material and contain topics normally covered initially in any
semiconductor device course. Chapter 5 discusses the electrostatics of the pn and
Schottky junctions. This material is necessary and sufficient for the understanding of
the MOS transistor presented in Chapters 6 and 7. There are two reasons for discussing
the MOS transistor at this point. First, since the MOS transistor is fundamental
to integrated circuits, this material is presented early enough in the course so
that it doesn＊t get ※short changed,§ as it might when covered at the end of a course.
Second, since a ※real§ semiconductor device is discussed fairly early in the course,
the reader may have more motivation to continue studying this course material.
After the MOS transistor is presented, the nonequilibrium characteristics of
the semiconductor material is presented in Chapter 8 and then the forward-biased
pn junction and Schottky diodes are discussed in Chapter 9. The bipolar transistor is
xv
xvi Preface
presented in Chapter 10. Chapter 11 covers additional devices such as junction fieldeffect
transistors and thyristors. Finally, optical devices are discussed in Chapter 12.
One possible disadvantage to this order of presentation is that the discussion of
the pn junction is ※interrupted.§ However, the author feels that a ※just-in-time§ approach
is justified. Some discussion of the pn junction is necessary before presenting
the MOS transistor. However, if the entire discussion of the pn junction, including
the discussion of nonequilibrium excess carriers, took place before the MOS transistor,
then much of the knowledge gained of forward-biased pn junctions would be lost
by the reader by the time the bipolar transistor is discussed.
The following table lists the textbook approach to the order of presentation of
topics. Unfortunately, because of time constraints, every topic in every chapter cannot
be covered in a one-semester course.
Textbook Approach
Chapter 1 Crystal structure
Chapter 2 Selected topics from quantum mechanics and theory of solids
Chapter 3 Semiconductor material physics
Chapter 4 Transport phenomena
Chapter 5 Electrostatics of the pn junction
Chapter 6 The MOS transistor
Chapter 7 Selected topics for advanced MOSFETs
Chapter 8 Selected topics from nonequilibrium semiconductor physics
Chapter 9 The pn junction diode
Chapter 10 The bipolar transistor
Chapter 11 Selected topics from other devices
Chapter 12 Selected topics from optical devices
Classical Approach
Chapter 1 Crystal structure
Chapter 2 Selected topics from quantum mechanics and theory of solids
Chapter 3 Semiconductor material physics
Chapter 4 Transport phenomena
Chapter 8 Selected topics from Nonequilibrium semiconductor physics
Chapter 5 Electrostatics of the pn junction
Chapter 9 The pn junction diode
Chapter 10 The bipolar transistor
Chapter 6 The MOS transistor
Chapter 7 Selected topics from advanced MOSFETs
Chapter 11 Selected topics from other devices
Chapter 12 Selected topics from optical devices
For those instructors who prefer the classical approach and wish to cover the
bipolar transistor before the MOS transistor, the following table lists the order of presentation.
The chapters are written so that this order of presentation is very plausible.
Preface xvii
USE OF THE BOOK
The text is intended for a one-semester course at the junior or senior level. As with
most textbooks, there is more material than can be conveniently covered in one semester;
this enables each instructor some flexibility in designing the course to his or
her own specific needs.
At the end of several chapters, there is a section dealing with fabrication technology.
In Chapter 1, this topic deals with the growth of semiconductor materials and
the oxidation process. In Chapter 3, this topic deals with the introduction of specific
impurities into the semiconductor by either diffusion or ion implantation. In later
chapters, this topic deals with the fabrication of specific devices. In each case, the
fabrication discussion is relatively short and intended only to give the reader a basic
understanding of the fabrication technology. These sections, as well as a few other
sections in the text, are denoted by the symbol in front of the section heading. The
symbol shows that reading these sections will aid in the total summation of the understanding
of semiconductor devices. However, a basic understanding of semiconductor
device physics can be accomplished without studying these sections in detail
during this first introductory course.
FEATURES OF THE BOOK
←Preview section: A preview section introduces each chapter. This preview links
the chapter to previous chapters and states the chapter＊s goals, that is, what the
reader should gain from the chapter.
←Historical and Present-Day Insights: A Historical Insight section relates the
chapter material to a few historical events and a Present-Day Insight section
relates the chapter material to current research and manufacturing events.
←Icon: , indicates sections that are to be read for understanding to increase the
total summation of knowledge of semiconductor devices. However, a detailed
study of these sections is not required during this first introductory course.
←Key terms in the margin: Key terms are listed in the margin of the text. Quickly
finding a key term adjacent to the text in which the material is discussed should
aid the student in reviewing the material.
←Examples: There are a liberal number of examples given in the text to reinforce
the theoretical concepts being developed. These examples contain all the details
of the analysis or design, so the reader does not have to fill in missing steps.
←Exercise problems: An exercise problem is given after each example. These
exercises are similar in scope to the preceding example. The ability to solve
these exercise problems should be an indication as to whether the student has
mastered the previous material. Answers to these problems are given.
←Test Your Understanding exercises: At the end of major sections, additional
exercise problems are given. These exercise problems tend to be more
comprehensive than the exercise problems given after each example. Answers
to these problems are also given.
xviii Preface
←Summary: A summary section follows the text of each chapter. This section
summarizes the overall results derived in the chapter and reviews the basic
concepts developed.
←Checkpoint: A checkpoint section follows the Summary section. This section
states the goals that should have been met and states the abilities the reader
should have gained. The Checkpoints will help assess progress before moving
to the next chapter.
←Review questions: A list of review questions is included at the end of each
chapter. These questions serve as a self-test to help the reader determine how
well the concepts developed in the chapter have been mastered.
←End-of-chapter problems: A substantial number of problems are provided at the
end of each chapter, organized according to the subject of each section. An
asterisk in front of a problem indicates a more difficult problem. Answers to a
selected number are provided in Appendix F.
←Reading list: A reading list finishes up each chapter. The references indicated
by an asterisk are at a more advanced level compared with this text.
←Answers to selected problems: Answers to selected problems are given in
Appendix F. Knowing the answer to a problem can aid and reinforce the
problem solving.
SUPPLEMENTS
ACKNOWLEDGMENTS
Peter John Burke, University of California, Irvine
Chris S. Ferekides, University of South Florida
Ashok K. Goel, Michigan Technological University
Lili He, San Jose State University
Erin Jones, Oregon State University
Yaroslav Koshka, Mississippi State University
Shrinivas G. Joshi, Marquette University
Gregory B. Lush, University of Texas, El Paso
A. James Mallmann, Milwaukee School of Engineering
Donald C. Malocha, University of Central Florida
Shmuel Mardix, University of Rhode Island
The Crystal Structure of Solids
This text deals with the electrical properties and characteristics of semiconductor
materials and devices. The electrical properties of solids are therefore of primary
interest. Since the semiconductor is in general a single-crystal material and
since the electrical properties of a single-crystal material are determined not only by
the chemical composition but also by the arrangement of atoms in the solid, a brief
study of the crystal structure of solids is warranted. This introductory chapter provides
the necessary background in single-crystal materials and crystal growth for a
basic understanding of the electrical properties of semiconductor materials and
devices.
1.0 | PREVIEW
In this chapter, we will
1. List and describe semiconductor materials.
2. Describe three classifications of solids: amorphous, polycrystalline, and single
crystal.
3. Describe basic crystal structures, crystal planes, and the diamond structure.
4. Discuss differences in atomic bonding between various solids.
5. Describe various single-crystal imperfections and impurities in solids.
6. Describe processes that are used to create single-crystal semiconductor
materials.
7. Describe the formation of an oxide on silicon.
C H A P T E R 1
1
Historical Insight
Materials have always been an integral part of electrical engineering, from finding
good conductors of electricity that can handle hundreds of amperes to finding good
insulators that can handle thousands of volts. Dielectric properties of materials are
fundamental in the design of capacitors and magnetic properties of materials are fundamental
in the design of electromagnets or permanent magnets. Creating highpurity
single-crystal semiconductor materials has been crucial to the development of
the vast semiconductor industry.
Present-Day Insight
Materials continue to be a fundamental component of electrical engineering.
Creating single-crystal silicon semiconductor wafers that are 12 inches in diameter
and, at the other end of the scale, creating layers of different semiconductor materials
that are on the order of tens of angstroms thick are continuing topics of research.
The properties of high-purity single-crystal materials are fundamental to the design
of the vast number of semiconductor devices.
1.1 | SEMICONDUCTOR MATERIALS
Objective: List and describe semiconductor materials.
Semiconductors are a group of materials having conductivities between those of metals
and insulators. One fundamental characteristic of a semiconductor material is that the
conductivity can be varied over several orders of magnitude by adding controlled
amounts of impurity atoms.The ability to control and change the conductivity of a semiconductor
material allows for the design of the vast number of semiconductor devices.
Two general classifications of semiconductors are the elemental semiconductor
materials, found in group IV of the periodic table, and the compound semiconductor
materials, most of which are formed from special combinations of group III and
group V elements. Table 1.1 shows a portion of the periodic table in which the more
common semiconductors are found, and Table 1.2 lists a few of the semiconductor
materials. (Semiconductors can also be formed from combinations of group II and
group VI elements, but in general these will not be considered in this text.)
The elemental materials, those that are composed of single species of atoms, are
silicon and germanium. Silicon dominates the semiconductor commercial market.
The vast majority of integrated circuits (ICs) are fabricated in silicon, so silicon will
be emphasized to a great extent in this text.
The two-element, or binary, compounds such as gallium arsenide or gallium
phosphide are formed by combining one group III and one group V element. Gallium
arsenide is one of the more common of the compound semiconductors. It is used to
make light-emitting diodes and laser diodes. GaAs is also used in specialized applications
in which, for example, very high speed is required.
We can also form a three-element, or ternary, compound semiconductor. An example
is AlxGa1?xAs, in which the subscript x indicates the fraction of the lower
atomic number element component. More complex semiconductors can also be
formed that provide flexibility when choosing material properties.
2 CHAPTER 1 The Crystal Structure of Solids
Elemental
semiconductor
Binary
semiconductor
Ternary
semiconductor
1.2 Types of Solids 3
1.2 | TYPES OF SOLIDS
Objective: Describe three classifications of solids: amorphous, polycrystalline,
and single crystal.
In Section 1.1, we simply listed various semiconductor materials. Since semiconductors
used in discrete device or IC fabrication are generally single-crystal materials, it is
worth while discussing various types of crystalline structures. We will describe the
spatial arrangement of atoms in crystals and attempt to visualize the three-dimensional
configurations. The arrangement of atoms, as well as the chemical composition, affect
the electrical properties of the material.
Amorphous, polycrystalline, and single crystal are the three general types of
solids. Each type is characterized by the size of an ordered region within the material.
Table 1.2 | A partial list of semiconductor materials
Elemental Semiconductors IV Compound Semiconductors
Si Silicon SiC Silicon carbide
Ge Germanium SiGe Silicon germanium
Binary III每V Compounds Binary II每VI Compounds
AlAs Aluminum arsenide CdS Cadmium sulfide
AlP Aluminum phosphide CdTe Cadmium telluride
AlSb Aluminum antimonide HgS Mercury sulfide
GaAs Gallium arsenide ZnS Zinc sulfide
GaP Gallium phosphide ZnTe Zinc telluride
GaSb Gallium antimonide
InAs Indium arsenide
InP Indium phosphide
Ternary Compounds Quaternary Compounds
AlxGa1xAs Aluminum gallium AlxGa1xAsySb1y Aluminum gallium arsenic
arsenide atimonide
GaAs1xPx Gallium arsenic GaxIn1xAs1yPy Gallium indium arsenic
phosphide phosphide
Table 1.1 | A portion of the periodic table showing elements used in semiconductor materials
Group
Period II III IV V VI
2 B C N O
Boron Carbon Nitrogen Oxygen
3 Al Si P S
Aluminum Silicon Phosphorus Sulfur
4 Zn Ga Ge As Se
Zinc Gallium Germanium Arsenic Selenium
5 Cd In Sn Sb Te
Cadmium Indium Tin Antimony Tellurium
6 Hg
Mercury
An ordered region is a spatial volume in which atoms or molecules have a regular geometric
arrangement or periodicity. Amorphous materials have order only within a few
atomic or molecular dimensions, while polycrystalline materials have a high degree of
order over many atomic or molecular dimensions. These ordered regions, or singlecrystal
regions, vary in size and orientation with respect to one another. The singlecrystal
regions are called grains and are separated from one another by grain
boundaries. Single-crystal materials, ideally, have a high degree of order, or regular
geometric periodicity, throughout the entire volume of the material. The advantage of
a single-crystal material is that, in general, its electrical properties are superior to
those of a nonsingle-crystal material, since grain boundaries tend to degrade the
electrical characteristics. Two-dimensional representations of amorphous, polycrystalline,
and single-crystal materials are shown in Figure 1.1.
1.3 | SPACE LATTICES
Objective: Describe basic crystal structures, crystal planes, and the diamond
structure.
Our primary concern will be the single crystal with its regular geometric periodicity
in the atomic arrangement. A representative unit, or group of atoms, is repeated at
regular intervals in each of the three dimensions to form the single crystal. The periodic
arrangement of atoms in the crystal is called the lattice.
1.3.1 Primitive and Unit Cell
Wecan represent a particular atomic array by a dot that is called a lattice point. Figure 1.2
shows part of an infinite two-dimensional array of lattice points. The simplest means of
repeating an atomic array is by translation. Each lattice point in Figure 1.2 can be translated
a distance a1 in one direction and a distance b1 in a second noncolinear direction to
generate the two-dimensional lattice. A third noncolinear translation will produce the
three-dimensional lattice. The translation directions need not be perpendicular.
Since the three-dimensional lattice is a periodic repetition of a group of atoms,
we do not need to consider the entire lattice, but only a fundamental unit that is being
4 CHAPTER 1 The Crystal Structure of Solids
(a) (b) (c)
Figure 1.1 | Two-dimensional schematics of three general types of solids: (a) amorphous,
(b) polycrystalline, and (c) single crystal.
Lattice
Lattice point
repeated. A unit cell is a small volume of the crystal that can be used to reproduce the
entire crystal. Aunit cell is not a unique entity. Figure 1.3 shows several possible unit
cells in a two-dimensional lattice.
The unit cell A can be translated in directions a2 and b2, the unit cell B can be
translated in directions a3 and b3, and the entire two-dimensional lattice can be constructed
by the translations of either of these unit cells. The unit cells C and D in
Figure 1.3 can also be used to construct the entire lattice by using the appropriate
translations. This discussion of two-dimensional unit cells can easily be extended to
three dimensions to describe a real single-crystal material.
A primitive cell is the smallest unit cell that can be repeated to form the lattice.
In many cases, it is more convenient to use a unit cell that is not a primitive cell. Unit
cells may be chosen that have orthogonal sides, for example, whereas the sides of a
primitive cell may be nonorthogonal.
A generalized three-dimensional unit cell is shown in Figure 1.4. The relationship
between this cell and the lattice is characterized by three vectors ‘a, ‘b, and ‘c,
which need not be perpendicular and which may or may not be equal in length. Every
equivalent lattice point in the three-dimensional crystal can be found using the vector
‘r = p ‘a + q ‘b + s ‘c (1.1)
where p, q, and s are integers. Since the location of the origin is arbitrary, we will let
p, q, and s be positive integers for simplicity.
1.3 Space Lattices 5
a1
b1
Figure 1.2 | Two-dimensional
representation of a single-crystal lattice.
b2
b4
b1
b3
a2
a4
a1
a3
A
B
D
C
Figure 1.3 | Two-dimensional representation of a single-crystal
lattice showing various possible unit cells.
c
b
a
Figure 1.4 | A generalized
primitive unit cell.
Unit cell
Primitive cell
1.3.2 Basic Crystal Structures
Before we discuss semiconductor crystals, let us consider the characteristics of three
basic cubic structures. Figure 1.5 shows the simple cubic, body-centered cubic, and facecentered
cubic structures. For these simple structures, we can choose unit cells such that
the general vectors ‘a, ‘b, and ‘c are perpendicular to each other and the lengths are equal.
The simple cubic (sc) structure has an atom located at each corner; the body-centered
cubic (bcc) structure has an additional atom at the center of the cube; and the facecentered
cubic (fcc) structure has an additional atom at the center of each face plane.
By knowing the crystal structure of a material and its lattice dimensions, we can
determine several characteristics of the crystal. For example, we can determine the
volume density of atoms.
6 CHAPTER 1 The Crystal Structure of Solids
(a) (b) (c)
Figure 1.5 | Three lattice types: (a) simple cubic, (b) body-centered cubic, and (c) face-centered cubic.
EXAMPLE 1.1
OBJECTIVE
Determine the volume density of atoms in a crystal.
Consider a single-crystal material that is a face-centered cubic with a lattice constant
a0 = 5 ～ A = 5 ℅ 10?8 cm. Each corner atom is shared by eight unit cells that meet at the corner,
so each corner atom effectively contributes one-eighth of its volume to each unit cell. The eight
corner atoms then contribute an equivalent of one atom to the unit cell. Each face atom is shared
by two unit cells that meet at each side, so each face atom effectively contributes one-half of its
volume to each unit cell. The six face atoms then contribute an equivalent of three atoms to the
unit cell. Each unit cell of a face-centered cubic then effectively contains four atoms.
← Solution
The volume density of atoms is then found by dividing the number of unit cell atoms by the
unit cell volume, or
Volume density =
4 atoms
a3
0 =
4
(5 ℅ 10?8)3
or
Volume density = 3.2 ℅ 1022 atoms per cm3
Simple cubic
Body-centered cubic
Face-centered cubic
1.3 Space Lattices 7
b
每
a每
c每
c
pa
qb
sc
b
a
2c
2b
2a
Figure 1.6 | General lattice plane
intercepting the ‘a, ‘b, and ‘c axes at p, q,
and s, respectively.
← Comment
This value of the volume density of atoms in a crystal represents the order of magnitude of density
for most materials. The actual density is a function of the crystal type and crystal structure
since the packing density〞number of atoms per unit cell〞depends on crystal structure.
Exercise Problem
EX1.1 The lattice constant of a body-centered cubic structure is a0 = 4.75 ～ A. Determine
the volume density of atoms.
1.3.3 Crystal Planes and Miller Indices
Since real crystals are not infinitely large, they eventually terminate at a surface.
Semiconductor devices are fabricated at or near a surface, so the surface properties
may influence the device characteristics. We would like to be able to describe
these surfaces in terms of the lattice. Surfaces, or planes through the crystal, can
be described by first considering the intercepts of the plane along the ‘a, ‘b, and ‘c
axes used to describe the lattice.
Figure 1.6 shows a general plane intercepting the ‘a, ‘b, and ‘c axes at points pa,
qb, and sc, where p, q, and s are integers. To describe the plane, we write the reciprocals
of the intercepts as
1
p
,
1
q
,
1
s (1.2)
(Ans. 1.87 ℅ 1022 cm3)
Multiplying by the lowest common denominator, we obtain a set of numbers such as
(hkl). The plane is then referred to as the (hkl) plane. The parameters h, k, and l are
referred to as the Miller indices.
8 CHAPTER 1 The Crystal Structure of Solids
Miller indices
EXAMPLE 1.2
OBJECTIVE
Describe the plane shown in Figure 1.7.
The lattice points in Figure 1.7 are shown along the ‘a, ‘b, and ‘c axes only.
← Solution
From Equation (1.1), the intercepts of the plane correspond to p = 2, q = 3, and s = 2. Write
the reciprocals of the intercepts, from Equation (1.2), as
1
2
,
1
3
,
1
2
Now multiply by the lowest common denominator, which in this case is 6, to obtain (3, 2, 3).
The plane in Figure 1.7 is then referred to as the (323) plane. The integers are referred to as the
Miller indices. We will refer to a general plane as the (hkl) plane.
← Comment
We can show that the same three Miller indices are obtained for any plane that is parallel to the
one shown in Figure 1.7. Any parallel plane is entirely equivalent to any other.
2c
3b
2a
Figure 1.7 | A crystal-lattice plane for Example 1.2.
Exercise Problem
EX1.2 Aplane in a simple cubic crystal is described as a (132) plane. (a) What are the intercepts
on the ‘a, ‘b, and ‘c axes. (b) Sketch the plane.
Three planes that are commonly considered in a cubic crystal are shown shaded
in Figure 1.8. The plane in Figure 1.8a is parallel to the ‘b and ‘c axes so the intercepts
are given as p = 1, q =﹢, and s =﹢. Taking the reciprocal, we obtain the Miller
indices as (1, 0, 0), so the plane shown in Figure 1.8a is referred to as the (100) plane.
Again, any plane parallel to the one shown in Figure 1.8a and separated by an integral
number of lattice constants is equivalent and is referred to as the (100) plane.
One advantage to taking the reciprocal of the intercepts to obtain the Miller indices
is that the use of infinity is avoided when describing a plane that is parallel to an axis.
If we were to describe a plane passing through the origin of our system, we would
obtain infinity as one or more of the Miller indices after taking the reciprocal of the
intercepts. However, the location of the origin of our system is entirely arbitrary and
so, by translating the origin to another equivalent lattice point, we can avoid the use
of infinity in the set of Miller indices.
For the simple cubic structure, the body-centered cubic, and the face-centered
cubic, there is a high degree of symmetry. The axes can be rotated by 90～ in each of the
three dimensions and each lattice point can again be described by Equation (1.1) as
‘r = p ‘a + q ‘b + s ‘c (1.1)
Each face plane of the cubic structure shown in Figure 1.8a is entirely equivalent.
These planes are grouped together and are referred to as the {100} set of planes.
We may also consider the planes shown in Figures 1.8b and 1.8c. The intercepts
of the plane shown in Figure 1.8b are p = 1, q = 1, and s =﹢. The Miller indices
[Ans. (a) p = 6, q = 2, s = 3]
1.3 Space Lattices 9
a 每 a 每 a 每
c每
b
每
(a) (b)
c每
b
每
(c)
c每
b
每
Figure 1.8 | Three lattice planes in a simple cubic lattice: (a) (100) plane, (b) (110) plane, and (c) (111) plane.
are found by taking the reciprocal of these intercepts and, as a result, this plane is
referred to as the (110) plane. In a similar way, the plane shown in Figure 1.8c is
referred to as the (111) plane.
One characteristic of a crystal that can be determined is the distance between
nearest equivalent parallel planes. Another characteristic is the surface concentration
of atoms, number per square centimeter (#/cm2), that are cut by a particular plane.
Again, a single-crystal semiconductor is not infinitely large and must terminate at
some surface. The surface density of atoms may be important, for example, in determining
how another material, such as an insulator, will ※fit§ on the surface of a semiconductor
material.
10 CHAPTER 1 The Crystal Structure of Solids
(a) (b)
a0 2
a0
a每
c每
b
每
a0
a0
a0
Figure 1.9 | (a) The (110) plane in a face-centered cubic and (b) the atoms cut by the (110)
plane in a face-centered cubic.
EXAMPLE 1.3
OBJECTIVE
Calculate the surface density of atoms on a particular plane in a crystal.
Consider the face-centered cubic structure and the (110) plane shown in Figure 1.9a.
Assume the atoms can be represented as hard spheres with the closest atoms touching each
other and that the lattice constant is a0 = 4.5 ～ A = 4.5 ℅ 10?8 cm. Figure 1.9b shows how the
atoms are cut by the (110) plane.
The atom at each corner is shared by four similar equivalent lattice planes, so each corner
atom effectively contributes one-fourth of its area to this lattice plane, as indicated in the
figure. The four corner atoms then effectively contribute one atom to this lattice plane. The
atom on each face plane is shared by two similar equivalent lattice planes, so each face atom
effectively contributes one-half of its area to this lattice plane as indicated in the figure. The
two face atoms then effectively contribute one atom to this lattice plane. The lattice plane in
Figure 1.9b, then, contains two atoms.
