CONTENTS
Preface xiii
PA R T 1
Materials 1
Chapter 1
Electron Energy and States in
Semiconductors 3
1.1 Introduction and Preview 3
1.2 A Brief History 4
1.3 Application to the Hydrogen Atom 5
1.3.1 The Bohr Model for the Hydrogen
Atom 5
1.3.2 Application to Molecules: Covalent
Bonding 11
1.3.3 Quantum Numbers and the Pauli
Exclusion Principle 13
1.3.4 Covalent Bonding in Crystalline
Solids 14
1.4 Wave-Particle Duality 20
1.5 The Wave Function 22
1.5.1 Probability and the Wave Function 22
1.6 The Electron Wave Function 23
1.6.1 The Free Electron in One
Dimension 23
1.6.2 The de Broglie Relationship 25
*1.6.3 The Free Electron in Three
Dimensions 26
1.6.4 The Quasi-Free Electron Model 27
1.6.5 Reflection and Tunneling 32
1.7 A First Look at Optical Emission and
Absorption 33
1.8 Crystal Structures, Planes,
and Directions 39
1.9 Summary 41
1.10 Reading List 42
1.11 References 42
1.12 Review Questions 42
1.13 Problems 43
Chapter 2
Homogeneous Semiconductors 48
2.1 Introduction and Preview 48
2.2 Pseudo-Classical Mechanics for Electrons
in Crystals 49
2.2.1 One-Dimensional Crystals 49
*2.2.2 Three-Dimensional Crystals 55
2.3 Conduction Band Structure 56
2.4 Valence Band Structure 58
2.5 Intrinsic Semiconductors 59
2.6 Extrinsic Semiconductors 62
2.6.1 Donors 62
2.6.2 Acceptors 66
2.7 The Concept of Holes 67
2.7.1 Hole Charge 67
*2.7.2 Effective Mass of Holes 69
2.8 Density-of-States Functions for Electrons
in Bands 71
2.8.1 Density of States and Density-of-States
Effective Mass 71
2.9 Fermi-Dirac Statistics 73
2.9.1 Fermi-Dirac Statistics for Electrons and
Holes in Bands 73
2.10 Electron and Hole Distributions with
Energy 76
*2.11 Temperature Dependence of Carrier
Concentrations in Nondegenerate
Semiconductors 89
*2.11.1 Carrier Concentrations at High
Temperatures 89
*2.11.2 Carrier Concentrations at Low
Temperatures (Carrier Freeze-out) 93
2.12 Degenerate Semiconductors 94
2.12.1 Impurity-Induced Band-Gap
Narrowing 94
2.12.2 Apparent Band-Gap Narrowing 97
2.12.3 Carrier Concentrations in Degenerate
Semiconductors 99
2.13 Summary 100
2.13.1 Nondegenerate Semiconductors 101
2.13.2 Degenerate Semiconductors 102
2.14 Reading List 103
2.15 References 103
2.16 Review Questions 103
2.17 Problems 104
Chapter 3
Current Flow in Homogeneous
Semiconductors 111
3.1 Introduction 111
3.2 Drift Current 111
3.3 Carrier Mobility 115
3.3.1 Carrier Scattering 119
3.3.2 Scattering Mobility 121
3.3.3 Impurity Band Mobility 122
3.3.4 Temperature Dependence
of Mobility 124
3.3.5 High-Field Effects 124
3.4 Diffusion Current 128
3.5 Carrier Generation and
Recombination 131
3.5.1 Band-to-Band Generation and
Recombination 133
3.5.2 Two-Step Processes 133
3.6 Optical Processes in Semiconductors 133
*3.6.1 Absorption 133
*3.6.2 Emission 137
vi Contents
3.7 Continuity Equations 139
3.8 Minority Carrier Lifetime 142
3.8.1 Rise Time 144
3.8.2 Fall Time 144
3.9 Minority Carrier Diffusion Lengths 147
3.10 Quasi Fermi Levels 149
3.11 Summary 152
3.12 Reading List 154
3.13 References 154
3.14 Review Questions 154
3.15 Problems 155
Chapter 4
Nonhomogeneous
Semiconductors 159
4.1 Constancy of the Fermi Level at
Equilibrium 159
4.2 Graded Doping 161
4.2.1 The Einstein Relation 165
4.2.2 A Graded-Base Transistor 166
*4.3 Nonuniform Composition 170
*4.4 Graded Doping and Graded Composition
Combined 173
4.5 Summary 175
4.6 Reading List 175
4.7 References 175
4.8 Review Questions 176
4.9 Problems 176
Supplement to Part 1
Materials 179
Supplement 1A
Introduction to Quantum
Mechanics 180
S1A.1 Introduction 180
S1A.2 The Wave Function 180
S1A.3 Probability and the Wave Function 182
*S1A.3.1 Particle in a One-Dimensional
Potential Well 182
Contents vii
S1A.4 Schroedinger¡¯s Equation 184
S1A.5 Applying Schroedinger¡¯s Equation to
Electrons 185
S1A.6 Some Results from Quantum
Mechanics 187
S1A.6.1 The Free Electron 187
S1A.6.2 The Quasi-Free Electron 188
S1A.6.3 The Potential Energy
Well 189
S1A.6.4 The Infinite Potential Well in
One Dimension 191
S1A.6.5 Reflection and Transmission at a
Finite Potential Barrier 194
S1A.6.6 Tunneling 196
S1A.6.7 The Finite Potential Well 203
S1A.6.8 The Hydrogen Atom
Revisited 205
S1A.6.9 The Uncertainty Principle 206
S1A.7 Summary 210
S1A.8 Review Questions 211
S1A.9 Problems 211
Supplement 1B
Additional Topics on Materials 215
S1B.1 Measurement of Carrier Concentration
and Mobility 215
S1B.1.1 Resistivity Measurement 215
S1B.1.2 Hall Effect 216
S1B.2 Fermi-Dirac Statistics for Electrons in
Bound States 219
S1B.3 Carrier Freeze-out in
Semiconductors 222
S1B.4 Phonons 223
*S1B.4.1 Carrier Scattering
by Phonons 228
S1B.4.2 Indirect Electron
Transitions 230
S1B.5 Summary 232
S1B.6 Reading List 232
S1B.7 References 232
S1B.8 Review Questions 232
S1B.9 Problems 233
PA R T 2
Diodes 235
Chapter 5
Prototype pn Homojunctions 239
5.1 Introduction 239
5.2 Prototype pn Junctions
(Qualitative) 241
5.2.1 Energy Band Diagrams of Prototype
Junctions 241
5.2.2 Description of Current Flow in a pn
Homojunction 248
5.3 Prototype pn Homojunctions
(Quantitative) 253
5.3.1 Energy Band Diagram at Equilibrium
(Step Junction) 253
5.3.2 Energy Band Diagram with Applied
Voltage 256
5.3.3 Current-Voltage Characteristics of
pn Homojunctions 263
5.3.4 Reverse-Bias Breakdown 284
5.4 Small-Signal Impedance of Prototype
Homojunctions 286
5.4.1 Junction Resistance 286
5.4.2 Junction Capacitance 288
5.4.3 Stored-Charge
Capacitance 290
5.5 Transient Effects 294
5.5.1 Turn-off Transient 294
5.5.2 Turn-on Transient 297
5.6 Effects of Temperature 301
5.7 Summary 301
5.7.1 Built-in Voltage 302
5.7.2 Junction Width 302
5.7.3 Junction Current 303
5.7.4 Junction Breakdown 304
5.7.5 Capacitance 305
5.7.6 Transient Effects 305
5.8 Reading List 305
5.9 Review Questions 306
5.10 Problems 306
viii Contents
Chapter 6
Additional Considerations
for Diodes 311
6.1 Introduction 311
6.2 Nonstep Homojunctions 311
*6.2.1 Linearly Graded Junctions 314
6.2.2 Hyperabrupt Junctions 317
6.3 Semiconductor Heterojunctions 317
6.3.1 The Energy Band Diagrams of
Semiconductor-Semiconductor
Heterojunctions 317
6.3.2 Effects of Interface States 327
*6.3.3 Effects of Lattice Mismatch on
Heterojunctions 329
6.4 Metal-Semiconductor Junctions 331
6.4.1 Ideal Metal-Semiconductor Junctions
(Electron Affinity Model) 331
6.4.2 Influence of Interface-Induced
Dipoles 331
6.4.3 The Current-Voltage Characteristics of
Metal-Semiconductor Junctions 334
6.4.4 Ohmic (Low-Resistance)
Contacts 337
6.4.5 I -Va Characteristics of Heterojunction
Diodes 339
*6.5 Capacitance in Nonideal Junctions and
Heterojunctions 339
6.6 Summary 340
6.7 Reading List 340
6.8 References 340
6.9 Review Questions 341
6.10 Problems 341
Supplement to Part 2
Diodes 346
S2.1 Introduction 346
S2.2 Dielectric Relaxation Time 346
S2.2.1 Case 1: Dielectric Relaxation Time for
Injection of Majority Carriers 347
S2.2.2 Case 2: Injection of Minority
Carriers 349
S2.3 Junction Capacitance 350
S2.3.1 Junction Capacitance in a Prototype
(Step) Junction 350
S2.3.2 Junction Capacitance in a Nonuniformly
Doped Junction 352
S2.3.3 Varactors 353
S2.3.4 Stored-Charge Capacitance of Short-
Base Diodes 354
S2.4 Second-Order Effects in Schottky
Diodes 356
S2.4.1 Tunneling Through Schottky
Barriers 357
S2.4.2 Barrier Lowering in Schottky Diodes
Due to the Image Effect 359
S2.5 SPICE Model for Diodes 361
S2.5.1 The Use of SPICE as a
Curve Tracer 362
S2.5.2 Transient Analysis 365
S2.6 Summary 368
S2.7 Reading List 368
S2.8 References 369
S2.9 Problems 369
PA R T 3
Field-Effect Transistors 373
Chapter 7
The MOSFET 385
7.1 Introduction 385
7.2 MOSFETs (Qualitative) 385
7.2.1 Introduction to MOS Capacitors 386
7.2.2 MOSFETs at Equilibrium
(Qualitative) 390
7.2.3 MOSFETs Not at Equilibrium
(Qualitative) 392
7.3 MOSFETs (Quantitative) 403
7.3.1 Long-Channel MOSFET Model with
Constant Mobility 404
7.3.2 More Realistic Long-Channel Models:
Effect of Fields on the Mobility 417
*7.3.3 Series Resistance 432
Contents ix
7.4 Comparison of Models with
Experiment 434
7.5 Summary 435
7.6 Reading List 438
7.7 References 438
7.8 Review Questions 438
7.9 Problems 439
Chapter 8
Additional Considerations
for FETs 442
8.1 Introduction 442
8.2 Measurement of Threshold Voltage and
Low-Field Mobility 443
8.3 Subthreshold Leakage Current 445
8.4 Complementary MOSFETs (CMOS) 448
8.4.1 Operation of the Inverter 449
*8.4.2 Matching of CMOS devices 450
8.5 Switching in CMOS Inverter
Circuits 452
8.5.1 Effect of Load Capacitance 452
8.5.2 Propagation (Gate) Delay in Switching
Circuits 454
8.5.3 Pass-through Current in CMOS
Switching 457
8.6 MOSFET Equivalent Circuit 457
8.6.1 Small-Signal Equivalent
Circuit 458
8.6.2 CMOS Amplifiers 463
8.7 Unity Current Gain Cutoff
Frequency fT 463
*8.8 Short-Channel Effects 464
8.8.1 Dependence of Effective Channel
Length on VDS 464
8.8.2 Dependence of Threshold Voltage on
the Drain Voltage 466
8.9 MOSFET Scaling 467
8.10 Silicon on Insulator (SOI) 469
8.11 Other FETs 473
8.11.1 Heterojunction Field-Effect
Transistors (HFETs) 473
8.11.2 MESFETs 476
8.11.3 Junction Field-Effect Transistors
(JFETs) 481
8.11.4 Bulk Channel FETs:
Quantitative 482
8.12 Summary 485
8.13 Reading List 486
8.14 References 486
8.15 Review Questions 487
8.16 Problems 487
Supplement to Part 3
Field-Effect Transistors 491
S3.1 Introduction 491
S3.2 Comments on the Formulation for the
Channel Charge Qch 491
S3.2.1 Effect of Varying Depletion Width on
the Channel Charge 491
S3.2.2 Dependence of the Channel
Charge Qch on the Longitudinal
Field L 493
S3.3 Threshold Voltage for MOSFETs 495
S3.3.1 Fixed Charge 497
S3.3.2 Interface Trapped Charge 497
S3.3.3 Bulk Charge 498
S3.3.4 Effect of Charges on the Threshold
Voltage 498
S3.3.5 Flat Band Voltage 499
S3.3.6 Threshold Voltage Control 502
*S3.3.7 Channel Quantum Effects 504
S3.4 Universal Relations for Low-Field
Mobility 507
S3.5 Measurement of VT 509
*S3.6 Alternative Method to Determine VT and
¦Ìlf Applicable to Long-Channel
MOSFETs 513
S3.7 MOS Capacitors 514
S3.7.1 Ideal MOS Capacitance 515
S3.7.2 The C-VG Characteristics of Real
MOS Capacitors 520
S3.7.3 Parameter Analyses from C-VG
Measurements 521
x Contents
*S3.8 MOS Capacitor Hybrid
Diagrams 521
*S3.8.1 Dynamic Random-Access Memories
(DRAMs) 525
*S3.8.2 Charge-Coupled Devices
(CCDs) 527
*S3.9 Device Degradation 530
*S3.9.1 Lightly Doped Drain (LDD)
MOSFETs 534
*S3.10 Low-Temperature Operation of
MOSFETs 535
*S3.11 Applications of SPICE to
MOSFETs 538
S3.11.1 Examples of the Use of SPICE with
MOSFETs 539
S3.11.2 Determining the Transient
Characteristics of a CMOS Digital
Inverter 543
S3.12 Summary 545
S3.13 Reading List 546
S3.14 References 546
S3.15 Review Questions 547
S3.16 Problems 547
PA R T 4
Bipolar Junction
Transistors 551
Chapter 9
Bipolar Junction Devices:
Statics 557
9.1 Introduction 557
9.2 Output Characteristics
(Qualitative) 561
9.3 Current Gain 563
9.4 Model of a Prototype BJT 564
9.4.1 Collection Efficiency M 567
9.4.2 Injection Efficiency ¦Ã 568
9.4.3 Base Transport Efficiency ¦ÁT 570
9.5 Doping Gradients in BJTs 575
9.5.1 The Graded-Base Transistor 578
9.5.2 Effect of Base Field on ¦Â 582
9.6 The Basic Ebers-Moll DC Model 583
9.7 Current Crowding and Base Resistance
in BJTs 586
9.8 BaseWidth Modulation (Early Effect) 590
9.9 Avalanche Breakdown 594
9.10 High Injection 594
9.11 Base Push-out (Kirk) Effect 595
9.12 Recombination in the Emitter-Base
Junction 597
9.13 Summary 598
9.14 Reading List 599
9.15 References 599
9.16 Review Questions 600
9.17 Problems 601
Chapter 10
Time-Dependent Analysis
of BJTs 607
10.1 Introduction 607
10.2 Ebers-Moll AC Model 607
10.3 Small-Signal Equivalent Circuits 609
10.3.1 Hybrid-Pi Models 611
10.4 Stored-Charge Capacitance
in BJTs 615
10.5 Frequency Response 620
10.5.1 Unity Current Gain
Frequency fT 621
10.5.2 Base Transit Time 623
10.5.3 Base-Collector Transit Time, tBC 624
10.5.4 Maximum Oscillation
Frequency fmax 625
10.6 High-Frequency Transistors 625
10.6.1 Double Poly Si Self-Aligned
Transistor 625
10.7 BJT Switching Transistor 628
10.7.1 Output Low-to-High Transition
Time 629
Contents xi
10.7.2 Schottky-Clamped
Transistor 631
10.7.3 Emitter-Coupled Logic 632
10.8 BJTs, MOSFETs, and BiMOS 635
10.8.1 Comparison of BJTs and
MOSFETs 635
10.8.2 BiMOS 636
10.9 Summary 638
10.10 Reading List 639
10.11 References 639
10.12 Review Questions 639
10.13 Problems 639
Supplement to Part 4
Bipolar Devices 642
S4.1 Introduction 642
S4.2 Heterojunction Bipolar
Transistors (HBTs) 642
S4.2.1 Uniformly Doped HBT 644
S4.2.2 Graded-Composition HBT 646
S4.3 Comparison of Si-Base, SiGe-Base, and
GaAs-Base HBTs 649
S4.4 Thyristors (npnp Switching
Devices) 650
S4.4.1 Four-Layer Diode Switch 650
S4.4.2 Two-Transistor Model of an npnp
Switch 652
S4.5 Silicon Controlled Rectifiers (SCRs) 654
S4.6 Parasitic pnpn Switching in CMOS
Circuits 658
S4.7 Applications of SPICE to BJTs 658
S4.7.1 Parasitic Effects 661
S4.7.2 Low to Medium Currents 661
S4.7.3 High Currents 663
S4.8 Examples of the Application of SPICE
to BJTs 664
S4.9 Summary 669
S4.10 References 670
S4.11 Review Questions 670
S4.12 Problems 671
PA R T 5
Optoelectronic Devices 673
Chapter 11
Optoelectronic Devices 675
11.1 Introduction and Preview 675
11.2 Photodetectors 675
11.2.1 Generic Photodetector 675
*11.2.2 Solar Cells 683
11.2.3 The p-i-n (PIN) Photodetector 689
11.2.4 Avalanche Photodiodes 691
11.3 Light-Emitting Diodes 692
11.3.1 Spontaneous Emission in a Forward-
Biased Junction 692
*11.3.2 Isoelectronic Traps 694
11.3.3 Blue LEDs and White
LEDs 696
11.3.4 Infrared LEDs 696
11.4 Laser Diodes 702
11.4.1 Optical Gain 703
11.4.2 Feedback 706
11.4.3 Gain + Feedback Laser 709
11.4.4 Laser Structures 710
11.4.5 Other Semiconductor Laser
Materials 714
11.5 Image Sensors 715
11.5.1 Charge-Coupled Image
Sensors 715
11.5.2 MOS Image Sensors 717
11.6 Summary 718
11.7 Reading List 719
11.8 References 719
11.9 Review Questions 719
11.10 Problems 720
Appendices
Appendix A Constants 724
Appendix B List of Symbols 725
PREFACE
This is a textbook on the operating principles of semiconductor devices. It
is appropriate for undergraduate (junior or senior) or beginning graduate
students in electrical engineering as well as students of computer engineering,
physics, and materials science. It is also useful as a reference for practicing
engineers and scientists who are involved with modern semiconductor
devices.
Prerequisites are courses in chemistry and physics and in basic electric circuits,
which are normally taken in the freshman and sophomore years.
This text is appropriate for a two-semester course on semiconductor devices.
However, it can be used for a one-semester course by eliminating some of the
more advanced material and assigning some of the sections as ¡°read only.¡± The
authors have attempted to organize the material so that some of the detailed
derivation sections can be skipped without affecting the comprehension of other
sections. Some of these sections are marked with an asterisk.
This book is divided into five parts:
1. Materials
2. Diodes
3. Field-Effect Transistors
4. Bipolar Transistors
5. Optoelectronic Devices
The first four parts are followed by ¡°Supplements¡± that, while not required
for an understanding of the basic principles of device operation, contain related
material that may be assigned at the discretion of the instructor. For example, the
use of SPICE for device and circuit analysis is briefly discussed for diodes, fieldeffect
transistors, and bipolar transistors. While SPICE is normally taught in
courses on electric circuits, it is useful to know the origin of the various parameters
used to characterize devices. This material on SPICE is relegated to supplements,
since not all schools cover SPICE in courses on electron circuit analysis
and such courses may be taught before, concurrently with, or after the course on
semiconductor devices.
Part 1, ¡°Materials,¡± contains four chapters and two supplements. The first
two chapters contain considerable review material from the prerequisite
courses. This material is included since it is used extensively in later chapters to
explain the principles of device operations. Depending on the detailed content
of the prerequisite courses, much of these chapters can be relegated to reading
assignments.
xiii
xiv Preface
The level of quantum mechanics to be covered in a course like this varies
widely. In this book some basic concepts are included in the main chapters of
Part 1; those wishing to cover quantum mechanics in more detail will find more
extensive material in Supplement A to Part 1.
The basic operating principles of large and small devices of a particular type
(e.g., diodes, field-effect transistors, bipolar junction transistors, photodetectors)
are the same. However, the relative importance of many of the parameters involved
in device operation depends on the device dimensions. In this book the
general behavior of devices of large dimensions is treated first. We treat, in each
case, ¡°prototype¡± devices (such as step junctions and long-channel field-effect
transistors), from which the fundamental physics can be learned, and then develop
more realistic models considering ¡°second-order effects.¡± These secondorder
effects can have significant influence on the electrical characteristics of
modern, small-geometry devices. The instructor can go into as much depth as desired
or as time permits.
Topics treated that are typically omitted in undergraduate texts are:
¡ö The differences between the electron and hole effective masses as used in
density-of-state calculations and conductivity calculations.
¡ö The differences in electron and hole mobilities (and thus diffusion
coefficients) if they are majority carriers or minority carriers.
¡ö The effects of doping gradients in the base of bipolar junction transistors
(and/or the composition in heterojunction BJTs) on the current gain and
switching speed.
¡ö Band-gap reduction in degenerate semiconductors. While this has little
effect on the electrical characteristics of diodes or field-effect transistors,
its effect in the emitter of bipolar junction transistors reduces the current
gain by an order of magnitude.
¡ö The velocity saturation effects due to the longitudinal field in the channel
of modern field-effect transistors with submicrometer channel lengths
reduces the current by an order of magnitude compared with that calculated
if this effect is neglected.
While the major emphasis is on silicon and silicon devices, the operation of
compound semiconductor devices, alloyed devices (e.g., SiGe, AlGaAs) and
heterojunction devices (junctions between semiconductors of different composition)
are also considered because of the increased performance that is possible
with such band-gap engineering.
Many of the seminal publications on semiconductor devices cited in the references
at the end of each chapter through 1990 are reprinted in Semiconductor
Devices: Pioneering Papers, edited by S. M. Sze, World Scientific Publishing
Co., Singapore, 1991.
Fabrication, while an important part of semiconductor engineering, is often
skipped in the interest of time. This material is introduced in Appendix C, and
can be assigned as read-only material if desired.
Preface xv
ACKNOWLEDGEMENTS
We would like to thank, first and foremost, our spouses Bill and Claire for their
love, support, patience, and help. We are also grateful to the anonymous manuscript
reviewers for their comments and suggestions, as well as the staff at
McGraw-Hill for all their help. We thank our students for valuable feedback on
the manuscript. Finally, we would like to thank all the companies and individuals
that provided photographs and data for this book.
Anderson & Anderson
P A R T 1
Materials
Semiconductors form the basis of most modern electronic systems (e.g.,
computers, communication networks, control systems). While there are
applications for other materials in electronics (e.g., magnetic materials
in hard drives), this book concentrates on electronic devices that are based on
semiconductors.
Understanding the operation and design of semiconductor devices begins
with an understanding of the materials involved. In Part 1 of this book, we investigate
the behavior of electrons in materials, starting with the atoms themselves.
Then we progress to electrons in crystalline semiconductors.
We will see that classical mechanics does not provide a complete picture of
electron activity in solids. In principle one should instead use quantum mechanics
to predict the electrons¡¯ behavior, but the application of quantum mechanics
is not as simple as the more familiar classical or Newtonian mechanics. We will
therefore introduce pseudo-classical mechanics, which modifies familiar classical
equations to account for some quantum mechanical effects.
Some basic quantum mechanical concepts important for the understanding of
device operation are covered in Chapter 1. (A more thorough discussion is contained
in Supplement A to Part 1, found after Chapter 4.) In Chapter 2, we cover
pseudo-classical mechanics, which allows us to predict the reaction of electrons
to complicated fields, while using simple and intuitive pseudo-classical equations.
The use of pseudo-classical mechanics will also allow us to draw and use
energy band diagrams. These diagrams are indispensable for understanding and
predicting the motion of the electrons and holes, and thus the current in
semiconductors.
In Chapter 3, we will see that conductivity of semiconductors is controlled
by the number of charge carriers available to carry current. The charge carriers in
semiconductors are electrons and holes. Their numbers are controlled by the concentrations
of impurity elements that are intentionally added to the material. The
carrier concentrations also depend on temperature, and if light is shining on the
sample, the concentrations can also vary.
2 PART 1 Materials
It will emerge that there are two major forms of current in semiconductors,
drift current and diffusion current. Drift current is caused by the presence of an
electric field, whereas diffusion current arises when the carrier concentrations
vary with position.
Chapter 4 covers nonhomogeneous semiconductor materials, in which the
doping or the material composition itself may vary. These variations can lead to
internal electric fields that can enhance device performance. Most modern semiconductor
devices have regions of such nonhomogeneous materials.
The Supplements to Part 1 contain additional topics relevant to semiconductor
materials, including a more detailed discussion of quantum mechanics, the
statistics of electrons in bound states, and phonons.
We will start with electrons in atoms. ¡ö
C H A P T E R 1
Electron Energy and
States in Semiconductors
1.1 INTRODUCTION AND PREVIEW
We begin our study of semiconductors with some fundamental physics of how
electrons behave in matter. The ability to control the movement of electrons in
solids is the basis of semiconductor device engineering. In order to understand
the electronic properties of these devices, it is necessary to understand the electronic
properties of the materials from which they are made, and how those properties
are affected by impurities (intentional and unintentional), temperature,
applied voltages, device structures, and optical radiation.
Since solids are composed of atoms, we start by examining the electronic
properties of atoms, and then extending those results to simple molecules and
solids. In particular, the results for silicon (Si) and gallium arsenide (GaAs) are
emphasized, currently the two most commonly used semiconductors in integrated
circuits and semiconductor devices. Several other semiconductors and
semiconductor alloys important in modern devices are also discussed.
As we investigate the atom, we¡¯ll be using quantum mechanics, a branch of
science that is needed to accurately describe the behavior of very small objects
such as atoms and electrons.We will see as we go along that quantum mechanics
is based on the idea that energy can exist only in discrete packets, or quanta. The
size of a quantum is so small that it doesn¡¯t affect one¡¯s results when one is computing
the momentum or velocity of large objects such as automobiles or dust particles,
but the quantum description is extremely important for electrons and atoms.
An understanding of quantum mechanics is not simple to obtain, and its use
to calculate properties of more than a few systems in closed form is difficult. Fortunately,
however, in semiconductors the behavior of electrons of interest can be
determined by pseudo-classical mechanics, in which classical formulas such as
Newton¡¯s laws and the Lorentz equation can be used, with the true electron mass
replaced by an effective mass. As a result, in this section, a minimal discussion of
3
4 PART 1 Materials
Figure 1.1 (a) The Thompson model of an atom, in which the positive charge is
uniformly distributed in a sphere and the electrons are considered to be negative
point charges embedded in it; (b) the Bohr model, in which the positive charge
is concentrated in a small nucleus and the electrons orbit in circles; (c) the
Wilson-Sommerfeld model, which allows for elliptical orbits.
(a)
(b)
(c)
quantum mechanics is presented. A somewhat greater discussion of quantum
mechanics appropriate to some of the electronic processes in semiconductor
devices is presented in Supplement A to Part 1, after Chapter 4.
The key to understanding semiconductors is to appreciate the physical interpretation
of the mathematical results. Physical understanding is emphasized in
this book.
1.2 A BRIEF HISTORY
In the early twentieth century, scientists were trying to develop models that would
explain the results observed from such experiments as the scattering of X-rays, the
photoelectric effect, and the emission and absorption spectra of atoms. In 1910,
J. J. Thompson proposed a model of the atom in which a sphere of continuous
positive charge is embedded with electrons, as shown in Figure 1.1a. Ernest
Rutherford, in 1911, offered an improvement to the Thompson model: in the
Rutherford model of the atomic structure, all of the positive charge and virtually all
of the atom¡¯s mass were assumed to be concentrated in a small region in the center
of the atom. This nucleus is often treated as a sphere with a radius on the order of
10?14 meters. The negatively charged electrons were assumed to orbit about the
positively charged nucleus, much as planets orbit the sun or satellites orbit the
earth.
In 1913, Neils Bohr assumed that the electrons in the Rutherford model of the
atom orbited the nucleus in circles, as shown in Figure 1.1b. From this, he predicted
that for the atom to be stable, the electrons could have only certain energies,
or that the energies would be quantized. Energy and many other observables
(properties that can be directly measured) are expressed in terms of Planck¡¯s constant.
Planck¡¯s constant, h, has the value 6.63 ¡Á 10?34 joule¨Cseconds. The energies
Bohr predicted for electrons in atoms were in excellent agreement with the
experimental results obtained from spectroscopic data.
CHAPTER 1 Electron Energy and States in Semiconductors 5
In 1916, Wilson and Sommerfeld generalized the Bohr model to apply it to
any physical system in which a particle¡¯s motion is periodic with time. This modification
allows for the possibility of elliptical orbits, as shown in Figure 1.1c.
1.3 APPLICATION TO THE HYDROGEN ATOM
In this section, we briefly review the Bohr model of the hydrogen atom. The
hydrogen atom is emphasized because hydrogen-like impurities are important in
semiconductor devices, and these impurities can be treated in a manner analogous
to the Bohr model. In Supplement Ato Part 1, we will compare these results
to those obtained using quantum mechanics as represented by Schroedinger¡¯s
equation.
1.3.1 THE BOHR MODEL FOR THE HYDROGEN ATOM
We start with the Bohr model, in which the electrons revolve around the nucleus
in circular paths. Because the mass of the nucleus is 1.67 ¡Á 10?27 kg, 1830 times
that of the electron, the nucleus is considered to be fixed in space.
We consider as an example the neutral hydrogen atom, which has one orbiting
electron, and we treat the electron and nucleus both as point charges. The
Coulomb force between two particles with charges Q1 and Q2 is
F =
Q1Q2
4¦Ð¦Å0r 2 = ?q2
4¦Ð¦Å0r 2 (1.1)
where r is the distance between the two charges and ¦Å0 = 8.85 ¡Á 10?12 farads/
meter is the permittivity of free space (because there is only free space between
the nucleus and the electron). The expression at the far right-hand side of
Equation (1.1) is obtained by recognizing that the hydrogen nucleus has only one
proton, so Q1 is equal to +q = 1.602 ¡Á 10?19 coulombs, the elemental charge,
and the charge of the electron Q2 is equal to ?q. The resulting negative sign in
Equation (1.1) indicates that the force is attractive.
We now have an expression for the attractive (centripetal) force between the
two particles, and we recall from classical mechanics that the force F on a particle
is equal to minus the gradient of the potential energy, or
F = ??EP = ?
dEP
dr
(1.2)
In the last expression, the gradient is taken in the r direction, and EP is the potential
energy of the electron at position r. Equation (1.2) with the aid of (1.1) can be
rewritten as
dEP = dEP(r ) = ?Fdr =
q2 dr
4¦Ð¦Å0r 2 (1.3)
One can integrate both sides to obtain EP, but there will be a constant of integration.
The actual value of the potential energy is arbitrary (as is the choice of
6 PART 1 Materials
Figure 1.2 Potential energy diagram
for an electron in the vicinity of a
single positive point charge. The
electron is considered to be a point
charge.
Evac
EP
F F
r
r 0
the constant), since the value of the potential energy depends entirely on one¡¯s
choice of reference. We can choose a convenient reference by noting that the
Coulomb force at infinite distance is zero. It makes sense for this case, then, to
choose r =¡Þ as a reference point, so we define the potential energy at r =¡Þ as the vacuum level, Evac:
EP(r =¡Þ) = Evac (1.4)
This is the energy required to free the electron from the influence of the nucleus,
essentially by moving the electron infinitely far away from it. If the electron is in-
finitely far from the nucleus, it cannot really be considered part of the atom¡ªit
is now a free electron in vacuum.
Now we can solve Equation (1.3) for a given value of r:
Evac
EP
dEP =
¡Þ
r
q2 dr
4¦Ð¦Å0r 2 (1.5)
where EP is the electron potential energy at some distance r from the nucleus.
Integrating both sides and rearranging, we obtain
EP = Evac ?
q2
4¦Ð¦Å0r
(1.6)
Figure 1.2 shows a plot of the r dependence of EP. From Equation (1.1), and
since the force is equal to minus the gradient (slope) of the potential energy, we
see that the force on the electron is directed toward the nucleus, or the Coulomb
force is centripetal.
Since the electron is revolving in a circle of radius r around the nucleus, we
know from Newtonian mechanics that its centrifugal force is equal to
F =
mv2
r
(1.7)
CHAPTER 1 Electron Energy and States in Semiconductors 7
For the atom to be stable, the net force on the electron must be zero. Equating
our previous expression for the centripetal force due to the Coulomb attraction
[Equation (1.1)] to the centrifugal force [Equation (1.7)], we can write
mv2
r ?
q2
4¦Ð¦Å0r 2 = 0 (1.8)
Bohr also postulated that the integral of the angular momentum around one
complete orbit is an integer multiple of Planck¡¯s constant h:
P¦È d¦È =
2¦Ð
0
mvr d¦È = nh (1.9)
where n is an integer. Since the orbit is assumed circular in the Bohr model, r is
a constant, and so are the potential energy EP and the speed v. Therefore, the
integral becomes
2¦Ðmrv = nh (1.10)
There is a solution for each integer value of n, so we write
mvnrn = n
h
2¦Ð = n¡¥h (1.11)
Here we have introduced a new symbol; it turns out that engineers and physicists
(and now you) use the quantity h/2¦Ð so much that there is a special character for
it, ¡¥h, pronounced ¡°h-bar.¡± The subscripts n in Equation (1.11) indicate the particular
orbital radius or speed associated with a specific quantum number n.
If we simultaneously solve Equations (1.8) and (1.11), we can derive an
expression for the Bohr radius of the nth state, where by ¡°state¡± we mean the
properties associated with a particular value of n:
rn =
4¦Ð¦Å0n2¡¥h2
mq2 (1.12)
and the speed of the electron in that particular state is
vn =
q2
4¦Ð¦Å0n¡¥h
(1.13)
Our primary goal, however, is to find the energies associated with these
states. We know that the total energy of a system is equal to the kinetic energy
plus the potential energy. The kinetic energy of the nth energy level is
EKn =
1
2
mv2
n =
mq4
2(4¦Ð¦Å0)2n2¡¥h2 (1.14)
8 PART 1 Materials
EXAMPLE 1.1
For the nth energy level, we can find rn from Equation (1.12) and use that in
Equation (1.6) to write for the potential energy
EPn = Evac ?
mq4
2(4¦Ð¦Å0)2n2¡¥h2 EPn = Evac ?
mq4
(4¦Ð¦Å0)2n2¡¥h2 (1.15)
Thus, the total energy En is
En = EKn + EPn = Evac ?
mq4
2(4¦Ð¦Å0)2n2¡¥h2 (1.16)
We say that the energy is quantized. It can have only discrete values associated
with the quantum number n.
Find the energies and radii for the first four orbits in the hydrogen atom.
¡ö Solution
En = Evac ?
mq4
2(4¦Ð¦Å0)2n2¡¥h2
= Evac ?
(9.11 ¡Á 10?31 kg)(1.60 ¡Á 10?19 C)4
(2)(4)2(3.1416)2(9.85 ¡Á 10?12 F/m)2(1.05 ¡Á 10?34 J ¡¤ s)2 1
n2
(1.17)
En = Evac ? 1
n2
(2.18 ¡Á 10?17 J)
= Evac ? 1
n2
(13.6 eV)
Here a new unit of energy is introduced, the electron volt (eV). The electron volt
is defined as the amount of energy acquired by an electron when it is accelerated
through 1 volt of electric potential. To convert between S.I. (International System)
units ( joules) and electron volts, use
Electron volts are not S.I. units, and therefore must be used with care in calculations.
The Bohr radii can be calculated from Equation (1.12):
rn =
4¦Ð¦Å0n2¡¥h2
mq2 =
(4)(3.1416)(8.85 ¡Á 10?12 F/m)(1.05 ¡Á 10?34 J ¡¤ s)2
(9.11 ¡Á 10?31 kg)(1.60 ¡Á 10?19 C)2 ¡Á n2
rn = 0.053n2 nm
(1.18)
The energies and Bohr radii of the first four energy levels are given in Table 1.1. These
energies and radii are plotted in Figures 1.3 and 1.4, respectively.
1 eV = 1.60 ¡Á 10?19 joules
CHAPTER 1 Electron Energy and States in Semiconductors 9
r1 0.053
r2 0.212 r3 0.477 r4 0.848 nm
Radius
Figure 1.4 Radii of the first four atomic orbits of the hydrogen atom,
according to the Bohr model.
Electron energy
E Evac
E4 Evac 0.85 eV
E3 Evac 1.5 eV
E2 Evac 3.4 eV
E1 Evac 13.6 eV
Ground state
Figure 1.3 Allowed energies in the hydrogen
atom. Higher energies occur increasingly close to
each other, approaching the vacuum level.
Table 1.1 The first four Bohr energies and orbital radii for
the hydrogen atom
En rn
E1 = Evac ? 13.6 eV r1 = 0.053 nm
E2 = Evac ? 3.4 eV r2 = 0.212 nm
E3 = Evac ? 1.51 eV r3 = 0.477 nm
E4 = Evac ? 0.85 eV r4 = 0.848 nm
10 PART 1 Materials
There are several things to notice in this example. First, the differences between
the vacuum energy level and the allowed energies vary as 1/n2. Thus the
higher the quantum number, and therefore the energy, the closer together (in
energy) the energy levels are. Second, the Bohr radius varies as n2. This means
that the higher the energy level, the farther the electron is from the atomic
nucleus. If the electron has energy greater than Evac, the Coulombic force is not
enough to keep the electron bound to the atom. In this case, a hydrogen ion (H+)
is created as the electron leaves the atom.
Also, notice that we do not give a number for the energy of a particular state,
but rather we express the energies as so many electron volts from some reference
level (in this case Evac). It is pointless to say, ¡°This level is at 10 eV,¡± since 10 eV
could be anywhere, depending on your choice of reference. This point cannot be
emphasized enough. Potential energy and thus total energy are arbitrary.
EXAMPLE 1.2
1It is called a quantum well because the potential energy forms a ¡°well¡± with quantized energy states.
The energy of a state must always be expressed as an energy difference¡ªthe
difference between the energy of the state and some known reference, for example,
Evac ? E.
Finally, it should be pointed out that although the number of possible states
is infinite, once an electron occupies one of these states the hydrogen atom becomes
neutral and other electrons are not attracted.
Consider a particle in a one-dimensional universe, oscillating in the parabolic potential
energy shown in Figure 1.5. This represents an approximation to an electron in a
modern quantum well laser.1 The potential energy function is a parabola, and the particle
is a simple harmonic oscillator. Explain the motion of the particle using the
energy diagram, paying attention to where the kinetic energy is largest, smallest, and
the directions of the forces.
¡ö Solution
Conservation of energy dictates that the particle must remain at a constant energy.
Thus, it oscillates back and forth at this particular energy. When the particle is in the
center, it has the smallest potential energy and the largest kinetic energy, and thus the
largest velocity. Recall that the force is equal to the negative of the slope of the potential
energy [Equation (1.2)]. Therefore, as the particle travels through the center, e.g.,
to the right, the force is to the left (the slope is positive to right of center). Thus, the
particle decelerates. It continues to slow as it moves to the right. The total energy is
constant, but since the potential energy is increasing, the kinetic energy decreases.
When the particle gets to the edge, it will have zero kinetic energy. The particle stops
for an instant, but the force is still accelerating it to the left. It picks up speed until it
passes through the center, at which point the force is to the right, and the particle begins
to slow once again.
1It is called a quantum well because the potential energy forms a ¡°well¡± with quantized energy states.
