Fundamentals Of Semiconductors Physics And Materials Properties Advanced Texts In Physics

When the film is deposited on a substrate of similar structure but different chemical composition such as a GaAs film on a Si substrate, the growth process is known as hetero-epitaxy.. Fig

Fundamentals of Semiconductors

Peter Y Yu Manuel Cardona

Fundamentals

of Semiconductors Physics and Materials Properties

Third, Revised and Enlarged Edition

With 250 Two-Color Figures,

52 Tables and 116 Problems

123

Professor Dr Peter Y Yu

University of California, Department of Physics

CA 94720-7300 Berkeley, USA

email: pyyu@lbl.gov

Professor Dr., Dres h.c Manuel Cardona

Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1

70569 Stuttgart, Germany

email: cardona@cardix.mpi-stuttgart.mpg.de

3rd, Corrected Printing 2005

ISBN 3-540-25470-6

Springer Berlin Heidelberg New York

ISBN 3-540-41323-5 3rd Edition, 2nd Corrected Printing

Springer Berlin Heidelberg New York

Library of Congress Cataloging-in-Publication Data.

Yu, Peter Y., 1944 – Fundamentals of semiconductors: physics and materials properties /Peter Y Yu, Manuel Cardona – 3rd, rev and enlarged ed p cm Includes bibliographical references and index ISBN 3540413235 (alk paper) 1 Semiconductors 2 Semiconductors–Materials I Cardona, Manuel, 1934 – QC611.Y88 2001 537.6'22–dc21 2001020462

This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication

or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965,

in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable for prosecution under the German Copyright Law.

Springer is a part of Springer Science+Business Media

Cover picture: The crystal structure drawn on the book cover is a “wallpaper stereogram” Such stereograms

are based on repeating, but offset, patterns that resolve themselves into different levels of depth when viewed properly They were first described by the English physicist Brewster more than 100 years ago See:

Superstereograms (Cadence Books, San Francisco, CA, 1994)

Production editor: C.-D Bachem, Heidelberg

Typesetting: EDV-Beratung F Herweg, Hirschberg

Computer-to-plate and printing: Mercedes-Druck, Berlin

Binding: Stein+Lehmann, Berlin

The support for our book has remained high and compliments from readersand colleagues have been most heart-warming We would like to thank all ofyou, especially the many students who have continued to send us their com-ments and suggestions We are also pleased to report that a Japanese transla-tion appeared in 1999 (more details can be obtained from a link on our Web

site: http://pauline.berkeley.edu/textbook) Chinesea) and Russian translationsare in preparation

Semiconductor physics and material science have continued to prosper and

to break new ground For example, in the years since the publication of thefirst edition of this book, the large band gap semiconductor GaN and relatedalloys, such as the GaInN and AlGaN systems, have all become important ma-terials for light emitting diodes (LED) and laser diodes The large scale pro-duction of bright and energy-efficient white-light LED may one day changethe way we light our homes and workplaces This development may even im-pact our environment by decreasing the amount of fossil fuel used to produceelectricity In response to this huge rise in interest in the nitrides we haveadded, in appropriate places throughout the book, new information on GaNand its alloys New techniques, such as Raman scattering of x-rays, have givendetailed information about the vibrational spectra of the nitrides, availableonly as thin films or as very small single crystals An example of the progress

in semiconductor physics is our understanding of the class of deep defect ters known as the DX centers During the preparation of the first edition, thephysics behind these centers was not universally accepted and not all its pre-dicted properties had been verified experimentally In the intervening yearsadditional experiments have verified all the remaining theoretical predictions

cen-so that these deep centers are now regarded as cen-some of the best understooddefects It is now time to introduce readers to the rich physics behind thisimportant class of defects

The progress in semiconductor physics has been so fast that one problem

we face in this new edition is how to balance the new information with the oldmaterial In order to include the new information we had either to expand thesize of the book, while increasing its price, or to replace some of the existingmaterial by new sections We find either approach undesirable Thus we havecome up with the following solution, taking advantage of the Internet in this

a The Chinese version was published in 2002 by Lanzhou University Press (see

www.onbook.com.cn)

VI Preface to the Third Edition

new information age We assume that most of our readers, possibly all, are

“internet-literate” so that they can download information from our Web site.Throughout this new edition we have added the address of Web pages whereadditional information can be obtained, be this new problems or appendices

on new topics With this solution we have been able to add new informationwhile keeping the size of the book more or less unchanged We are sure theowners of the older editions will also welcome this solution since they canupdate their copies at almost no cost

Errors seem to decay exponentially with time We thought that in the ond edition we had already fixed most of the errors in the original edition.Unfortunately, we have become keenly aware of the truth contained in thistimeless saying: “to err is human” It is true that the number of errors discov-ered by ourselves or reported to us by readers has dropped off greatly sincethe publication of the second edition However, many serious errors still re-mained, such as those in Table 2.25 In addition to correcting these errors inthis new edition, we have also made small changes throughout the book toimprove the clarity of our discussions on difficult issues

sec-Another improvement we have made in this new edition is to add manymore material parameters and a Periodic Table revealing the most commonelements used for the growth of semiconductors We hope this book will benot only a handy source for information on topics in semiconductor physicsbut also a handbook for looking up material parameters for a wide range ofsemiconductors We have made the book easier to use for many readers whoare more familiar with the SI system of units Whenever an equation is dif-ferent when expressed in the cgs and SI units, we have indicated in red thedifference In most cases this involves the multiplication of the cgs unit equa-tion by(4Â0)⫺1 whereÂ0is the permittivity of free space, or the omission of

a factor of (1/c) where c is the speed of light.

Last but not least, we are delighted to report that the Nobel Prize inPhysics for the year 2000 has been awarded to two semiconductor physicists,Zhores I Alferov and Herbert Kroemer (“for developing semiconductor het-erostructures used in high-speed- and opto-electronics”) and a semiconductordevice engineer, Jack S Kilby (“for his part in the invention of the integratedcircuit”)

We have so far received many comments and feedback on our book from allquarters including students, instructors and, of course, many friends We aremost grateful to them not only for their compliments but also for their valu-able criticism We also received many requests for an instructor manual andsolutions to the problems at the end of each chapter We realize that semicon-ductor physics has continued to evolve since the publication of this book andthere is a need to continue to update its content To keep our readers informed

of the latest developments we have created a Web Page for this book Its

ad-dress (as of the writing of this preface) is: http://pauline.berkeley.edu/textbook.

At this point this Web Page displays the following information:

1) Content, outline and an excerpt of the book

2) Reviews of the book in various magazines and journals

3) Errata to both first and second printing (most have been corrected inthe second edition as of this date)

4) Solutions to selected problems

5) Additional supplementary problems

The solutions in item (4) are usually incomplete They are supposed to serve

as helpful hints and guides only The idea is that there will be enough leftfor the students to do to complete the problem We hope that these solutionswill satisfy the need of both instructors and students We shall continue to addnew materials to the Web Page For example, a list of more recent references

is planned The readers are urged to visit this Web Page regularly to find outthe latest information Of course, they will be welcomed to use this Web Page

to contact us

While the present printing of this book was being prepared, the 1998 national Conference on the Physics of Semiconductors (ICPS) was being held

Inter-in Jerusalem (Israel) It was the 24th Inter-in a biannual series that started Inter-in 1950

in Reading (U.K.), shortly after the discovery of the transistor by Shockley,Bardeen and Brattain in 1948 The ICPS conferences are sponsored by the In-ternational Union of Pure and Applied Physics (IUPAP) The proceedings ofthe ICPS’s are an excellent historical record of the progress in the field andthe key discoveries that have propelled it Many of those proceedings appear

in our list of references and, for easy identification, we have highlighted inred the corresponding entries at the end of the book A complete list of allconferences held before 1974, as well as references to their proceedings, can

VIII Preface to the Second Edition

be found in the volume devoted to the 1974 conference which was held inStuttgart [M H Pilkuhn, editor (Teubner, Stuttgart, 1974) p 1351] The nextICPS is scheduled to take place in Osaka, Japan from Sept 18 to 22 in theyear 2000

The Jerusalem ICPS had an attendance of nearly 800 researchers from 42different countries The subjects covered there represent the center of the cur-rent interests in a rapidly moving field Some of them are already introduced

in this volume but several are still rapidly developing and do not yet lendthemselves to discussion in a general textbook We mention a few keywords:Fractional quantum Hall effect and composite fermions

Mesoscopic effects, including weak localization

Microcavities, quantum dots, and quantum dot lasers

III–V nitrides and laser applications

Transport and optical processes with femtosecond resolution

Fullerites, C60-based nanotubes

Device physics: CMOS devices and their future

Students interested in any of these subjects that are not covered here, willhave to wait for the proceedings of the 24th ICPS Several of these topics arealso likely to find a place in the next edition of this book

In the present edition we have corrected all errors known to us at thistime and added a few references to publications which will help to clarify thesubjects under discussion

I, who one day was sand but am today a crystal

by virtue of a great fire and submitted myself to the demanding rigor

of the abrasive cut, today I have the power

to conjure the hot flame.

Likewise the poet, anxiety and word:

sand, fire, crystal, strophe, rhythm.

– woe is the poem that does not light a flame David Jou, 1983

(translated from the Catalan original)

The evolution of this volume can be traced to the year 1970 when one of us(MC) gave a course on the optical properties of solids at Brown Universitywhile the other (PYY) took it as a student Subsequently the lecture noteswere expanded into a one-semester course on semiconductor physics offered

at the Physics Department of the University of California at Berkeley Thecomposition of the students in this course is typically about 50 % from thePhysics Department, whereas the rest are mostly from two departments in theSchool of Engineering (Electrical Engineering and Computer Science; Mate-rials Science and Mineral Engineering) Since the background of the studentswas rather diverse, the prerequisites for this graduate-level course were kept

to a minimum, namely, undergraduate quantum mechanics, electricity andmagnetism and solid-state physics The Physics Department already offers atwo-semester graduate-level course on condensed matter physics, therefore itwas decided to de-emphasize theoretical techniques and to concentrate onphenomenology Since many of the students in the class were either growing

or using semiconductors in device research, particular emphasis was placed onthe relation between physical principles and device applications However, toavoid competing with several existing courses on solid state electronics, discus-sions of device design and performance were kept to a minimum This coursehas been reasonably successful in “walking this tight-rope”, as shown by thefact that it is offered at semi-regular intervals (about every two years) as aresult of demands by the students

One problem encountered in teaching this course was the lack of an equate textbook Although semiconductor physics is covered to some extent

ad-in all advanced textbooks on condensed matter physics, the treatment rarelyprovides the level of detail satisfactory to research students Well-establishedbooks on semiconductor physics are often found to be too theoretical by ex-perimentalists and engineers As a result, an extensive list of reading materialsinitially replaced the textbook Moreover, semiconductor physics being a ma-ture field, most of the existing treatises concentrate on the large amount of

X Preface to the First Edition

well-established topics and thus do not cover many of the exciting new opments Soon the students took action to duplicate the lecture notes, whichdeveloped into a “course reader” sold by the Physics Department at cost Thisvolume is approximately “version 4.0” (in software jargon) of these lecturenotes

devel-The emphasis of this course at Berkeley has always been on simple ical arguments, sometimes at the expense of rigor and elegance in mathemat-ics Unfortunately, to keep the promise of using only undergraduate physicsand mathematics course materials requires compromise in handling specialgraduate-level topics such as group theory, second quantization, Green’s func-tions and Feynman diagrams, etc In particular, the use of group theory nota-tions, so pervasive in semiconductor physics literature, is almost unavoidable.The solution adopted during the course was to give the students a “five-minutecrash course” on these topics when needed This approach has been carriedover to this book We are fully aware of its shortcomings This is not too seri-ous a problem in a class since the instructor can adjust the depth of the sup-plementary materials to satisfy the need of the students A book lacks suchflexibility The readers are, therefore, urged to skip these “crash courses”, es-pecially if they are already familiar with them, and consult the references forfurther details according to their background

phys-The choice of topics in this book is influenced by several other factors.Most of the heavier emphasis on optical properties reflects the expertise of theauthors Since there are already excellent books emphasizing transport prop-erties, such as the one by K H Seeger, our book will hopefully help to fill

a void One feature that sets this book apart from others on the market isthat the materials science aspects of semiconductors are given a more impor-tant role The growth techniques and defect properties of semiconductors arerepresented early on in the book rather than mentioned in an appendix Thisapproach recognizes the significance of new growth techniques in the devel-opment of semiconductor physics Most of the physics students who took thecourse at Berkeley had little or no training in materials science and hence abrief introduction was found desirable There were some feelings among thosephysics students that this course was an easier way to learn about materialsscience! Although the course offered at Berkeley lasted only one semester,the syllabus has since been expanded in the process of our writing this book

As a result it is highly unlikely that the volume can now be covered in onesemester However, some more specialized topics can be omitted without loss

of continuity, such as high field transport and hot electron effects, dynamiceffective ionic charge, donor–acceptor pair transitions, resonant Raman andBrillouin scattering, and a few more

Homework assignment for the course at Berkeley posed a “problem” cuse our pun) No teaching assistant was allocated by the department to helpwith grading of the problem sets Since the enrollment was typically over thirtystudents, this represented a considerable burden on the instructor As a “so-lution” we provide the students with the answers to most of the questions.Furthermore, many of the questions “lead the student by the hand” through

(ex-the calculation O(ex-thers have hints or references where fur(ex-ther details can befound In this way the students can grade their own solutions Some of thematerial not covered in the main text is given in the form of “problems” to beworked out by the student.

In the process of writing this book, and also in teaching the course, wehave received generous assistance from our friends and colleagues We are es-pecially indebted to: Elias Burstein; Marvin Cohen; Leo Esaki; Eugene Haller;Conyers Herring; Charles Kittel; Neville Smith; Jan Tauc; and Klaus von Klitz-ing for sharing their memories of some of the most important developments inthe history of semiconductor physics Their notes have enriched this book bytelling us their “side of the story” Hopefully, future students will be inspired

by their examples to expand further the frontiers of this rich and productivefield We are also grateful to Dung-Hai Lee for his enlightening explanation

of the Quantum Hall Effect

We have also been fortunate in receiving help from the over one hundredstudents who have taken the course at Berkeley Their frank (and anonymous)comments on the questionnaires they filled out at the end of the course havemade this book more “user-friendly” Their suggestions have also influencedthe choice of topics Many postdoctoral fellows and visitors, too numerous toname, have greatly improved the quality of this book by pointing out errorsand other weaknesses Their interest in this book has convinced us to continue

in spite of many other demands on our time The unusually high quality of theprinting and the color graphics in this book should be credited to the follow-ing people: H Lotsch, P Treiber, and C.-D Bachem of Springer-Verlag,Pauline Yu and Chia-Hua Yu of Berkeley, Sabine Birtel and Tobias Ruf ofStuttgart Last but not the least, we appreciate the support of our families.Their understanding and encouragement have sustained us through many dif-ficult and challenging moments PYY acknowledges support from the John S.Guggenheim Memorial Foundation in the form of a fellowship

XII Preface to the First Edition

A SEMI-CONDUCTOR

1 Introduction

1.1 A Survey of Semiconductors 2

1.1.1 Elemental Semiconductors 2

1.1.2 Binary Compounds 2

1.1.3 Oxides 3

1.1.4 Layered Semiconductors 3

1.1.5 Organic Semiconductors 4

1.1.6 Magnetic Semiconductors 4

1.1.7 Other Miscellaneous Semiconductors 4

1.2 Growth Techniques 5

1.2.1 Czochralski Method 5

1.2.2 Bridgman Method 6

1.2.3 Chemical Vapor Deposition 7

1.2.4 Molecular Beam Epitaxy 8

1.2.5 Fabrication of Self-Organized Quantum Dots by the Stranski–Krastanow Growth Method 11

1.2.6 Liquid Phase Epitaxy 13

Summary 14

Periodic Table of “Semiconductor-Forming” Elements 15

2 Electronic Band Structures 2.1 Quantum Mechanics 18

2.2 Translational Symmetry and Brillouin Zones 20

2.3 A Pedestrian’s Guide to Group Theory 25

2.3.1 Definitions and Notations 25

2.3.2 Symmetry Operations of the Diamond and Zinc-Blende Structures 30

2.3.3 Representations and Character Tables 32

2.3.4 Some Applications of Character Tables 40

2.4 Empty Lattice or Nearly Free Electron Energy Bands 48

2.4.1 Nearly Free Electron Band Structure in a Zinc-Blende Crystal 48

2.4.2 Nearly Free Electron Energy Bands in Diamond Crystals 52 2.5 Band Structure Calculations by Pseudopotential Methods 58

2.5.1 Pseudopotential Form Factors in Zinc-Blende- and Diamond-Type Semiconductors 61 2.5.2 Empirical and Self-Consistent Pseudopotential Methods 66

2.6 The k·p Method of Band-Structure Calculations 68

2.6.1 Effective Mass of a Nondegenerate Band Using the k·p Method 69

2.6.2 Band Dispersion near a Degenerate Extremum: Top Valence Bands in Diamond- and Zinc-Blende-Type Semiconductors 71

2.7 Tight-Binding or LCAO Approach to the Band Structure of Semiconductors 83

2.7.1 Molecular Orbitals and Overlap Parameters 83

2.7.2 Band Structure of Group-IV Elements by the Tight-Binding Method 87

2.7.3 Overlap Parameters and Nearest-Neighbor Distances 94

Problems 96

Summary 105

3 Vibrational Properties of Semiconductors, and Electron–Phonon Interactions 3.1 Phonon Dispersion Curves of Semiconductors 110

3.2 Models for Calculating Phonon Dispersion Curves of Semiconductors 114

3.2.1 Force Constant Models 114

3.2.2 Shell Model 114

3.2.3 Bond Models 115

3.2.4 Bond Charge Models 117

3.3 Electron–Phonon Interactions 121

3.3.1 Strain Tensor and Deformation Potentials 122

3.3.2 Electron–Acoustic-Phonon Interaction at Degenerate Bands 127

3.3.3 Piezoelectric Electron–Acoustic-Phonon Interaction 130

3.3.4 Electron–Optical-Phonon Deformation Potential Interactions 131

3.3.5 Fröhlich Interaction 133

3.3.6 Interaction Between Electrons and Large-Wavevector Phonons: Intervalley Electron–Phonon Interaction 135

Problems 137

Summary 158

4 Electronic Properties of Defects 4.1 Classification of Defects 160

4.2 Shallow or Hydrogenic Impurities 161

4.2.1 Effective Mass Approximation 162

4.2.2 Hydrogenic or Shallow Donors 166

4.2.3 Donors Associated with Anisotropic Conduction Bands 171 4.2.4 Acceptor Levels in Diamond- and Zinc-Blende-Type Semiconductors 174

XIV Contents

4.3 Deep Centers 180

4.3.1 Green’s Function Method for Calculating Defect Energy Levels 183

4.3.2 An Application of the Green’s Function Method: Linear Combination of Atomic Orbitals 188

4.3.3 Another Application of the Green’s Function Method: Nitrogen in GaP and GaAsP Alloys 192

4.3.4 Final Note on Deep Centers 197

Problems 198

Summary 202

5 Electrical Transport 5.1 Quasi-Classical Approach 203

5.2 Carrier Mobility for a Nondegenerate Electron Gas 206

5.2.1 Relaxation Time Approximation 206

5.2.2 Nondegenerate Electron Gas in a Parabolic Band 207

5.2.3 Dependence of Scattering and Relaxation Times on Electron Energy 208

5.2.4 Momentum Relaxation Times 209

5.2.5 Temperature Dependence of Mobilities 220

5.3 Modulation Doping 223

5.4 High-Field Transport and Hot Carrier Effects 225

5.4.1 Velocity Saturation 227

5.4.2 Negative Differential Resistance 228

5.4.3 Gunn Effect 230

5.5 Magneto-Transport and the Hall Effect 232

5.5.1 Magneto-Conductivity Tensor 232

5.5.2 Hall Effect 234

5.5.3 Hall Coefficient for Thin Film Samples (van der Pauw Method) 235

5.5.4 Hall Effect for a Distribution of Electron Energies 236

Problems 237

Summary 241

6 Optical Properties I 6.1 Macroscopic Electrodynamics 244

6.1.1 Digression: Units for the Frequency of Electromagnetic Waves 247

6.1.2 Experimental Determination of Optical Functions 247

6.1.3 Kramers–Kronig Relations 250

6.2 The Dielectric Function 253

6.2.1 Experimental Results 253

6.2.2 Microscopic Theory of the Dielectric Function 254

6.2.3 Joint Density of States and Van Hove Singularities 261

6.2.4 Van Hove Singularities in Âi 262

6.2.5 Direct Absorption Edges 268

6.2.6 Indirect Absorption Edges 269

6.2.7 “Forbidden” Direct Absorption Edges 273

6.3 Excitons 276

6.3.1 Exciton Effect at M0Critical Points 279

6.3.2 Absorption Spectra of Excitons 282

6.3.3 Exciton Effect at M1Critical Points or Hyperbolic Excitons 288

6.3.4 Exciton Effect at M3Critical Points 291

6.4 Phonon-Polaritons and Lattice Absorption 292

6.4.1 Phonon-Polaritons 295

6.4.2 Lattice Absorption and Reflection 298

6.4.3 Multiphonon Lattice Absorption 299

6.4.4 Dynamic Effective Ionic Charges in Heteropolar Semiconductors 303

6.5 Absorption Associated with Extrinsic Electrons 305

6.5.1 Free-Carrier Absorption in Doped Semiconductors 306

6.5.2 Absorption by Carriers Bound to Shallow Donors and Acceptors 311

6.6 Modulation Spectroscopy 315

6.6.1 Frequency Modulated Reflectance and Thermoreflectance 319

6.6.2 Piezoreflectance 321

6.6.3 Electroreflectance (Franz–Keldysh Effect) 322

6.6.4 Photoreflectance 329

6.6.5 Reflectance Difference Spectroscopy 332

6.7 Addendum (Third Edition): Dielectric Function 333

Problems 334

Summary 343

7 Optical Properties II 7.1 Emission Spectroscopies 345

7.1.1 Band-to-Band Transitions 351

7.1.2 Free-to-Bound Transitions 354

7.1.3 Donor–Acceptor Pair Transitions 356

7.1.4 Excitons and Bound Excitons 362

7.1.5 Luminescence Excitation Spectroscopy 369

7.2 Light Scattering Spectroscopies 375

7.2.1 Macroscopic Theory of Inelastic Light Scattering by Phonons 375

7.2.2 Raman Tensor and Selection Rules 378

7.2.3 Experimental Determination of Raman Spectra 385

7.2.4 Microscopic Theory of Raman Scattering 394

7.2.5 A Detour into the World of Feynman Diagrams 395

7.2.6 Brillouin Scattering 398

7.2.7 Experimental Determination of Brillouin Spectra 400

XVI Contents

7.2.8 Resonant Raman and Brillouin Scattering 401

Problems 422

Summary 426

8 Photoelectron Spectroscopy 8.1 Photoemission 431

8.1.1 Angle-Integrated Photoelectron Spectra of the Valence Bands 440

8.1.2 Angle-Resolved Photoelectron Spectra of the Valence Bands 443

8.1.3 Core Levels 451

8.2 Inverse Photoemission 456

8.3 Surface Effects 457

8.3.1 Surface States and Surface Reconstruction 457

8.3.2 Surface Energy Bands 458

8.3.3 Fermi Level Pinning and Space Charge Layers 460

Problems 465

Summary 468

9 Effect of Quantum Confinement on Electrons and Phonons in Semiconductors 9.1 Quantum Confinement and Density of States 470

9.2 Quantum Confinement of Electrons and Holes 473

9.2.1 Semiconductor Materials for Quantum Wells and Superlattices 474

9.2.2 Classification of Multiple Quantum Wells and Superlattices 478

9.2.3 Confinement of Energy Levels of Electrons and Holes 479 9.2.4 Some Experimental Results 489

9.3 Phonons in Superlattices 494

9.3.1 Phonons in Superlattices: Folded Acoustic and Confined Optic Modes 494

9.3.2 Folded Acoustic Modes: Macroscopic Treatment 499

9.3.3 Confined Optical Modes: Macroscopic Treatment 500

9.3.4 Electrostatic Effects in Polar Crystals: Interface Modes 502

9.4 Raman Spectra of Phonons in Semiconductor Superlattices 511

9.4.1 Raman Scattering by Folded Acoustic Phonons 511

9.4.2 Raman Scattering by Confined Optical Phonons 516

9.4.3 Raman Scattering by Interface Modes 518

9.4.4 Macroscopic Models of Electron–LO Phonon (Fröhlich) Interaction in Multiple Quantum Wells 521

9.5 Electrical Transport: Resonant Tunneling 525

9.5.1 Resonant Tunneling Through a Double-Barrier Quantum Well 526

9.5.2 I–V Characteristics of Resonant Tunneling Devices 529

9.6 Quantum Hall Effects in Two-Dimensional Electron Gases 533

9.6.1 Landau Theory of Diamagnetism in a Three-Dimensional Free Electron Gas 534

9.6.2 Magneto-Conductivity of a Two-Dimensional Electron Gas: Filling Factor 537

9.6.3 The Experiment of von Klitzing, Pepper and Dorda 538

9.6.4 Explanation of the Hall Plateaus in the Integral Quantum Hall Effect 541

9.7 Concluding Remarks 545

Problems 546

Summary 551

Appendix: Pioneers of Semiconductor Physics Remember… Ultra-Pure Germanium: From Applied to Basic Research or an Old Semiconductor Offering New Opportunities By Eugene E Haller 555

Two Pseudopotential Methods: Empirical and Ab Initio By Marvin L Cohen 558

The Early Stages of Band-Structures Physics and Its Struggles for a Place in the Sun By Conyers Herring 560

Cyclotron Resonance and Structure of Conduction and Valence Band Edges in Silicon and Germanium By Charles Kittel 563

Optical Properties of Amorphous Semiconductors and Solar Cells By Jan Tauc 566

Optical Spectroscopy of Shallow Impurity Centers By Elias Burstein 569

On the Prehistory of Angular Resolved Photoemission By Neville V Smith 574

The Discovery and Very Basics of the Quantum Hall Effect By Klaus von Klitzing 576

The Birth of the Semiconductor Superlattice By Leo Esaki 578

Physical Parameters of Tetrahedral Semiconductors (Inside Front Cover) Table of Fundamental Physical Constants (Inside Back Cover)

Table of Units (Inside Back Cover)

XVIII Contents

C O N T E N T S

1.1 A Survey of Semiconductors 21.2 Growth Techniques 5Summary 14

In textbooks on solid-state physics, a semiconductor is usually defined rather

loosely as a material with electrical resistivity lying in the range of 10⫺2 ⫺

109ø cm.1Alternatively, it can be defined as a material whose energy gap (to

be defined more precisely in Chap 2) for electronic excitations lies betweenzero and about 4 electron volts (eV) Materials with zero bandgap are met-als or semimetals, while those with an energy gap larger than 3 eV are morefrequently known as insulators There are exceptions to these definitions Forexample, terms such as semiconducting diamond (whose energy gap is about

6 eV) and semi-insulating GaAs (with a 1.5 eV energy gap) are frequentlyused GaN, which is receiving a lot of attention as optoelectronic material inthe blue region, has a gap of 3.5 eV

The best-known semiconductor is undoubtedly silicon (Si) However, there

are many semiconductors besides silicon In fact, many minerals found in ture, such as zinc-blende (ZnS) cuprite (Cu2O) and galena (PbS), to name just

na-a few, na-are semiconductors Including the semiconductors synthesized in lna-abo-ratories, the family of semiconductors forms one of the most versatile class ofmaterials known to man

labo-Semiconductors occur in many different chemical compositions with alarge variety of crystal structures They can be elemental semiconductors,such as Si, carbon in the form of C60 or nanotubes and selenium (Se) orbinary compounds such as gallium arsenide (GaAs) Many organic com-pounds, e g polyacetylene (CH)n, are semiconductors Some semiconductorsexhibit magnetic (Cd1⫺xMnxTe) or ferroelectric (SbSI) behavior Others be-come superconductors when doped with sufficient carriers (GeTe and SrTiO3)

Many of the recently discovered high-Tc superconductors have nonmetallicphases which are semiconductors For example, La2CuO4 is a semiconductor(gap
2 eV) but becomes a superconductor when alloyed with Sr to form

(La1⫺xSrx)2CuO4

1 ø cm is a “hybrid” SI and cgs resistivity unit commonly used in science and ing The SI unit for resistivity should be ø m

ger-of diamond and·-tin (a zero-gap semiconductor also known as “gray” tin) In

this structure each atom is surrounded by four nearest neighbor atoms (each

atom is said to be four-fold coordinated), forming a tetrahedron These

tetra-hedrally bonded semiconductors form the mainstay of the electronics industry

and the cornerstone of modern technology Most of this book will be devoted

to the study of the properties of these tetrahedrally bonded semiconductors.Some elements from the groups V and VI of the periodical table, such asphosphorus (P), sulfur (S), selenium (Se) and tellurium (Te), are also semi-conductors The atoms in these crystals can be three-fold (P), two-fold (S, Se,Te) or four-fold coordinated As a result, these elements can exist in severaldifferent crystal structures and they are also good glass-formers For example,

Se has been grown with monoclinic and trigonal crystal structures or as a glass(which can also be considered to be a polymer)

1.1.2 Binary Compounds

Compounds formed from elements of the groups III and V of the periodictable (such as GaAs) have properties very similar to their group IV counter-parts In going from the group IV elements to the III–V compounds, the bond-ing becomes partly ionic due to transfer of electronic charge from the groupIII atom to the group V atom The ionicity causes significant changes in thesemiconductor properties It increases the Coulomb interaction between theions and also the energy of the fundamental gap in the electronic band struc-ture The ionicity becomes even larger and more important in the II–VI com-pounds such as ZnS As a result, most of the II–VI compound semiconductorshave bandgaps larger than 1 eV The exceptions are compounds containingthe heavy element mercury (Hg) Mercury telluride (HgTe) is actually a zero-bandgap semiconductor (or a semimetal) similar to gray tin While the largebandgap II–VI compound semiconductors have potential applications for dis-plays and lasers, the smaller bandgap II–VI semiconductors are important ma-terials for the fabrication of infrared detectors The I–VII compounds (e g.,CuCl) tend to have even larger bandgaps (⬎3 eV) as a result of their higher

ionicity Many of them are regarded as insulators rather than semiconductors.Also, the increase in the cohesive energy of the crystal due to the Coulombinteraction between the ions favors the rock-salt structure containing six-foldcoordinated atoms rather than tetrahedral bonds Binary compounds formedfrom group IV and VI elements, such as lead sulfide (PbS), PbTe and tin sul-fide (SnS), are also semiconductors The large ionicity of these compounds alsofavors six-fold coordinated ions They are similar to the mercury chalcogenides

in that they have very small bandgaps in spite of their large ionicity Thesesmall bandgap IV–VI semiconductors are also important as infrared detectors.GaN, a large bandgap III–V compound, and the mixed crystals Ga1⫺xInxN arebeing used for blue light emitting diodes and lasers [1.1]

1.1.3 Oxides

Although most oxides are good insulators, some, such as CuO and Cu2O, arewell-known semiconductors Since cuprous oxide (Cu2O) occurs as a mineral(cuprite), it is a classic semiconductor whose properties have been studied ex-tensively In general, oxide semiconductors are not well understood with re-gard to their growth processes, so they have limited potential for applications

at present One exception is the II–VI compound zinc oxide (ZnO), which hasfound application as a transducer and as an ingredient of adhesive tapes andsticking plasters However, this situation has changed with the discovery of su-perconductivity in many oxides of copper

The first member of these so-called high-Tc superconductors, discovered

by M ¨uller and Bednorz2, is based on the semiconductor lanthanum copperoxide (La2CuO4), which has a bandgap of about 2 eV Carriers in the form

of holes are introduced into La2CuO4 when trivalent lanthanum (La) is placed by divalent barium (Ba) or strontium (Sr) or when an excess of oxygen

re-is present When sufficient carriers are present the semiconductor transformsinto a superconducting metal So far the highest superconducting transition

temperature at ambient pressure (Tc
135 K) found in this family of

mate-rials belongs to HgBaCa2Cu3O8⫹‰ HgBaCa2Cu3O8⫹‰ reaches a Tc
164 K

under high pressure [1.2] At the time this third edition went into print thisrecord had not yet been broken

1.1.4 Layered Semiconductors

Semiconducting compounds such as lead iodide (PbI2), molybdenum disulfide(MoS2) and gallium selenide (GaSe) are characterized by their layered crys-tal structures The bonding within the layers is typically covalent and muchstronger than the van der Waals forces between the layers These layered semi-conductors have been of interest because the behavior of electrons in the lay-ers is quasi-two-dimensional Also, the interaction between layers can be mod-

2 For this discovery, Bednorz and M ¨uller received the Physics Nobel Prize in 1987.

be changed more easily than those of inorganic semiconductors to suit the plication by changing their chemical formulas Recently new forms of carbon,such as C60 (fullerene), have been found to be semiconductors One form ofcarbon consists of sheets of graphite rolled into a tube of some nanometers

ap-in diameter known as nanotubes [1.3,4] These carbon nanotubes and their

“cousin”, BN nanotubes, hold great promise as nanoscale electronic circuitelements They can be metals or semiconductors depending on their pitch

1.1.6 Magnetic Semiconductors

Many compounds containing magnetic ions such as europium (Eu) and ganese (Mn), have interesting semiconducting and magnetic properties Ex-amples of these magnetic semiconductors include EuS and alloys such as

man-Cd1⫺xMnxTe Depending on the amount of the magnetic ion in these alloys,the latter compounds exhibit different magnetic properties such as ferromag-netism and antiferromagnetism The magnetic alloy semiconductors containinglower concentrations of magnetic ions are known as dilute magnetic semicon-ductors These alloys have recently attracted much attention because of theirpotential applications Their Faraday rotations can be up to six orders of mag-nitude larger than those of nonmagnetic semiconductors As a result, thesematerials can be used as optical modulators, based on their large magneto-optical effects The perovskites of the type La0.7Ca0.3MnO3 undergo metal–semiconductor transitions which depend strongly on magnetic field, giving rise

to the phenomenon of collossal magneto-resistance (CMR) [1.5].

1.1.7 Other Miscellaneous Semiconductors

There are many semiconductors that do not fall into the above categories Forexample, SbSI is a semiconductor that exhibits ferroelectricity at low temper-atures Compounds with the general formula I–III–VI2 and II–IV–V2 (such asAgGaS2, interesting for its nonlinear optical properties, CuInSe2, useful forsolar cells, and ZnSiP2) crystallize in the chalcopyrite structure The bonding

in these compounds is also tetrahedral and they can be considered as analogs

of the group III–V and II–VI semiconductors with the zinc-blende structure.Compounds formed from the group V and VI elements with formulas such

as As2Se3 are semiconductors in both the crystalline and glassy states Many

of these semiconductors have interesting properties but they have not yet ceived much attention due to their limited applications Their existence showsthat the field of semiconductor physics still has plenty of room for growth andexpansion

re-1.2 Growth Techniques

One reason why semiconductors have become the choice material for theelectronics industry is the existence of highly sophisticated growth techniques.Their industrial applications have, in turn, led to an increased sophistication

of these techniques For example, Ge single crystals are nowadays amongstthe purest elemental materials available as a result of years of perfecting theirgrowth techniques (see Appendix by E.E Haller in p 555) It is now possible

to prepare almost isotopically pure Ge crystals (natural Ge contains five ferent isotopes) Nearly perfect single crystals of Si can be grown in the form

dif-of ingots over twelve inches (30 cm) in diameter Isotopically pure28Si crystalshave been shown to have considerably higher thermal conductivity than their

natural Si counterparts [1.6] Dislocation densities in these crystals can be as

low as 1000 cm⫺3, while impurity concentrations can be less than one part pertrillion (1012)

More recent developments in crystal growth techniques have made conductors even more versatile Techniques such as Molecular Beam Epitaxy(MBE) and Metal-Organic Chemical Vapor Deposition (MOCVD) allow crys-tals to be deposited on a substrate one monolayer at a time with great pre-cision These techniques have made it possible to synthesize artificial crystal

semi-structures known as superlattices and quantum wells (Chap 9) A recent

ad-vance in fabricating low-dimensional nanostructures takes advantage of eitheralignment of atoms with the substrate or strain between substrate and epi-

layer to induce the structure to self-organize into superlattices or quantum

dots Although a detailed discussion of all the growth techniques is beyondthe scope of this book, a short survey of the most common techniques willprovide background information necessary for every semiconductor physicist.The references given for this chapter provide further background material forthe interested reader

1.2.1 Czochralski Method

The Czochralski method is the most important method for growing bulk

crys-tals of semiconductors, including Si The method involves melting the raw

SiO2crucible Susceptor (graphite)

Si melt

Fig 1.1 Schematic diagram

of a Czochralski furnace for growing Si single crystals

terial in a crucible A seed crystal is placed in contact with the top, coolerregion of the melt and rotated slowly while being gradually pulled from themelt Additional material is solidified from the melt onto the seed The most

significant development in the Czochralski technique [1.7] (shown cally in Fig 1.1) is the discovery of the Dash technique [1.8,9] for growing

schemati-dislocation-free single crystals of Si even when starting with a dislocated seed.Typical growth speed is a few millimeters per minute, and the rotation ensuresthat the resultant crystals are cylindrical Silicon ingots grown by this methodnow have diameters greater than 30 cm

The crucible material and gas surrounding the melt tend to contribute

to the background impurities in the crystals For example, the most mon impurities in bulk Si are carbon (from the graphite crucible) and oxy-gen Bulk GaAs and indium phosphide (InP) crystals are commonly grown bythe Czochralski method but with the melt isolated from the air by a layer ofmolten boron oxide to prevent the volatile anion vapor from escaping Thismethod of growing crystals containing a volatile constituent is known as the

com-Liquid-Encapsulated Czochralski (LEC) Method As expected, LEC-grown

GaAs often contains boron as a contaminant

1.2.2 Bridgman Method

In the Bridgman method a seed crystal is usually kept in contact with a melt,

as in the Czochralski method However, a temperature gradient is createdalong the length of the crucible so that the temperature around the seed crys-tal is below the melting point The crucible can be positioned either horizon-tally or vertically to control convection flow As the seed crystal grows, thetemperature profile is translated along the crucible by controlling the heaters

along the furnace or by slowly moving the ampoule containing the seed crystalwithin the furnace.

1.2.3 Chemical Vapor Deposition

Both the Czochralski and Bridgman techniques are used to grow bulk singlecrystals It is less expensive to grow a thin layer of perfect crystal than a largeperfect bulk crystal In most applications devices are fabricated out of a thinlayer grown on top of a bulk crystal The thickness of this layer is about 1Ìm

or less Economically, it makes sense to use a different technique to grow athin high quality layer on a lower quality bulk substrate To ensure that thisthin top layer has high crystalline quality, the crystal structure of the thin layershould be similar, if not identical, to the substrate and their lattice parameters

as close to each other as possible to minimize strain In such cases the atoms

forming the thin layer will tend to build a single crystal with the same lographic orientation as the substrate The resultant film is said to be deposited

crystal-epitaxially on the substrate The deposition of a film on a bulk single crystal of

the same chemical composition (for example, a Si film deposited on a bulk Si

crystal) is known as homo-epitaxy When the film is deposited on a substrate

of similar structure but different chemical composition (such as a GaAs film

on a Si substrate), the growth process is known as hetero-epitaxy.

Epitaxial films can be grown from solid, liquid or gas phases In general,

it is easier to precisely control the growth rate in gas phase epitaxy by trolling the amount of gas flow In Chemical Vapor Deposition (CVD) gases

con-containing the required chemical elements are made to react in the vicinity ofthe substrate The semiconductor produced as a result of the reaction is de-

posited as a thin film on a substrate inside the reactor The temperature of the

substrate is usually an important factor in determining the epitaxy and hencethe quality of the resultant film The most common reaction for producing a

Si film in this way is given by

SiH4

(silane)

+ 2H2↑ Si substrate ↓

heat

Highly pure Si can be produced in this way because the reaction by-product

H2 is a gas and can be easily removed Another advantage of this technique

is that dopants, such as P and As, can be introduced very precisely in theform of gases such as phosphine (PH3) and arsine (AsH3) III–V compoundsemiconductors can also be grown by CVD by using gaseous metal-organiccompounds like trimethyl gallium [Ga(CH3)3] as sources For example, GaAsfilms can be grown by the reaction

8 1 Introduction

Pump GaAs

RF heater Filter

PH 3 + H 2

AsH 3 + H2

Ga(CH3) 3 + H2

Exhaust

Vacuum pump

Quartz nozzle Rotating susceptor

Heater Substrate

Stainless

steel

chamber

Conical quartz tube

IR Radiation Thermometer

Main flow TMG+NH +H3 2

Subflow

N +H2 2

Main Flow TMG+NH +H 3 2

Subflow

N +H2 2

Fig 1.2 (a) Schematic diagram of a MOCVD apparatus [1.10] (b) Details of

two-flow MOCVD machine introduced by Nakamura and co-workers for growing GaN.

(c) Schematic diagram of the gas flows near the substrate surface [1.11]

This method of growing epitaxial films from metal-organic gases is known as

Metal-Organic Chemical Vapor Deposition (MOCVD), and a suitable growth

apparatus is shown schematically in Fig 1.2a A recent modification duced for growing GaN is shown in Fig 1.2b Figure 1.2c shows the details

intro-of interaction between the two gas flows near the substrate [1.11]

1.2.4 Molecular Beam Epitaxy

In CVD the gases are let into the reactor at relatively high pressure cally higher than 1 torr) As a result, the reactor may contain a high concen-tration of contaminants in the form of residual gases This problem can be

(typi-avoided by growing the sample under UltraHigh Vacuum (UHV) conditions.

(Pressures below 10⫺7torr are considered high vacuum, and a base pressure

Fig 1.3 Schematic diagram of an effusion (Knudsen) cell [1.10]

around 10⫺11torr is UHV See Sect 8.1 for further discussion of UHV tions and the definition of torr.) The reactants can be introduced in the form

condi-of molecular beams A molecular beam is created by heating a source

mate-rial until it vaporizes in a cell with a very small orifice Such a cell is known

as an effusion (or Knudsen) cell and is shown schematically in Fig 1.3 As

the vapor escapes from the cell through the small nozzle, its molecules (oratoms) form a well-collimated beam, since the UHV environment outside thecell allows the escaping molecules (or atoms) to travel ballistically for meterswithout collision Typically several molecular beams containing the necessaryelements for forming the semiconductor and for doping the sample are aimed

at the substrate, where the film grows epitaxially Hence this growth technique

is known as Molecular Beam Epitaxy (MBE).

Figure 1.4 shows the construction of a typical MBE system In principle,

it is difficult to control the concentration of reactants arriving at the strate, and hence the crystal stoichiometry, in MBE growth The techniqueworks because its UHV environment makes it possible to utilize electrons andions as probes to monitor the surface and film quality during growth The

sub-ion-based probe is usually mass spectrometry Some of the electron based techniques are Auger Electron Spectroscopy (AES), Low Energy-Electron

Diffraction (LEED), Reflection High-Energy Electron Diffraction (RHEED),

and X-ray and Ultraviolet Photoemission Spectroscopy (XPS and UPS) These

techniques will be discussed in more detail in Chap 8 The one most monly used in MBE systems is RHEED

com-A typical RHEED system consists of an electron gun producing a

high-energy (10–15 keV) beam aimed at a very large angle of incidence (grazing

incidence) to the substrate surface (see Fig 1.4) The reflected electron

diffrac-tion pattern is displayed on a phosphor screen (labeled RHEED screen inFig 1.4) on the opposite side This diffraction pattern can be used to establishthe surface geometry and morphology In addition, the intensity of the zeroth-order diffraction beam (or specular beam) has been found to show damped

Sample exchange mechanism

Fig 1.5 Oscillations in the intensity of the specularly reflected electron beam in the

RHEED pattern during the growth of a GaAs or AlAs film on a GaAs(001) substrate One period of oscillation corresponds precisely to the growth of a single layer of GaAs

or AlAs [1.10]

oscillations (known as RHEED oscillations) that allow the film growth rate

to be monitored in situ Figure 1.5 shows an example of RHEED oscillations

measured during the growth of a GaAs/AlAs quantum well Quantum wells

are synthetic structures containing a very thin layer (thickness less than 10 nm)

of semiconductor sandwiched between two thin layers of another tor with a larger bandgap (see Chap 9 for further discussion) Each oscillation

semiconduc-in Fig 1.5 corresponds to the growth of a ssemiconduc-ingle molecular layer of GaAs orAlAs

To understand how such perfectly stoichiometric layers can be grown, wenote that the Ga or Al atoms attach to a GaAs substrate much more readilythan the As atoms Since arsenic is quite volatile at elevated temperatures,any arsenic atoms not reacted with Ga or Al atoms on the substrate will not

be deposited on a heated substrate By controlling the molecular beams withshutters and monitoring the growth via RHEED oscillations, it is possible togrow a thin film literally one monolayer at a time

The MBE technique is used for the growth of high-quality quantum wells.The only drawback of this technique in commercial applications is its slowthroughput and high cost (a typical MBE system costs a least US $ 500 000)

As a result, the MBE technique is utilized to study the conditions for growinghigh-quality films in the laboratory but large-scale commercial production ofthe films uses the MOCVD method

1.2.5 Fabrication of Self-Organized Quantum Dots

by the Stranski–Krastanow Growth Method

The epitaxial growth of a thin film A on a substrate B can occur in one

of three main growth modes: (1) monolayer or two-dimensional growth; (2)three-dimensional growth or Volmer–Weber mode and (3) Stranski– Krastanow mode [1.12] In mode (1) the atoms of A are attracted to the sub-

strate more strongly than to each other As a result the atoms first aggregate

to form monolayer islands which then expand and coalesce to form the firstmonolayer In mode (2) the atoms of A are attracted more strongly to eachother than to the substrate Thus they will first aggregate to form islands and

as deposition continues these islands will grow and finally form a continuousfilm In case of mode (3) the atoms of A will first grow two-dimensionally toform either a single monolayer or a small number of monolayers thin film.However, when growth proceeds further the additional atoms of A start toform three-dimensional islands on top of the thin film as in the Volmer–Weber

mode The continuous thin film is often referred to as the wetting layer.

One important factor which controls the growth of an epitaxial film is the

lattice mismatch between the epitaxial layer A and the substrate B Let us

as-sume that the lattice mismatch between A and B is not too large, say onlyaround 1% of their lattice constants There are at least two possible ways for

a thin film of A to grow on B The first possibility is for atoms of A to line

up on top of the corresponding atoms of B and to take on the lattice

con-12 1 Introduction

stant of B In this case the film A is strained but pseudomorphic (a

pseudo-morph is an altered crystal form whose outward appearance is the same of

another crystal species) In the second possibility the atoms of A retain theirbulk lattice constant and therefore are out of registry with the substrate atoms

To minimize this mismatch between the two kinds of atoms, the thin film A

will develop a kind of lattice defect known as a dislocation (see also Chap 4)

[1.13] For example, if the lattice constant of A is smaller than B then the match can be compensated by periodically inserting an extra plane of atoms

mis-of A into the film A to bring its atoms into alignment with the substrate atoms

again This kind of dislocation is known as a misfit dislocation Since the lattice

mismatch between A and B occurs in two directions lying within the surface,these dislocations form a two-dimensional network The competition betweenthese two growth modes for lattice-mismatched systems was studied by Frankand van der Merwe in 1949 [1.14] The trade-off is between the strain energy

in the strained pseudomorphic film and the energy required to form misfit locations in the unstrained film The strain energy increases with the volume

dis-of the film while the dislocation energy depends only on the area dis-of the film

As a result pseudomorphic growth dominates when the film thickness is small.However, as the film thickness increases it will become energetically more fa-vorable for dislocations to form One may expect this “cross-over” to occur at

some critical layer thickness The calculation of this critical thickness [1.13] is

beyond the scope of this book

The above consideration may suggest that the Stranski–Krastanow growthmode is undesirable for achieving an epitaxial film of uniform thickness Re-cently it was found that this growth mode is a convenient and inexpensive way

to produce nanometer structures known as quantum dots [1.15] In this case

the lattice constant of the epi-layer A has to be larger than that of the strate B The atoms of A can relax the tensile strain by “buckling” to formislands The principle behind this island formation is similar to the buckling of

sub-a bi-metsub-allic bsub-ar with incresub-ase in tempersub-ature, sub-an effect used in msub-aking perature sensors and thermostats Since these quantum dots are formed spon-taneously and can also be formed coherently, their formation is an example of

tem-a phenomenon in crysttem-al growth known tem-as self-orgtem-aniztem-ation Figure 1.6 shows

a plane-view transmission electron-microscope (TEM) image of a single sheet

of InAs (film A with lattice constant 6.06 ˚A) quantum dots grown on GaAssubstrate (B with lattice constant 5.64 ˚A)

Figure 1.7 shows the cross-sectional TEM image of a 25 layers thick stack

of InGaAs quantum dots (the thicker part of the dark regions) grown on aGaAs substrate Notice that the quantum dots are connected within the layers

by thin dark regions representing the wetting layers The various layers areseparated by GaAs represented by the lighter regions The quantum dots inFig 1.7 are aligned on top of each other to form arrays by the tensile strainwhich is transmitted through the thin GaAs layers The coherence is gradu-ally lost as the layers get farther from the substrate In addition to quantumdot arrays, monolayer superlattices, such as GaP/InP, can also be grown

by self-organization

Fig 1.6 A plane-view transmission

electron-microscope (TEM) image of a single sheet of InAs quantum dots grown on a [100]-oriented GaAs substrate Reproduced from [1.15]

Fig 1.7 A cross-sectional TEM image

of a 25-layer thick stack of InGaAs quantum dots (the thicker dark regions) grown on GaAs substrate (lighter area near the bottom of the picture) The lighter regions surrounding the InGaAs layers are also GaAs Reproduced from [1.15]

1.2.6 Liquid Phase Epitaxy

Semiconductor films can also be grown epitaxially on a substrate from the

liq-uid phase This Liqliq-uid Phase Epitaxy (LPE) growth technique has been very

successful in growing GaAs laser diodes Usually a group III metal, such as Ga

or In, is utilized as the solvent for As When the solvent is cooled in contactwith a GaAs substrate it becomes supersaturated with As and nucleation of

14 1 Introduction

Thermocouple

Graphite Slider

Ga melt with As, Al and dopants GaAs substrate

Fig 1.8 Setup for LPE crystal growth

GaAs starts on the substrate By using a slider containing several different lutes (as shown in Fig 1.8), successive epitaxial layers (or epilayers in short)

so-of different compositions and/or different dopants can be grown The tage of LPE is that the equipment required is inexpensive and easy to set up.However, it is difficult to achieve the level of control over the growth condi-tions possible with the MBE technique

advan-In summary, different techniques are employed to grow bulk single crystalsand thin epilayers of semiconductors The Czochralski or Bridgman techniquesare used to grow bulk crystals When feasible, the LPE method is prefered forgrowing thin films because of its low cost and fast growth rate When epilay-ers of thickness less than 100 nm are required, it is necessary to utilize theMOCVD or MBE techniques

In recent years optical mirror furnaces have become very popular for thegrowth of oxide semiconductors [1.16]

S U M M A R Y

In this chapter we have introduced the wide class of materials referred to

as semiconductors and we have mentioned the large range of structural and

physical properties they can have Most of the semiconductors used in ence and modern technology are single crystals, with a very high degree

sci-of perfection and purity They are grown as bulk three-dimensional crystals

or as thin, two-dimensional epitaxial layers on bulk crystals which serve assubstrates Among the techniques for growing bulk crystals that we havebriefly discussed are the Czochralski and Bridgman methods Epitaxial tech-niques for growing two-dimensional samples introduced in this chapter in-clude chemical vapor deposition, molecular beam epitaxy, and liquid phaseepitaxy

grown with epitaxial techniques

Self-organized two-dimensional lattices of quantum dots can also be

Periodic Table of “Semiconductor-Forming” Elements

2 2s 2 2p 2s 2 2p 2

2

Ge322

Sn505s 2 5p 2

Pb826s 2 6p 2

Fe264s23d

Ni284s 2 3d

Sc214s 2 3d

Xe5465s25p

Rn866 6s 2 6p

Zr40

5s Rh455s Pd46

Elements which crystallize as Semiconductors

Elements forming Binary III-VI Semiconductors

4d 7 4d8 4d 10

Cd48

5s 2

In49 5p 5s2 1 Sb51

5s25p3 Te525s 2 5p 4 I535s 2 5p5

Y395s 2 4d

Nb41

5s4 d 6

Mo425s4d5

Hg80

6s 5d10Tl81

6s 6p 2 2

Bi836s 2 6p3

Ba56

Series 57-71

Ta73 Re75Os76 Ir77

5d 9 Pt786s 5d 9 6s 2 5d 6

6s 2 5d 5 6s 2 5d 4 6s 2 5d 3 6s 2 5d 2 4f 14

Elements forming Binary II-VI Semiconductors

Elements forming Binary IV-VI Semiconductors

2 Elements forming II-VI-V Chalcopyrite Semiconductors2

Elements forming I-III-VI Chalcopyrite Semiconductors

Elements forming Binary III-V Semiconductors

Po846s 2 6p 4 At85 6s26p5

Elements forming Binary I-VII Semiconductors

2 Electronic Band Structures

C O N T E N T S

2.1 Quantum Mechanics 182.2 Translational Symmetry and Brillouin Zones 202.3 A Pedestrian’s Guide to Group Theory 252.4 Empty Lattice or Nearly Free Electron Energy Bands 482.5 Band Structure Calculations by Pseudopotential Methods 582.6 The k ·p Method of Band-Structure Calculations 68

2.7 Tight-Binding or LCAO Approach to the Band Structure

of Semiconductors 83Problems 96Summary 105

The property which distinguishes semiconductors from other materials cerns the behavior of their electrons, in particular the existence of gaps intheir electronic excitation spectra The microscopic behavior of electrons in asolid is most conveniently specified in terms of the electronic band structure.The purpose of this chapter is to study the band structure of the most com-mon semiconductors, namely, Si, Ge, and related III–V compounds We willbegin with a quick introduction to the quantum mechanics of electrons in acrystalline solid

con-The properties of electrons in a solid containing 1023atoms/cm3 are verycomplicated To simplify the formidable task of solving the wave equations forthe electrons, it is necessary to utilize the translational and rotational symme-

tries of the solid Group theory is the tool that facilitates this task However,

not everyone working with semiconductors has a training in group theory, so

in this chapter we will discuss some basic concepts and notations of group ory Our approach is to introduce the ideas and results of group theory whenapplied to semiconductors without presenting the rigorous proofs We will putparticular emphasis on notations that are often found in books and researcharticles on semiconductors In a sense, band structure diagrams are like mapsand the group theory notations are like symbols on the map Once the mean-ing of these symbols is understood, the band structure diagrams can be used

the-to find the way in exploring the electronic properties of semiconducthe-tors

We will also examine several popular methods of band structure tation for semiconductors All band structure computation techniques involveapproximations which tend to emphasize some aspects of the electronic prop-

compu-erties in semiconductors while, at the same time, de-emphasizing other aspects.Therefore, our purpose in studying the different computational methods is tounderstand their advantages and limitations In so doing we will gain insightinto the many different facets of electronic properties in semiconductors.

We note also that within the past two decades, highly sophisticated

tech-niques labeled “ab initio” have been developed successfully to calculate many

properties of solids, including semiconductors These techniques involve veryfew assumptions and often no adjustable parameters They have been applied

to calculate the total energy of crystals including all the interactions betweenthe electrons and with the nuclei By minimization of this energy as a function

of atomic spacing, equilibrium lattice constants have been predicted Otherproperties such as the elastic constants and vibrational frequencies can also

be calculated Extensions of these techniques to calculate excited-state erties have led to predictions of optical and photoemission spectra in goodagreement with experimental results It is beyond the scope of the presentbook to go into these powerful techniques Interested readers can consult ar-ticles in [2.1]

j

P2

j 2M j ⫹12

expression r i denotes the position of the ith electron, R j is the position of

the jth nucleus, Z j is the atomic number of the nucleus, p i and P j are themomentum operators of the electrons and nuclei, respectively, and ⫺e is the

with-electrons into two groups: valence with-electrons and core with-electrons The core

elec-trons are those in the filled orbitals, e g the 1s2, 2s2, and 2p6electrons in thecase of Si These core electrons are mostly localized around the nuclei, so they

can be “lumped” together with the nuclei to form the so-called ion cores As a

result of this approximation the indices j and j in (2.1) will, from now on,

de-note the ion cores while the electron indices i and i will label only the valence

2.1 Quantum Mechanics 19electrons These are electrons in incompletely filled shells and in the case of Si

include the 3s and 3p electrons.

The next approximation invoked is the Born–Oppenheimer or adiabatic

approximation The ions are much heavier than the electrons, so they move

much more slowly The frequencies of ionic vibrations in solids are typicallyless than 1013s⫺1 To estimate the electron response time, we note that theenergy required to excite electrons in a semiconductor is given by its funda-mental bandgap, which, in most semiconductors, is of the order of 1 eV There-fore, the frequencies of electronic motion in semiconductors are of the order

of 1015s⫺1 (a table containing the conversion factor from eV to various otherunits can be found in the inside cover of this book) As a result, electrons canrespond to ionic motion almost instantaneously or, in other words, to the elec-trons the ions are essentially stationary On the other hand, ions cannot followthe motion of the electrons and they see only a time-averaged adiabatic elec-tronic potential With the Born-Oppenheimer approximation the Hamiltonian

in (2.1) can be expressed as the sum of three terms:

Ᏼ ⫽ Ᏼions(R j)⫹ Ᏼe(r i , R j0)⫹ Ᏼe⫺ion(r i,‰R j), (2.2)where Ᏼion(R j) is the Hamiltonian describing the ionic motion under the in-fluence of the ionic potentials plus the time-averaged adiabatic electronic po-tentials.Ᏼe(r i , R j0) is the Hamiltonian for the electrons with the ions frozen in

their equilibrium positions R j0, andᏴe⫺ion(r i,‰R j) describes the change in theelectronic energy as a result of the displacements ‰R j of the ions from theirequilibrium positions.Ᏼe⫺ion is known as the electron–phonon interaction and

is responsible for electrical resistance in reasonably pure semiconductors atroom temperature The vibrational properties of the ion cores and electron-phonon interactions will be discussed in the next chapter In this chapter wewill be mainly interested in the electronic HamiltonianᏴe

The electronic HamiltonianᏴeis given by

semi-known as the mean-field approximation Without going into the justifications,

which are discussed in many standard textbooks on solid-state physics, we will

assume that every electron experiences the same average potential V(r) Thus

the Schr ¨odinger equations describing the motion of each electron will be tical and given by

where Ᏼ1e,º n (r) and E ndenote, respectively, the one-electron Hamiltonian,

and the wavefunction and energy of an electron in an eigenstate labeled by n.

We should remember that each eigenstate can only accommodate up to two

electrons of opposite spin (Pauli’s exclusion principle).

The calculation of the electronic energies E ninvolves two steps The first

step is the determination of the one-electron potential V(r) Later in this ter we will discuss the various ways to calculate or determine V(r) In one method V(r) can be calculated from first principles with the atomic numbers

chap-and positions as the only input parameters In simpler, so-called semi-empiricalapproaches, the potential is expressed in terms of parameters which are de-termined by fitting experimental results After the potential is known, it takesstill a complicated calculation to solve (2.4) It is often convenient to utilizethe symmetry of the crystal to simplify this calculation Here by “symmetry”

we mean geometrical transformations which leave the crystal unchanged

2.2 Translational Symmetry and Brillouin Zones

The most important symmetry of a crystal is its invariance under specific

trans-lations In addition to such translational symmetry most crystals possess some

rotational and reflection symmetries It turns out that most semiconductors

have high degrees of rotational symmetry which are very useful in reducingthe complexity of calculating their energy band structures In this and the nextsections we will study the use of symmetry to simplify the classification of elec-tronic states Readers familiar with the application of group theory to solidscan omit these two sections

When a particle moves in a periodic potential its wavefunctions can be

ex-pressed in a form known as Bloch functions To understand what Bloch

func-tions are, we will assume that (2.4) is one-dimensional and V(x) is a periodic

function with the translational period equal to R We will define a translation

operator T R as an operator whose effect on any function f (x) is given by

Next we introduce a functionº k (x) defined by

where uk(x) is a periodic function with the same periodicity as V, that is,

u k (x ⫹ nR) ⫽ uk (x) for all integers n When º k (x) so defined is multiplied

by exp [⫺iˆt], it represents a plane wave whose amplitude is modulated by the periodic function u k (x) º k (x) is known as a Bloch function By definition, when x changes to x ⫹ R, ºk (x) must change in the following way

It follows from (2.7) that º k (x) is an eigenfunction of T R with the

eigen-value exp (ikR) Since the Hamiltonian Ᏼ1e is invariant under translation by

R,Ᏼ commutes with T Thus it follows from quantum mechanics that the

2.2 Translational Symmetry and Brillouin Zones 21eigenfunctions ofᏴ1ecan be expressed also as eigenfunctions of T R We there-fore conclude that an eigenfunctionº(x) of Ᏼ1e can be expressed as a sum ofBloch functions:

where the Ak are constants Thus the one-electron wavefunctions can be

in-dexed by constants k, which are the wave vectors of the plane waves forming

the “backbone” of the Bloch function A plot of the electron energies in (2.4)

versus k is known as the electronic band structure of the crystal.

The band structure plot in which k is allowed to vary over all possible

values is known as the extended zone scheme From (2.6) we see that the

choice of k in indexing a wave function is not unique Both k and k⫹(2n/R), where n is any integer, will satisfy (2.6) This is a consequence of the trans- lation symmetry of the crystal Thus another way of choosing k is to replace

k by k  ⫽ k ⫺ (2n/R), where n is an integer chosen to limit k  to the

inter-val [⫺/R, /R] The region of k-space defined by [⫺/R, /R] is known as the

first Brillouin zone A more general definition of Brillouin zones in three

di-mensions will be given later and can also be found in standard textbooks [2.2]

The band structure plot resulting from restricting the wave vector k to the first

Brillouin zone is known as the reduced zone scheme In this scheme the wave

functions are indexed by an integer n (known as the band index) and a wave

vector k restricted to the first Brillouin zone.

In Fig 2.1 the band structure of a “nearly free” electron (i e., V → 0)

moving in a one-dimensional lattice with lattice constant a is shown in both

schemes for comparison Band structures are plotted more compactly in thereduced zone scheme In addition, when electrons make a transition from one

state to another under the influence of a translationally invariant operator, k

is conserved in the process within the reduced zone scheme (the proof of this

Fig 2.1 The band structure of a free particle shown in (a) the extended zone scheme and

(b) the reduced zone scheme

statement will be presented when matrix elements of operators in crystals are

discussed, Sect 2.3), whereas in the extended zone scheme k is conserved only

to a multiple of (i e modulo) 2 /R Hence, the reduced zone scheme is almost

invariably used in the literature

The above results, obtained in one dimension, can be easily generalized

to three dimensions The translational symmetries of the crystal are now

ex-pressed in terms of a set of primitive lattice vectors: a1, a2, and a3 We canimagine that a crystal is formed by taking a minimal set of atoms (known as a

basis set) and then translating this set by multiples of the primitive lattice

vec-tors and their linear combinations In this book we will be mostly concernedwith the diamond and zinc-blende crystal structures, which are shown in Fig.2.2a In both crystal structures the basis set consists of two atoms The ba-

Fig 2.2 (a) The crystal structure of diamond and zinc-blende (ZnS) (b) the fcc lattice

showing a set of primitive lattice vectors (c) The reciprocal lattice of the fcc lattice shown

with the first Brillouin zone Special high-symmetry points are denoted by °, X, and L, while high-symmetry lines joining some of these points are labeled as § and ¢

2.2 Translational Symmetry and Brillouin Zones 23sis set in diamond consists of two carbon atoms while in zinc-blende the twoatoms are zinc and sulfur The lattice of points formed by translating a point

by multiples of the primitive lattice vectors and their linear combinations is

known as the direct lattice Such lattices for the diamond and zinc-blende structures, which are basically the same, are said to be face-centered cubic

(fcc) see Fig 2.2b with a set of primitive lattice vectors In general, the choice

of primitive lattice vectors for a given direct lattice is not unique The tive lattice vectors shown in Fig 2.2b are

primi-a1⫽ (0, a/2, a/2),

a2⫽ (a/2, 0, a/2),

and

a3⫽ (a/2, a/2, 0),

where a is the length of the side of the smallest cube in the fcc lattice This

smallest cube in the direct lattice is also known as the unit cube or the

crys-tallographic unit cell.

For a given direct lattice we can define a reciprocal lattice in terms of

three primitive reciprocal lattice vectors: b1, b2, and b3, which are related to

the direct lattice vectors a1, a2, and a3by

b i ⫽ 2 (a j × a k)

(a1× a2)· a3

where i, j, and k represent a cyclic permutation of the three indices 1, 2, and 3

and (a1×a2)·a3is the volume of the primitive cell The set of points generated

by translating a point by multiples of the reciprocal lattice vectors is known asthe reciprocal lattice The reason for defining a reciprocal lattice in this way is

to represent the wave vector k as a point in reciprocal lattice space The first

Brillouin zone in three dimensions can be defined as the smallest polyhedronconfined by planes perpendicularly bisecting the reciprocal lattice vectors It iseasy to see that the region [⫺/R, /R] fits the definition of the first Brillouin

zone in one dimension

Since the reciprocal lattice vectors are obtained from the direct lattice tors via (2.9), the symmetry of the Brillouin zone is determined by the sym-metry of the crystal lattice The reciprocal lattice corresponding to a fcc lattice

vec-is shown in Fig 2.2c These reciprocal lattice points are said to form a

body-centered cubic (bcc) lattice The primitive reciprocal lattice vectors b1, b2, and

b3as calculated from (2.9) are

b1⫽ (2/a) (⫺1, 1, 1),

b2⫽ (2/a) (1, ⫺1, 1),

and

b ⫽ (2/a) (1, 1, ⫺1).

[Incidentally, note that all the reciprocal lattice vectors of the fcc lattice havethe form (2/a)(i, j, k), where i, j, and k have to be either all odd or all even].

The first Brillouin zone of the fcc structure is also indicated in Fig 2.2c Thesymmetry of this Brillouin zone can be best visualized by constructing a modelout of cardboard A template for this purpose can be found in Fig 2.27

In Fig 2.2c we have labeled some of the high-symmetry points of this louin zone using letters such as X and ° We will conform to the convention

Bril-of denoting high symmetry points and lines inside the Brillouin zone by Greek letters and points on the surfaces of the Brillouin zone by Roman letters The

center of the Brillouin zone is always denoted by° The three high-symmetrydirections [100], [110], and [111] in the Brillouin zone of the fcc lattice aredenoted by:

examina-as a 90˚ rotation about axes parallel to the edges of the body-centered cube

in Fig 2.2c In addition it is invariant under reflection through certain planes

containing the center of the cube These operations are known as symmetry

operations of the Brillouin zone The symmetry of the Brillouin zone results

from the symmetry of the direct lattice and hence it is related to the try of the crystal This symmetry has at least two important consequences for

symme-the electron band structure First, if two wave vectors k and k  in the louin zone can be transformed into each other under a symmetry operation ofthe Brillouin zone, then the electronic energies at these wave vectors must beidentical Points and axes in reciprocal lattice space which transform into each

Bril-other under symmetry operations are said to be equivalent For example, in

the Brillouin zone shown in Fig 2.2c there are eight hexagonal faces ing the point labeled L in the center These eight faces including the L pointsare equivalent and can be transformed into one another through rotations by90˚ Therefore it is necessary to calculate the energies of the electron at onlyone of the eight equivalent hexagonal faces containing the L point The secondand perhaps more important consequence of the crystal symmetry is that wavefunctions can be expressed in a form such that they have definite transforma-tion properties under symmetry operations of the crystal Such wave functions

contain-are said to be symmetrized A well-known example of symmetrized wave

func-tions is provided by the standard wave funcfunc-tions of electrons in atoms, which

are usually symmetrized according to their transformation properties under

rotations and are classified as s, p, d, f , etc For example, an s wave tion is unchanged by any rotation The p wave functions are triply degenerate and transform under rotation like the three components of a vector The d

func-wave functions transform like the five components of a symmetric and less second-rank tensor By classifying the wave functions in this way, some