Physics of Optoelectronic Devices (Wiley Series in Pure and Applied Optics)

physics of Optoelectronic Devices SHUN LIEN CMUA.NG Professor o f Electrical and Computer Engineering t'riiversi ty of Illinois a t Urbana-Chazpzizn

Wiley S e r i ~ sin Pure and Applied Optics T h e Wiley Series in P u r e and Applied Optics p~iblisliesoutstanding books in the field of optics. T h e nature of these books m a y b e basic ("pure" optics) o r practical ("applied" optics). T h e books are directed towards o n e o r more of the following nucliences: researchers i n universities. govern-. ment, o r industrial laboratories; practitioners o f optics in industry; o r graduate-level courses in universities. T h e eniphasis is o n the quality of the book anti its impol-tance to the discipline of optics. This tc'xt is printed o n acid-free paper. Copyright @ 1995 by J o h n Wiley & Sons. In(,. All rights reserved. Published siniultaneously irl C a n l ~ d a . Reproduction o r translariun of a n y part of this work heyond that prrnlitted by Section 107 o r 108 of the 1976 Unjteii . . Sti1rt.s Copyright Act without the pernlission of the copyright owner is unlawful. Requests for pcrmission o r further int'orn~:~tion should be adclrcssrd to the Permissions D e p a r t n ~ z n t . . e w York. NY J u h n Wilry & Sons, Inc.. 605 T h i r d A v e ~ l u z N 10155-0012.

Lihraq of Congress Catuloging-in-Publication D U ~ L I : C h u ~ l n g S. , L. Physics of o p t o e l ~ a t r o n i cdevices / S.L. Chunng. cm. - (Wiley series in pure ancl applied optics) p. "Wilzy-Interscience publication." lSHN 0-17 1-10939-8 ( ~ { l kp.i l p ~ ~ . ) I. Elzctrooptics. 2. Elt.ctl.ooplica1 c\evicr.s. 3. Si.~!~iic?ncluctori. I . Title'. I!. S e r ~ e s . C)C673.C4S 19?5 i14-24'70 I 62 1.35 1'(i.15--~lc20 I.'rinteJ i n the IIniteJ Stares

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Physics of Optoelectronic Devices

Preface This textbook is intended for graduate students and advanced undergraduate students in electrical engineering. physics, and materials science. It also provides an overview of the theoretica1 background for professional researchers in optoelectronic industries a n d research organizations. T h e book deals with the fundamental principles in semiconductor electronics, physics, and electromagnetic~,a n d then systematically presents practical optoelectronic devices, including semiconductor lasers, optical waveguides, directional couplers, optical modulators, and photodetectors. Both bulk a n d quantumwell semico~lductordevices are discussed. Rigorous derivations are presented and the author attempts to make the theories self-contained. Research o n optoelectronic devices has been advancing rapidly. To keep up with the progress in optoelectronic devices, it is important to grasp the fundanle~ltalphysical principles. Only through a solid understanding of fundamental physics are we able to develop new concepts and design novel devices with superior performances. T h e physics ofoptoelectronic devices is a broad field with interesting applications based o n electromagnetics, s e n ~ i c o ~ i d ~ l c t o rphysics, and quantum meclianics. I have developed this book fbr a course 011 optoelectronic devices which 1, have taught at the University of Illinois at rbana-Champaign for the past ten years. Many of our students are stiin~.~lated by the practical applications of quantum mechanics in sen~iconductoroptoelectronic devices because many quantum phenomena can be observed directly using artifical materials such as quantum-,well heterostnlctures with absorption or emission wavelengths determined by the quantized energy levels. '

Scope

This book emphasizes the theory of' semiconductor optoelectror~icdevices. Comparisons between theoretical and experimental results are also shown. The book starts with the fundanlentals, including Maxwell's equations, the continuity equation, ar.d the basic semiconductor equations of solidLstate slcctronics. These equations are 'essetltial in learning scn,icond~ictorphysics applied to optoclectronics, We then disoass t'r~t.p/*ti,r~lg~ltiolz. yc.rzt*ratiouz,inod~lIntion, curd rl~~tecriotl ~,J'li,ght. which z.1 re tllr keys to .ii nclerstanding the p11 ysics knowledge of the behind the operation cf optoelect.ronic rlevict:~.Fc~r~xanlpl.ct, ceneration anti propng:;ition of 1 i i ~ I l is t (:ri!ci;:il for -u~~derstar;ciing h o w a semi.. concl~iciurlaser t,pci-ntrs. 1 11;: t h c o r ~vf' ga.i:.! ~ ( j z f i i c i e i t rl j f scrujconductorlvsers shows hotv Iicrl.lt is ii1~1i3jj[ i c ~ ;; i,i k i wi1;'t"giii~1?theory shows how light is L.

r.

r .

Cc

.

-

1

.#

confincd to the w:~veeuiclein a laser cavi ly.An undel-standing of the modulation of light is useful in designing optical switches and nlodulators. The absorption coefficient or bulk a n d cliianturn-well se~niconcluctorsdernonstrates how light is detected and leads to a discussion on the operating principles of photodetectors.

Features lnlportant topics such as semiconductor heterojunctions a n d b a n d structure calculations near the band edges for both bulk a n d quantum-well senliconductors are presented. Both Kane's model, assuming parabolic bands and-Luttinger-Kohn's model, with valence-band mixingeffects in quailtun] wells, are presented. Optical dielectric waveguide t h e o ~ yis discussed and applied to semiconductor lasers, dil-ectional couplers, and electrooptic modulators. Basic optical transitions. absorption, and gain are discussed with the tinle-dependent perturbation theory. The general theory for gain and absorption is then applied to studying interband and intersubband transitions in b u l k a n d quantum-well semicondtictors. Inlportant se~niconductorlasers such as double-heterostructure, stripegeometry gain-guided sernicond~ictorlasers. quantum-well lasers, distributed feedback lasers, coupled laser arrays, a n d s~lrfiice-emittinglasers are discussed in great detail. High-speed modulation of semiconductor lasers using both linear and nonlinear gains is investigated systen:atically. T h e analytical theory for the laser spectral linewidth enhancenlerlt factor is derived. New subjects such as theories o n the band structures ofstrained semiconductors and strained quantum-well lasers are investigated. T h e electroabsorptions, in bulk (Franz-Keldysh effccts) a n d quantumwell sen~iconductors(quantum confined Starkeffects). are discussed systen~aticaltyincluding exciton effects. Both the bound a n d continuum states o f escitons ilsing the hydrogen atom model are d i s c ~ ~ s s e d . Intersubband transitions in quantum wells, in addition to conventional interband absorptions for fc.r-infrared photodetector applications, are presented. Courses

A few possibie courses for thz use of this book are listed. Some backgl-oiind i n unciergruduate electrornagnetics anci ~~~~~~~11 physics is assumed. A back-zrouncl in cluantuln mechanics will be I?tlprt~Ibut is not required, since all of the essentials a r e coverrcl i l l rile chapt-rs on Fundiirnental:;. *

Oveniesv of Optoelectronic Devi.crs: Chapter 1. Chapter- 2, Chapter 3 (3.1, ? -3 .. 5 . .-'1.7). Chapter -i (7. ! -7.-?. 7.6). CI-lapt$r 3, Chapter 9 (9.I-9.6), Chap3. .-, ?' (-1: . > ,.r... .t.3*;. .- , , t . k , l - s :l:>J C'!~:IJJ!L.I. 14. !

Optoelectronic Device Physics: Chapters 1-4, Chapter 7 (7.1, 7.5, 7.6), Chapters 9-14. Electrornagnetics and Optical Device Applications: Chapter 2 (2.1-2.4), Chapters 5-8, Chapter 9 (9.1-9.3, 9.5, 9.6), Chapter 10 (10.1-10.3, 10.510.7), Chapter 12, Chapter 14.

The entire book (except for some advanced sections) c a n also be used for a two-semester course. Acknowledgments

!

After receiving a rigorous training in my Pl1.D. work on electromag~leticsat Massachusetts Institute of Technology, I became interested in semiconductor optoelectronics because of recent developments in quantum-well devices with many applications in wave mechanics. I thank J. A. Kong, my Ph.D. thesis adviser, and many of my professors for their inspiration and insight. Because of the sigilificant i l u n ~ b e rof research results appearing in the literature, it is difiicult to list all ofthe irnport-tantcontributions in the field. For a textbook. only the f~lndamentalprinciples are emphasized. I thank those colleagues who granted nle pern~issionto reproduce their figures. I apologize to all of my colleagues whose important corltributions have not been cited. I atn grateful to many colleagues and friends in the field, especially D. A. B. Miller, W. H. K~lox,M. C. Nuss, A. F. J. Levi. J. O'Gorman, D. S. Chemla, and the late S. Schmitt-Rink, with whom I had many stimulating discussions on quantum-well physics during and afte;. my sabbatical leave at AT&T Bell Laboratories. I would also like to thank many of my students who provided valuable comments, especially .C. S. Chang and W. Fang, who proofread the h especially D. Ahn, C. Y. P ~naiiuscript.I t h a ~ ~ k m a n y o f m y r e s e a r cassistants, c h a o , and S. P. Wn, for their interaction o n research siibjecb related to this book. The support of my research on quantum-well optoelectronic devices by the Office ofNaval Research during the past years is greatly appreciated. I am grateful to L. Beck for reading the whole nlanuscript a n d K. C. Voyles for typing many revisions of the nlanuscript in the past years. The constant support and encouragement of my wife. Shu-Jung, are deeply appreciated. Teaching and cond~ictingresearch have beell the stimulus for writing this book; it was an enjoyable learning experience.

Contents Chapter 1. Introduction

1.1 Basic Concepts 1.2 Overview Problems References Bibliography

PART I FUNDAMENTALS Chapter 2.

-..

Basic Semiconductor Electronics

21

2.1 Maxwell's Equations a n d Boundary Conditions 2.2 Semicondilctor Electronics Equations 2.3 Generation a n d Recombination in Set-niconductors 3.4 Examples a n d Applications to Optoelectronic Devices 2.5 Semiconductor p-N and 12-PHeteroj~inutions 2.6 Semiconductor n-N Heter0junf:tion.s a n d Metal-Semiconductor Junctic-. .s Problems References .

chapter 3 . . Basic Quan turn Mechanics !

3.1 Schrodinger Equation 3.2 The Square Well 3.3 The Harmonic Oscillator 3.4 The Hydrogen Atom (3D a n d 2 0 Exciton Bound and Continuum States) 3.5 Time-Independent Perturbation Theory 3.6 Lowdin's Renormalization Method 3.7 Tinle-Dependent Perturbation Theory Problems References

Chapter 4.

4.1 4.2

Theory a j f FI:iechrt!!.;ic Band 5trncfures i a Semiconductors ~

'The Bloch '-Yhec-;lt-c:ni;~.!:d t l ~ cb, . p T~,lsthodhiSimple Bands Kanc's TvI~dc:! Gir Btir~iiStr~icture: 3r . p Flettlod . - 1.fi !:? iSi,icij (1) !,.I. ).i{i tll [I3:t: c; !.> l l 4 . i ~ 1 13

124 I24

1 29

xii

CONTENTS

4.3 Luttinger-Kohn's Model: The k p Method for Degenerate Bands 4.4 The Effective Mass 'Tlleory for a S i n ~ l cBand and Degenerate Bands 4.5 Strain Effects on Band Structures 4.6 Electronic States in a n Arbitrary One-Dimensiox~al Potential 4.7 Kronig-Penney Model for a Superlattice 4.8 Band Stntctures of Sen~iconductorQuantum Wells 4.9 Band Structures of Strained Semiconcluctor Quantum Wells Problems References Chapter 5.

Electromagnetics

5.1 General Solutions to Maxwell's Equations a n d Gauge Transbr~mations 5.2 Time-Harmonic Fields a n d Duality Principle 5.3 Plane Wave Reflection From a Layered Medium 5.4 Radiation a n d Far-Field Pattern Problems References

PART I1 Chapter 6.

PROPAGATION OF LIGHT Light Propagation in Various Media

6.1 Plane Wave ~ o l u t i q n sfor Maxwell's Equations in Homogeneous Media 6.2 Light Propagation in Isotropic Media 6.3 Light Propagation in Uniaxial Media Problems References Chapter 7.

Optical Waveguide Theory

7.1 Symmetric Dielectric Slab Wavesuides 7.2 ~ s ~ n ~ l n e tDielectric ric Slab Waveguides Ray Optics Approach to the Waveguide Problems 7.3 7.4 Rectangular Dielectric Waveg~liclcs 7.5 The Effective Index Method 7.6 Wiive Guiclance in a Lossy or- Gain Pvlediurn Problenls Rekrences

137 141 144

157 166 175 185 190 195 200

-" 200 203 205 214 219 220

xiii

I."OI\ITENTS

Chapter 8.

253

kVaveguidc Couplers and Coup1,ed-Mode Theory

8.1 8.2 8.3 8.4

bVaveguidc Couplers Coupling of Modes in the Time Domain Coupled Optical Waveguides Improved Coupled-Mode Theory and Its Applications 8.5 Applications of Optical Waveguide.Coup1ers 8.6 Distributed Feedback Structures Problems References

PART 111 GENERATION OF LIGHT Chapter 9. Optical Processes in Semiconductors

337

Optical Transitions Using Fermi's Golden Rule Spontaneous and Stimulated Emissions Interband Absorption and Gain Interband Absorption and Gain in a Quantum-Well Structure 9.5 Momentum Matrix Elements o.f Bulk and Quantum-Well Semiconductors 9.6 Intersubband Absorption 9.7 Gain Spectrum in a Quantum-Well LAaserwith Valence-Band-Mixing Effects Problems References

9.1 9.2 9.3 9.4

Chapter 10. Semiconductor Lasers

10.1 Double Hcterojunction Semiconductor Lasers 10.2 Gain-Guided and Index-Guided Semiconductor Lasers 10.3 Quantu m-Well Lasers 10.4 Strained Quantum-Well Lasers 10.5 Co~lpledLaser Arrays 10.6 Distributed Feedback Lasers 10.7 Surface-Emitting Lasers Problems References

Chapter I I. 1 1.1

Direct Wlodnla ti:,!] of S e m i c o n d ~ e t o Lasers r

Rate E q u a t i c t ~ s:incl Linear G;rin Analysis

3 95 412 42i 437 449 457 444 47 1 4'73

;

:,

A.

TIlc Hydrogen Atom (3D and 2D Exciton Bound and Continuum States)

B.

Proof of the Effective Mass Theory

C.

Derivations of the Pikus-Bir Harniltonian for a Strained Semiconductor

D.

Semiconductor Heterojunction Band Lineups in the Model-Solid Theory

E.

Kramers-Mronig Relations

F.

Poynting's Theorem and Reciprocity Theorem

G.

Light Yn~pagationin Gyro tropic Media-Magnetoop tic Effects

-,..

W. Formulation of the Improved Coupled-Mode Theory ]I.

Density-Matrix Formulation of Optical Susceptibility

J.

Optical Constants of GaAs and InP

K.

Electronic Properties of Si, Ge, and a Few 'Binary, Ternary, and Quarternary Compounds

Introduction Semiconductor optoelectronic devices, such as laser diodes, light-emitting diodes, optical waveguides, directional couplers, electrooptic modulators, and photodetectors, have important applications in optical communication systems. To understand the physics and the operational characteristics of these optoeIectronic devices, we have to understand the fundamental principIes. In this chapter, we discuss some of the basic concepts of optoelectronic devices, then present the overview of this book. I

BASIC CONCEPTS

?'he basic idea is that for a semiconductor, such as GaA.s or InP, many interesting optical properties occur near the band edges. For example, Table 1.1 shows part of the periodic tabIe with many of the elements that are important for semiconductors [I, 21, including group IV, 111-V, and 11-VI compounds. For a 111-V compound semicon 'rlctor mlch as GaAs, the gallium (Ga) and arsenic (As) atoms form a zinc-biei;de structure, which consisls of two interpenetrating face-centered cubic Iattices, one made of gallium atoms and the other made of arsenic atoms (Fig. 1.1). The Ga atom h+"an atomic nun-tber 31, which has an [Ar] 3d1"s'-tp1 configuration, i.e., three valence electrons on the outermost shell. (Here [Ar] denotes the configbration of Ar, which has an atomic nu~nber 18, and the 15 electrons are clistribrlted as ls22s22p63s23p6.) The As atom has an atomic number 33 with an [Ar] 3d104s24p3configuration or five valence electrons in the outermost shell. For a simplified view, we show a planar bonding diagram [3, 41 in Fig. 1.2a, where each bond between two nearby atoms is indicated rvith two dots representing two valence electrons. These valence electrons are contributed by either G a or As atoms. The bonding diagram shows that each atom, such as Ga, is connected to four nearby As atoms by four valence bonds. If we assume that none of the bonds is broken, tve have all of the electrons in the valence band and no free electrons in the concf~.rctionband. 'The energy band diagram as a function of positiog is shown in Fig. 1.2b, where E, is the band edge of the cor~cluc:irjnb:in.(:i 3.11~1 is the band edge of the valerrcc band. When a light with an, optic;~ientcgv hrj abirve t h e h ; ~ n d g a pE , is inciderrt on the semicvnductor. c;ptical nbsorptior: is significmt. Here / I Is tl~e.Planck I?!,

Table 1.1 Part of the Periodic Table Containing Group I1 to VI Elements -

Group I I A

B

B

A

I [Ne] 3

Group VI

Group V

Group IV

Group I11 A B

A

I [Ne] 3s'?p3

~ ~ 3 ~ '

B

A

[

[Ne] 3 ~ ~ 3 ~ '

32 Ge

33 As

34 Se

[Ar] 3d1'

[Ar]3d1"

[Ar] 36'" 4s'4p4

4s24p'

3 ~ ~ 4 ~ "

50 ~n

51 Sb

52 Te

[Kr] 4d "

[k4d1? ]

I

48 Cd

49 In

[Kr] 46'''

[lncluctorlaser, a n d the quantum e-fficiencyfur converting cp'iic;ll pol&or to current iil st photodetector, havc to be investigated 11s;rig tt~esi:sen-!iconductor electronic equations These fundamental equations xcturil!y are i : ~ ~ ~ p If.0 e d eiic11 other, and the 7.

most complete solution would require a self-consistent scheme, which would require heavy co1nput;itions. Fortunately, with good understalldings of most of the device physics, various approximation rnetllods, such as the depletion approximation and p e r t ~ ~ r b a t i otheories p for various device operation conditions, are possible. The validity of the models can be checked with full numerical solutions and confirmed with experimental observations. After we explore the fundamentals, we investigate optoelectronic devices for propagation, generation, modulation, and detection of light. Below, we list some of the major issues of study in this book.

Part I Fundamentals

-

Basic Semiconductor Electronics (Chapter 2) What are the fundamental semiconductor electronics equations governing the electron and hole concentrations and their corresponding current densities? How are the carrier densities affected by the presence of optical illumination or current injection? How are the energy-band diagrams drawn for heterojunctions such as P-n, N-p, p-N, or n-P junctions? Here a capital letter such as P refers to a wide-band-gap nzaterial doped P type, and a small p refers to a sn~allerband-gap semiconductor doped p type. Knowing the energy band bending, carrier injectior;, and current flow is a very important step to understanding the device operation characteristics. Basic Q ~ l a n t u mMechanics-(Chapter 3) What are the eieenvrrlues and eigenfunctions of a rectangular quantum-well potential? These have important applications to semiconductor quantum-well devices. What are the bound- and continuum-state soll~tionsof a hydrogen atom? The spherical harmonics solutions are clsefu1 when we talk about the band structures of the heavy-hoIe and light-hole bands of bulk and q&nturn-we~l semiconductors. T h c radial functions are useful for describing an exciton, formed by an electron-hole pair bounded b,y the Coulomb attractive potential, which is exactly the hycl rogen model. What is ~ e r m i ' sgolderi r u l e for thc transition rate of a semiconducto~in the presence of optical excitiitiun'? This rule is the basis fur deriving the optical absorption ;ind p i n in semi.conductor devices. V/h;~t are the time-ir~dept.ni;!cnf r)e!.tur!~ationmethod and Lijwdin's h;!ve applications to Kane's medel pcrt~lrbation ~ n e ~ l ~ o c'T'hzsc l'? ;111cl Luttinger-lic~iln's l i l ~ c l c lio!- *::ilc2~i.-band s t r u c t ~ ~ r e s .

*

Theory of Electronic Band Structures in Semic~onductors(Chapter 4) What are the band structures near the band edges of a direct band-gap semiconductor'? What is the effective-mass theory and how is it used to study. the electronic properties of a semiconductor quantum-well structure? How do we calculate the electron and hole subband energies in a quantum well and their in-plane dispersion curves? What are the band structures of a strained bulk semiconductor and a strained quantum well? Electromagnetics (Chapter 5 ) What is the far-field radiation pattern if the modal fie1.d on the facet of a diode laser is known?

Part 11 Propagation of Light

Light Propagation in Various Media (Chapter 6) What are the optical electric and magnetic fields of a laser light propagating in or reflected from a piece of semiconductor? What is a uniaxial medium? What are the basic concepts for a polaroid and quarter-wave plate? Optical Waveguide Theory (Chapter 7 ) How do we find the modes and their propagation constants in slab waveguides and rectangular dielectric waveguides? (a) A double-heterojunction semiconductor laser

Light ou [put

(b) A distributed feedback sen~iconductorlaser

Figrlrc 1.8. A cross secticn c.)f (a 1 a dr-:u!~lc'-ht'lr:.c~jui-:zti0i1~ , z t ~ ~ i c u n d u cIxsrr t c ~ ~st:iic:turz SpitCc anil ( b ) 11 dist~.il?utecllieriit~ncksernicot~ducror-la...:!-.

ir-r r e d

A directional couplcr modulator

Figure 1.9. A directional coupler modulator w l ~ o s eoutptlt light power Inay be switched by an electric field bias.

How do the propagation constants change in the presence of absorption or gain in the waveguide? These waveguide modes are very important for applications in a semiconductor laser (Fig. 1.8a) and a directional coupler (Fig. 1.9). Waveguide Couplers and Coupled-Mode Theory (Chapter 8) What is the coupled-model theory and how do we design-waveguide directional couplers? What are the solutions for the wave equation in a distributed feedback structure? T h e distributed feedback structure has applications in a semiconductor laser as well, Fig. 1.8b. Part 111 Generation of Light

Optical Processes in Sen~iconductors(Chapter 9) What are the basic formulas for optical absorption and gain in a semiconductor material? What is the differei~cein the optical absorption spectrum between a bulk and a qua'ntum-well semiconductor? What is the difference between an interband and intersubband transition in a quantum-well structure'? 10) ~ e m i c o n d u c t o r ' ~ a s e r(Chapter s What are the operational principles of different types of semiconductor lasers? What determines the quantum efficiency and the threshold current of a diode laser?

fart lV Modulation of Light Direct Modulation 01Szn11cond~:ctorLascrb (Chapter 1 I) How d o wz clirectly fi~odulatcthe I l g i ~ to ~ l t p u tfrom a semiconductor laser?

Photons

1

$ Photons

*

Figure 1.10, A p-n function yl-lotodiode with optical illumination from the top or from t h e bottom.

What determines the spectral linewidth of the semiconductor laser light? Electrooptic and Acoustooptic Modulators (Chapter 12) How do we modulate the intensity or phase of light? Elec troabsoi-ption kIodulators (Chapter 13) How do we control the transn-lission of light passing through a bulk or a quantum-well semiconductor with an electric field bias?

Part V Detection of Light Photodetectors (Chaptcr 14) What are the different types of photodetectors and their operational characteristics'? A simple example is a p-n junction photodiode as shown in Fig. 1.10. The absorption of photons arid tile conversion of optical energy to electric current will be in;. ;tigated. What are quant~im-wellintersubbancl photodetectors? Many of the above questions are still research issues that are &der intensive investigation. The materials presented in this book emphasize the fundamental principles and analytical skills on the essentials of the physics of optoelectronic devices. The book was written with the hope that the readers of this book will acquire enough analytical power and knowled'ge for the physics of optoelectronjc devices to analyze their research results, to understand more advanced materials in journal papers and research, monographs, and to generate novel designs of optoelcctrunir devices. Many hooks that are highly reconlmerlded for further reading 2re listed in the bibliography.

PROBLEMS 1.1

id Calculate the band gap v:;.iv;3!~:1.gth GaY a t 300 K.

f o r Si, GaAs, InAs, lnP, and

,A,,

Use the [3aj;c'l gap erle;,gil;-sj i : , A , , i 7 p f : j l i i 1 ~ K. (b) Find the optic:ii cjlcri;;: !j.:::Sl.l..:;~~!i!.;j; i-rl !.he wavelength l.3prn and 1.55j.t m. i .

INTRODU! ITION

14

1.2

(a) Find t 1 1 ~galliii~nmole i'raction x for I n , -,Ga,, As compound semiconductor such that its lattice constant equals that of InP. The Iatticc constants of a few binary compounds are listed in Appendix K. (b) Find the allunlinum mole fraclion s for A1,In, -,t As such that its lattice constant is the same as that of InP.

1.3

Calculate the band edge discontinuities A E , and AE,. for GaAs/ Al,Ga,-,As heterojunction if x = 0.2 and x = 0.3.

REFERENCES 1. C. Kittel, Introdlrctiorz to Solid Stale Physics, Wiley, New York, 1976. 2. N. W. Ashcroft arid N. D. Mcrmin, Solici Sftrtc Physics, Holt, Rillehart & Winston, Saunders College, Philadelphi;i, 1976. 3. B. G. Streetman, Solicl State Electroriic Dei~ices,Prentice Hall, Englewood Cliffs, NJ, 1980. 4. R . F. Pierret. Ser?licond~icto~. F~indtlmelztals,Vol. 1 in R. F. Pierret and G. W. Neudeck, Eds., rtlod~ilarSeries on Solid State De~iices,Addison-Wesley, Reading, MA, 1983. 5 . W. T. Tsang, Volume Ed., Lightwaue Corntnliniccrtions Technology, Vol. 22, Parts A-E, in R . K. Willardson and A. C. Beer, Eds., Sernicondrictor ariii Setnilrletals, Academic, New York, 1985. 6. H. C. Casey, Jr., and M. B. I'anish, Hetrrostr~ict~ire Lasers, Parts A and B, Academic, Orlando, FL, 1978. 7. T. Miya, Y. Terunuma, T. Hosaka, and T. Mjyashita, " A n ultimate low loss single mode fiber at 1.55 pm," Elect~orz.Lett. 15, 106-108 (1979). 8. D . S. Chemla and A. Pinczuk, Guest Ed., Special issues on Semiconductor Quantum Wells and Superlattices: Physics and Applications, I E E E J. Qtlczntr~rn Electron. QE-22 (September 19S6). 9. P. S . Zory, Jr., Ed., Qitnntzirn I.!'ell Lasers, Academic, San Diego, 1993. 10. T. P. Pearsall, Volumc Ed., S r r a i ~ e dL n ~ e rSziperlcrttices: Physics, Vol. 32, 1990; and Stmiized Layer S~lperlatt~ces: Miter-ids Scietzce and Technology, Vol. 33, 1991, in R. K. Willardson and '4. C. Beer, Eds., Senzicondrictor rzrzd Srmimztals, Academic, New York.

General Semiconductor Optics: arid Electronic Physics -

.

I . S. M. Sze, Physics ;?f' Sc1r~rico!.;d!~clr!~; ~ L . I I : I ? . S . Wiley, Ncw York, 1951. 2. S. Wang, F~rr~clilr?~c.nrr~i; (-1:' Serrzicoitt6~~::1c)ii T / ~ ~ o l ~lrzd - ) l Dt.1:ic.e P/~ysics,Prentice I i u l l , Engle~vooc!C!itTs, N.1. 1989.

3. B. R. Nag, T h e o ~of Eieorrical Trcrrtspo?.i irl Scr71icorzdzlctors, Pergamon, Oxford, UK, 1972. 4. 73. K. Nag, EIe'ctr.oi; I rrrr~j.porti r l Coit y c > ~ u z c Senticoiltl~~r:t~r~s, f Springer, Hcrlin, 19SO. 5. B. K. Ridley, Qz~arrtrlrnPlac~~.ssc,s in Ser~~icc~~clzrcfor.~, 2d ed., Clarendon, Oxford, UK, 1988. 6. N. W. Ashcroft and N. D. Merrnin, Solid Stcrte E'/~y.si~s,Holl, Rinehard & \Vitlsto11, New York, 1976. 7, H. Haken, Iiglzt, Vol. 1, T+'~i~:cs, P/lorotzs, Atorrts, North-Holland, Amsterdam, 1986. 8. R. Loudon, The Quantum Theory of L,igIqt, 2d ed., Clarendon, Oxford, UK, 1986. 9. F. Bassani and G. I?. Parravicini, Electr-orzic States atld Optical Transitiorzs in solid.^, Pergamon, Oxford, IJK, 1975. Hetc~.ostrz~ct~~res, Halsted, 10. G. Bastard, Ware 1?1fcolziz!tics Appliecl to So~tzicortdii~tor New York, 1988. 1I. 1-1. Maug and S. W. Koct:, Qrln~ltrrrnTlicury ofthe Opticizl arzd ELectrorzic Properties uf'Sertlicondtictors, World Scientific, Singapore, 1990. 12. K. Hess, Aciucirzced Theory of Scrrricorrd~ictorDeuicds, Yrentice Hall, Englewood Cliffs, NJ, 1988. Pergamon, Oxford, UK, 13. I. M. ~ s d i l k o v s k i ,B~rnclStr-~ictttreof Scmicui~d~~crors, 1982. 14. K. Seeger, Setniootzd~~(.ror Pllysics, Springer, Berlin, 1982. 15. K. W. Beer, S L L Pof- Serrti~ui~~i~tctor L~ Physics7 Van Nostrand Reinhold, NEWYork, 1990. 16. C. Weisbuch and B. Vinter, Qi~trnlurnSenzicorzdrtctor Stjzrcticres, Academic, New York, 1991. 1.7. R . K. Williardson and A. C. Beer, Eds., Opticczl Properties of I![-V Compo~irzds, Vol. 3 in Sernico~tdirctorslrrzd Serninletals, Academic, New York, 1967. 18. K. K. Williardson and A. C. Beer, Eds., hfod~ilatiun TEC/E~L~CILLCS, VOI. 9 in Sernicoildiictors u~zdScnzinzetals, Academic, New York, 1972. 19. M. Cardona, l k f o ~ i ~ ~ l ~\Spcc~roscopy, ~tion in Solid State Phys., Suppl. 11, Academic, New York, 1969. - 7

General Optical or Quaritum Eiectronics 20. A . Yariv, Qriarrtrtr~rElectr-or&-::, 3d ed., LViley, New York, 1959.. 21. A. Yariv, Opticczl Elect,-unics, 3d ecl., Holt. Rinehart Sr Winston. New York, 1985. 211. H. A. Haus, P Y L ~ L : ~~zizd s F'ield.~zil~ OptoeI~~cfrortic'~~, Prentice Hall, Englewood Cliffs, NJ, 1983. . 2-3. .J. T. Vt:rdeq.cn, La.sr:r Eiectt.:~ir!i:.; f'rentic::~ Elail! Englewood Cliffs, NJ, 1989. 23. R. E. A. Saieh and LM. C, l.cic:l:, I''~l~!~/i~t:tc:fi~~~i~ {if Photnrrics, Wiley, New York. 1991.. ,1.3. A. E;. Cihatak ancl K. ?'::yag;t:.:.,jr!!7, .:.',!pllcr:/ .!7!~r,i-;;-~l:ic>r(zii(~n and Testing, Wiley, New York, 1990. 49. R. K. Willardson and A. C. Beer, Eds., Infrared Detcctc~rs,Vol. 5, in Serrriconductors cind Serniinetnls, Academic, New York, 1970. 50. R . K. Willardson and A. C. Beer, Eds., Infr~zred Detectors II, VoI. 12 in Sernicorzducbors arzci Set-zziilretnls, Academic, New York, 1977. 51. R. K. Willardson and A. C . Beer, Eds., Mercrlry Cndmi~imTell~iride,Vol. 18 i11 Semicond~lctorsnnd Seminzetnls, Academic, New York, 1981. 52. A. Rogalsi, Ed., Selected Papers on Sertzicorzd~~ctorIrzfr~zred Detectors, SPIE Milestone Series, Vol. MS66, SPIE Optical Engineering Press, Bellingham, WA, 1992. 53. M. 0. Manasreh, Ed., Semicondzrctor Q ~ ~ n n t ~ cWells r n ntzd S~dperlntticesfor LotzgPVnr!elength Irzfrared Detectors, Artech House, Boston, MA 1993.

~ o n l i n & r Optics 54. N. Bioembergen, No~zlittear Optics, Addison-Wesley, Redwood City, CA 1992. (Originally published by W.A. Benjamin, Inc., 1965). 55. R. W. Boyd, Nonlinear Optics, Academic, San Diego, 1992. 56. A. C. Newell and J. V. Moloney, Novllinenr Optics, Addison-Wesley, Redwood City, CA 1992. 57. Y. R. Shen, The Pi-itrclples of Norzlirlear Optics, Wiley, New York, 1984. 58. H. M. Gibbs, Oplicnl Bistnbilify: Controlling Light with Ligt.': Academic, San Diego, 1985. 59. H. Haug, Ed., Opticnl Nonlirtenl-ifies and lr~stnbilitiesilz Sernicond~~ctors, Academic, San Diego, 1988.

PART Fundamentals

Basic Semiconductor Electronics In the study of semiconductor devices such as diodes and transistors, the characteristics of the devices are described by the voitage-current relations. The injection of electrons and holes by a voltage bias and their transport properties are studied. When optical injection or emission is involved, such as in laser diodes and photodetectors, we are interested in the optical field in the device as well as the light-matter interaction. In this case, we loolc for the light output versus the device bias current for a laser diode, or the change in the voltage-current relation due to the illumination of light in a photodetector. In general, it is useful to know the voltage, current, or quasi-static potentials and electric field in the electronic devices, and the optical electric and magnetic fields in the optoelectronic devices. Thus, a full understanding of the basic equations for the modeling of these devices is very important. In this chapter, we review the basic Maxwell's equations, semiconductor electronics equations, and boundary conditions. We also study the generation and recombination of carriers in semiconductors. The g e n ~ r a ltheory for semiconductor heterojunctions and semiconductor/metal junctions is also invest?gated.

2.1

MAXWELL'S EQUATIONS AND BOUNDARY CONDITIONS [I, 21

Maxwell's equations are the fundamental equations in electromagnetics. They were first established by James Clerk Maxwell in 1873 and were verified experimentally [3] by ~ e i n r i c hHertz in 1888. Maxwell unified all knowledge of electricity and magnetism, 'added a displacement current density t e r ~ n JD/ar in Ampkre's law, and predicted electromagnetic wave motion. H e explained light propagation arci a n clectr~nlagnetic wave phenomenon. Heinrich Hertz demonstrated experimentally the electromagnetic wave phenomenon using a spark.-gap ger~er;itor as a !ransmittcr and a loop of' wire with a very small g a p as a receiver. He then set off' a spark in the transmitter and showed that a spark at the receiver was pr:c!clucecl. For' a histcxical account of the classical and qu.a-litrl!nthc.ory of i-ight,see Ref. 3.: for example.

i3AS!C SEMICC)I\~L>~J!e mobility. Since r I i l = ,u,[V/ t is the :ivex.apt. cli-ift ,L. v c i 3 of tile eiec:crons, we can write the photocurrerlt as

BASIC SEMICOP: DlJ(,T3II ELECT'RON[CS

45

where 7 , = P / L ! , , is the average transit tirn.2 of the elcctrons across the a r ~ d(C;,,A 1: is just the total number of photoconductor of a length generated electron-hole pairs in a volume A e. The ratio of the carrier recombination lifetime r,, to the transit time T, is the photoconductive gain, w m p i e A homogeneous gel-inanium photoconductor (Fig. 2.8) is illuminated with an optical beam with the wavelength h = 1.55 ,urn and an optical powel--P = 1 mW. Assume that the optical beam is absorbed uniformly by the photoconductor and each photon creates one electron-hole pair, that is, the intrinsic quantum efficiency q i is unity. The optical energy h v is

where h is in microns. T h e photon flux per second:

(I>

is the number of photons injected

Since we assume that all the photons are absorbed and the intrinsic quantum efficiency is unity, we have the generated rate per unit volume in (wd e):

We use

The injected "primary" photocurrent is q ~ ; = @ 1.6 X 10-'' = 1.25 rnA. T h e average electron trhnsit time is

e

T1,?=

0.1 a n

v =

=

7

I.,

\

-

,

\

a113t h e average transit tirrle Ifor holes is

X

7-81 X ~ O ' ~ A

2.56 X l o p 6 s

The photocurrent in the photoconductor is

where a photoconductive gain of around 570 occurs in this example. 2.4.2

I

Nonuniform Carrier Generation

If the carrier generation rate is not uniform, carrier diffusion will be important. For example, if the optical illumination on the semiconductor is limited to only a width S, instead of the length e (Fig. 2.9a), the excess carriers generated in the illumination region will diffuse in both directions, assuming there is no external field applied in the x direction [13]. Another example is in a stripe-geometry semiconductor laser [20] with an intrinsic (i) region as the active layer where the carriers are injected by a uniform current density within a stripe width S (Fig. 2.9b). We ignore the current spreading outside the region lyl IS/2.

(a)

Optical illumination

m O g

(c)

Semiconductor 0

Figure 2.9. (a) C)ptical in;ecrion into ;I :cmic3n:!i;rtorr -ri!t? iiti~rnination is .over a width S. (b) Current injection i n t o tile acti\i:: ( i i ;i::g,'i(>n w i t h a thickl-less d of a stripe-ge~rnetrj, sernicor,ducto~laser (c) Tile carriel. pe:?c~.;!*i;:iz;-at,- as a Fur!c:l;i>t") of the position. (d) The excess carrier distribution after diffusion.

-.

T h e continuity eclualion for the electrons in the bulk semiconductor in Fig. 2.9a or in the inti-insic region in Fig. 2.C)b:is

and the electron current,density is given only by diffusion:

A t steady state, d / J t

=

0 and

a2 D , , - p a ~-

st1 -

==

- G )I ( y )

(2.4.18)

7 , I

Here G,(y) is given by the optical intensity in Fig. 2.9a or, in the case of electric injection with a current density J o from the stripe S, Fig. 2 . 9 ~ :

where q = 1.6 x 10-I" C, and d is the thickness of the active region. T h e .. solution to Eq. (2.4.16) is

where L, = (D,,T,)'/' is the electron diffusion length. From the symmetry of the problem, we know that 6 r z ( y ) must be an eb'en function of y. Therefore, the coefficients A ancl B must be ecli;al. Another way to look at this is that the y cornponeni of thc electron c u r r e n t hensity J,,(y) must be zero ;it the symnletry plane 1) = 0. Therefore (d,/dy)i';fl(y) = O at y = 0, which also gives A = B. Matching the boundary conditions nt ; -= ,S,/2, in which 6 n ( y ) and .J,,(y) are continuous, we find R i i j ~ dC. T h e f i n d expressions for S n ( y ) can be pllt

in the form

Tlle total carrier concentration n( y) = n o + an( y) -- art( y); since the active region is undoped, n o is very small. This carrier distribution n ( y ) (Fig. 2.9d), is proportional to the spontaneous emission profiIe from the stripe-geometry semiconductor laser measured experimentally. In Chapter 10, we discuss in more detail the stripe-geometry, gain-guided semiconductor laser, and take into account the current spreading effects.

-__ 2.5

SENIICONDUCTOR p-N AND n-P IlETEROJUNCTIONS C20-251

When two crystals of semiconductors with difierent energy gaps are combined, a heterojunction is formed. The conductivity type of the smaller energy gap crystal is denoted by a lower case n or p and that of the larger energy gap crystal is denoted by an upper case N or P. Here we discuss the basic Anderson model for the heterojunctions [21, 221. It has been pointed out that a more fundamental approach using the bulk and interface propertics of the semiconductors should be used for the heterojunction model [33-251. In Appendix D, we also discuss a model-solid theoly for the band lineups of semiconductor heterojunctions including the strain effects, which are convenient for the estimation of band-edge discontinuities. 2.5.1

Senriconductor p-rV Heterojunction

Consider first a p-type narrow-gap semiconductor, such as GaAs, in contact with an N-type wide-band-gap semiconductor, such as AI,Ga,_,As. Let x be the eIectroil affinity,which is the energy required to take an electron from the conduction band edge to the vacuum level, and let @ be the work function, which is the energy difference between the vacuum level and the Fermi level. In each region, the Fermi IeveI is determined by the charge neutrality condition

where rr ancl p art: related tu the ilu~~si--F'i-rmi ieve!s f;;, and F,, respectively, through (2.2.39) or (2.2.47). F r ~ re.:itnlple, ir; tlle 17 region; y >is= 11, we may

denote PJLl = N - h

as thc "net" acceptor c(.,ncentration. If

1%

>+-ni,we

50

then have

which will determine the Fermi level Fp in the p region. Similarly, using ND = No+- NA- as the "net7' ionized donor concentration, we have

which wilI determine the bulk Fermi level FN in the N region before contact. When the two crystals are in contact, the Fermi level will line up to be a constant across the junction under thermal equilibrium conditions without any voltage bias. Thus there will be redistributions of eIectrons and hoIes such that a built-in electric field exists to prevent any current flow in the crystal. T o find the band bending, we use thc depletion approximation. -

Depletion Approximation for an Unbiased p-N Junction. Since the charge density is

and the free carriers y and the junction, we have

17

are depleted in the space charge region near

where again N,,is the net acceptor concentration in the p side, and the net donor concentration on the N side. From Gauss's law, we know that 'T . ( E E ) = p, which gives

N, is

where t,, and E , ~ , are the p ~ r ~ n i t t i v i tiny the 1 7 dild IV regions, respectively. Gauss's law states th;it: tilt: sli)pe c ~ f the E(.\: profile is given by the charge density dividc.cl by the permittibrity. T h w the electric field is given by two

2.5

SEM1::ONDUCTOR

p-hr AYJ3

ti-;-'

~$ET'ETT).JuN~~IONS

linear functions

in the depletion region and zero outside. The boundary condition states that the normal displacement vector D = EE is continuous at x = 0:

The electrostatic potential distribution 4 ( x ) across the junction is related to the electric field by

which means that the slope of the potential profile is given by the negative of the electric field profile. If we choose the reference potential to be zero for x < -x,, we have

where

I.' t.)

. == h.

!j

Pa$, '1

+I-

2 E t.,

!O

Thus

'0 q 'N, -(x 2 E,, E,.(x)=

(

X

'

7

.. q Ll\!.,

- A EL,-b 7 x ,i 1r , /

clb/cl - h .EL.

-X,,

-x,, I .K < 0

.Y,*)-

q'~,,,


0) and reverse bias (V, < 0) conditions are shown in Figs. 2.20a and b. Note that the positive electrode of the voltage source is always connected to the P side of the diode for the definition of the polarity. The I-V4 curve of the n-P diode is similar to (2.5.60),

where D,, L,, and p,,,, refer to the quantities in the n region, while D N , L N , and N,, are the quantities in the P region.

2.6.1

Semiconductor

11 -rY

Heterojuncrj!,ns

bit morz involved. With the development of t t ~ ehigll e!r=clrcin rnobilily !.r:li:.sistor (HEMT!, also called the

For an

hcteroj~lnction.the thzo~yI:;

; i

(a) Forward bias n-type

Y-type

- -Vacuum level

_ - - I - - - - - - -

Ip

/

#

------------Vacuum level

/

(b) Reverse bias

I I

/

/ /

Fp(x)

EC"

Fn

+---,, FP

'+ '-

FP

n /

F"(x)

EVP

EvnFigure 2.20. The energy band diagrams for an n-P heterojunction diode under (a) forward bias and (b) reverse bias.

rrlcjdulation cloped field effect trrrnsistc:!? (MODFET), the rr-N he terojunct ion theory is very ~ ~ s e f ufor l rrlodelinp these devices [%I. D e p e n d k g on the model, the results snay be cfiilkrei~t. A ~~~~~~~ative approach is shown in Fig. 2.21. When -the two crystals ivi th !?-type and rV-type dopings are brought in contact, the charge r-edisiributioil prociuces a space charge region. Some electrons will :;pill over fr'r-on-Ithe iLr r.eginn tc> t 1 1 i ~ 1region for the alignment

(a) Before contact

Step 3 ) Electric field lstribution

Step 4) Electrostatic potential

(b) After contact (VA= 0)

Step 1 ) Line up the Ferrni levels

E,"

F,------------------

E c ~ FN

Step 5) Band diagram Step 2) Space charge distribution

I-s $(x)

x.

( a ) A cl~luniurx!\v,-il tvirl; \,l/iilti~ L il!;cl di:ipersion i n tile X , or k,,space using Eq. 1.3.2.3). Figure 3.1.

;iji

ii~fniii: b,ii.rizr b"

-

;L.

(b) 'I'ht. encrg;c

.:Y

E;\S?C: (~UW?.UM /1iEt::M 4iJICS

yuantized electron subba11c1energies are

where

Therefore, we find thc subband energies are

Two-Dimensional Density of States. The electron concentration in a quantum well can be calculated using

bilere the silmmation is over all the occupied subbands. Since the electron energy is quantized in the k , quantum number, and the dependence on x and y is still plane-wave-like, we convert the sunlmation over k,and k, into integrations: ., 2

=

- k v k" =

. 2

dk.v

dk,

j.2?~/L:l2~,L,"

-

lld(k x2dk.v 4

- L.

where we have used d k , d i , = d(b,ii, d k , in the polar coordinate, k j = k,: + k t, d E = h'k, d k ,j r r l ' . We i n n then write the two-dimensional (2D) density of states as

Figure 3.2. The electron density of states P2D(E)solid line for a two-dimensional quantum-well structure is conipared with the three-dimensional density of states p,,(E) (dashed curve).

for x < 0. The electron density n is then given by

The density of states p,,(E) is plotted in Fig. 3.2. The step-like function with each step occurs wherever there is a new subband energy level E,, E , = 4E,, E j = 9E1, etc. It is interesting to compare the two-dimensional density of states with the three-dirnensionaI density of states (2.2.36):

wh,ich is the same as p2,(l.E = E,,). The three-dimensional density of states p,D(E) is shown as the d;ished c u n c in Fig. 3.2 with a smooth v / F behavior above the conduction [~c211(.! edge, v+,hitc the two-djn~ensicn;~l pz,(E) shows a sharp step-like behavi;r a!.!cve encli sulAj:~nd edge. c r'::e~tr(jfidensity of states B-rclmpk As a IILinlcric;l!.;:i;!n!pie:; ~ ~ b .~2 ; ! j ~ l l i a tthe fclr a bulk GaAs scmicr:,ri:j\:c'tc>~at an energy ( j . 1 eV rtbcve the conduction

band edge:

-

For a quantum well with L; = 100 A, we estimate the density of states of the first step (with E l = 56.5 rneV):

W e can also estimate the carrier concentration with an energy spread of kBT = 0.026 e V at 300 K and obtain

One-Dimensional Density of States. Similar to (3.2.10), the electron conckntration in a one-dimensional (ID) quantum wire along the z direction can be obtained from

Y n , , 17,

k,

wl-,:re the electron energy E is quantized in both the x and y directions,

-

n.,, n ,

=

1 , 2 ?3 , . . . , and k , is a continuous variable. Using

..

the 1D density of states p,,( E )is

3.2.2

Finite Barrier Mcdel

For a finite barrier quantum well, as shown in Fig. 3.3, we have

We solve the Schrodinger equation

where rn = m,, in the well and m = rn, in the barrier region. We consider the bound-state solutions, which have energies in the range between 0 and Yo*

For the even wave functions, we have a solution of the form

L

C , e-"(l'I-L/2)

2 -

#(z> =

C , cos kz

Izl




1.4
te~lti;il ~ cI~iet ; ~~ : i e;ipp!ietl !kid LJF~: i ~ ' t r e ; i t e das a per-lur.hation.

where H , describes an electron in the infinite quantum well wifhout the electric field. 'Treating the term eFz' as a perturbation, show that the energy shift due to the applied electric field, E - E:'), can be written as

where E ~ O ) and E(:) are the energies of the ground state and the n t h state of the infinite quantum well without an applied erectric field. Find a general expression for C,. All energies are measured from the center of the well. b. Evaluate C,, for n = 1, 2, and 3 numerically by keeping the first two nonzero terms i n the summation over the index rn. Show that C1 is negative while C2 and C, are positive. c. C o n ~ p a r enumerically C,,, n = 1,2,3, with those in part (b), where C,, is given by

a. W e treat H' = eFz as a perturbation from H,. We write down the unperturbed wave functions and eigenvalues from the infinite barrier model. Zeroth -order solrrtions

_-

Therefore, the first-order correcti:.n in the energy vanishes because of the syrnmet~yo f the origin31 qual~tlum-weilpotential with well-defined even and odd parities of rhe wave functic>ns. .

Secon.cE-order.pert~whatiort The energy to secoi~d-orderperturbation is given by (3.5.21b). We need to evaluate H,:,,, n # rn

where we have changed the variable from z to t , t Write

=

(z

and use 'l

We obtain

The perturbation to second order gives

M -1

for M

#

0

+ L/2)/L.

13/'2:i C QU..'LN'TULLIMECHANICS

112

The above results show that th:? energy shifts depend quadratically on the applied electric field, which is an important phenomenon when we study the q~~antum-confined Stark effects jn Chapter 13. b. If we only keep the first two nonvanishing terms in the above summation for C,,, we obtain

We see that E,decreases with increasing field F since C , < 0 and both E, and E , increase slightly with increasing field F. c. T h e above C,, in part (a) can be summed up to an analytical expression [ l 1-13]:

which are close to those in (b). These results using the second-order perturbation theory agree very well with those obtained from variaI tional methods [13-161.

3.5.2 Matrix Formulation Alternatively, the eigenvalue problem

can be solved directIy by letting

where

{~p:')

;ire the eigenfunctions of !he ilnperturbed Hamiltonian.

The second subscript 12 in n,,;,; is dropped for convenience. Here &I:), IT.^ = I., 2,. . . , N, may also be degenerate wave functioils. Substituting (3.5.31) directly into (3.5.30) and taking the inner product with respect to &iO), I\: = 1 , . . . , N, gives

if (4:)) form an orthonormal set. If (#$', m ma1 to each other,

=

1,2,. . . , N ) are not orthonor-

then, the above eigenequation becomes

The eigenequations (3.5.33) or (3.5.35) can be solved by letting detJI-i,, - ES,,(

=

0

for (3.5.33)

( 3 -5.36)

det(H,,,, - EN,,(

=

0

for (3.5 -35)

(3.5 -37)

The above procedure is equivalent to diagonalizing -the matrix H,,, in (3.5.33) or simultaneously diagonalizing H,, and N,, in (3.5.35) (noting that N is also Hermitian). Finally, one: finds the eigenvalues for E and the eigenvector a,,. The wave function is ':obtained from (3.5.31), which should be normalized. I

3.5.3

Variational Principle

The above equations can also be derived from the variational principle by minimizing the functional for the energy

assuming

for the eigenfunction. Using

6E if E

=A

SA - ESB =

B

/ B , we obtain

When the basis functions equation reduces to

(+I:)) are

orthonormal to each other, the above

where

Equation (3.5.42) is identical to (3.5.33).

3.6

LOWDIN-:; RENONVLALIZATION METHOD [91

A n interesting technique in perturbation theory is Lowdin's perturbation method. It may be useful to divide the eigenfunctions and energies into two classes, A and B. Suppose we are mainly interested in the-states in class A , and we look for a formula for class A with the influence from the states in B as a perturbation. \Ye start wit11 eige-nequation (3.5.42), assuming that the unperturbed states are orthonormalized. Equation (3.5.42) may be rewrit ten as '

I

.'I

a,,,

=

C

,l +L ,?I

.

6'

H,Ii,! C!,,

- Ht7,/11

-7

Ht,,',

2 t,!

E - H,,,1n

2:

f tu

(l o

where the first sun) on ;he right-ha.ncl sidc is e v e r the states in class .A only, while the second sum is over. tjic states in clsss B. Since we are interested i n t h e coeflicients (I,,, for rt2 in class ...I, we may e!inlinxte those i n cIass R by an

3.6

!. ~ W D I N ' SRENO!i!Mrhl~..l%ATIC~K h.11 .T;I013.

iteration procedure and obtain

and

Or, equivalently, we solve the eigenvalue problems for a , , ( n

E

A):

and

When the coeFcients n,, belonging to class A are determined from the eigenequation (. !.5),the coefficients n , in class B can be found from (3.6.6). A necessary condition for the expansion of (3.6.4) to be convergent is

In practice, to the second order in perturbation, we may replace E by EA in (3.6.4), where EA is an average energy of states in class A , and truncate the series (3.6.4) at the second-order term. For example, if class A consists of only a single nondegenerate state n. then class B consists of the rest. Equation (3.6.5) gives only one equation:

If we separate H into tj""'and

ri.

pL,-~:cil-b;:lti( :H T~-~!.-.C~:IE,~L~..A.P!D '?'HZ k '. p ' M F r H O U FOR SIIv4PLE RAP.!DS f'

1.55

For ar, electron i11 a periodic potential

where R = n,a, + n,a, + n,a,, and a,, a,, a, are the lattice vectors, and n,,n,, and n, are integers, the electron wave function satisfies the Schrodinger equation

?'he Hamiltonian is invariant under translation by the lattice vectors, r -+ r + R. If $(r) describes an electron moving in the crystal, $(r + R) will- also be a solution to (4.1.2). Thus, +(r + R) will differ from *(r) at most by a constant. This constant must have a unity magnitude; otherwise, the wave function may grow to infinity if we repeat the translation R indefinitely. ?'he general solution of the above equation is given by

where

is a periodic function. This result is the Bloch theorem. The wave function ~l/,,~(r) is usually called the Bloch function. The energy is given by

Here n refers to the band and k the wave vector of the electron. A full description of the band structure requires numerical methods [I, 21. An example of the GaAs band structure calculated [I51 by the pseudopotential method is shown in Fig. 4.1a, which represents the general energy bands along different k directions. Our interest will focus near the direct band gap (r* valley) between the valence t ~ a n dedges and the conduction band edge, as shown in Fig. 4.l.b, where the energy dispersions for a small k vector are considered. The k . p method is a usef:ll tecl-lnique for analyzing the band structure . . near a particular poi~itk , , , espscl:i~lq't\:hrn i: is Ilear an extrern~1.mof the band structure. Mere we consider t!mt fhe extrcmurn occurs at the zone l f ~ ill-.V r direct bandgap center where k,, = 0.Thii is a LC? L : ; ~ f ~ i case semicon~uctors.

L

A

A

P M

x

U.K

r

L

-.

e Vector 7;

Figure 4.1. (a) GaAs band structure cntculated by t h e pseudopotential method. (After Ref. 15.) (b) T h e band structure near the band edges of t h e direct band gap showing the conduction (C) heavy-hole (HH), light-hole (LH), and spin-orbit (SO) split-off bands.

Consider t h e gener.;l Schrodinger equation for a n electron wave function $,,,(r) in the tzth bani; with a wave vector k,

When written in terms of ~l,,(r), it becomes

T h e above equation can b e expanded near a particular point k , of interest in the band structure. W h e n k,, = 0, the above equation is expanded near E,,(U), ;ii [fi,, +

-

' -

1710

,1 . 1;

1

1 ,

1- =

I E,, I

ji'k" (

--

2/71,,

I

l!,lk(

4

(4.1 .h)

,I

Figure 4.2.

(a) A single-band model in the k

.

.

.,

--.

'

.*...

.S

., ,.--,r , > . r - , .

,

.

. . - ....

.

.r..

.

r...

. p theory. (b) The two-band model in the k . p

theory.

where

4.1.1 The k

. p Theory for a Single Band

If the band structure o, .nterest is near a single band, such as the band edge of a conduction band (Fig. 4.2a) and the coupling to other bands is negligible, then the perturbation theory and Lowdin's method (as discussed in Section 3.6) give the same results. Here the particular band of interest, labeled n, is called class A, and class R consists of the rest ofithe bands, nf # n. The time-independent perturbation theory, Eq. (3.5.21b) in Section 3.5, gives the energy to second order in perturbation: I

and the wave function to the first order in perturbation (3.5.16a):

.

.

where the monlentum matr;x elerrlents ate defined as

and r~,,,(r)'s are llormalized as

/

L L : ~ ( ~ ) LdL3 r~=~ S,l,Lv ~(~)

unit cell

If k, is at an extremum of E,(k), then Et,(k) must depend quadratically on k near k, and P,~,,= 0. That is wlzy we need to go to second-order perturbation theory for the energy correction and only the first-order correction is needed for- the wave function. Since we set k , to 0, we have

and

z. It should be noted that the DoP matrix in the where a,p = x , y , quadratic form has been defined to be symmetric. T h e matrix D " ~is the inverse effective mass in matrix form multiplied by h 2 / 2 . .*

4.1.2

The k

p Theory for Two-Band (or Nondegenerate Multibands) ~ o d e l

If only two (or multi-) strongly interacting nondegenerate bands are considzred, we call them class A , as shown in Fig. 4.2b. T o solve (4.1.6), we gssume

Substituting the above expression into (4.1.6) for u,,(r), ~l:,,(r) and integrating over a unit cell, we have

and multiplying by

whcre the ortho~onalityrelaiio:~. ~ .i : ~ ~ i ? d, ~'T ! , = 6,,,, has been used. For two equati(311can be solved from coupled bands, labeled by ii and I ? ' , the :-?.\?i>v? /'

the deter~ninantalequation:

The standard procedure is to find the eigenvalue E with the corresponding eigenvector. The two eigenvalues for (4.1.16) can also be compared with those obtained from a direct perturbation theory (see Problem 4.3).

KANE'S MODEL FOR BAND STRUCTURE: THE k WETH THE SPIN-ORBIT INTERACTION [7,81 4.2

p METHOD

In Kane's model for direct band semiconductors, the spin-orbit interaction is taken into account. Four bands-the conduction, heavy-hole, light-hole, and the spin-orbit split-off bands-are considered, which have double degeneracy with their spin counterparts (Fig. 4.3a). 4.2.1

The Schrodinger Equation for the Function n,,(r)

Clonsider the FIamiltori

.I

near the zone center k,

=

0.

where the second term in (4.2.1a) accounts for the spin-orbit inter,action, and a is the Pauli spin matrix with, components

w i ~ i c i when ~, operating

C)II

thc spins,

Figure 4.3. (a) T h e k . p method in Kane's model. Only a conduction band, a heavy-hole, a light-hole, and a spin-orbit split-off band with double degeneracy a r e considered. All other higher and lower bands a r e discarded. (b) Luttinger-Kohn's model. T h e heavy-hole, light-hole, and spin-split-off bands in double degeneracy are of interest and a r e called class A. All other bands a r e denoted as class B. T h e effects of bands in class B ort those in class A a r e taken into account in this model.

give

From the original Schrodinger equation for the BIoch function,

The Schrodinger equatiorl for the cell periodic function

ti,,(r!

is obtained:

) .:.,2 I...AT\JF,'Sfij()]:?dTdF(.:.R D A; :I:STRUCTURE

:31

~vhcre E' = E,(k) - h2k'/2rn,,. The last term on the left-hand side is a k-dependent spin-orbit interaction, which is small compared with the other terms because the crystal momentum fik is veiy small compared with the atomic momentum p in the far interior of the atom where most of the spin-orbit interaction occurs. Thus, only the first four terms on the left-hand side are considered:

4.2.2

Basis Functions and the Harniltonian Matrix

We look for the eigenvalue E' with corresponding eigenfunction

The band-edge filnctions u,,(r) are Conduction band: Valence band:

IS ), IS 1 ) with corresponding eigenenergy E, IX t ), IY t ), IZ t ), IX L ), IY 1 ), IZ L) with

eigenenergy E, where the wave functiorib in each band are degenerate with respect to H,. t ) = E,Ix t ), H,IY t) = H,IS t ) = E, I S T ), H,ISL ) = E,IS L I, H,IX E, I Y t ), H,I Z t ) = E, 1 Z T ), and so on. It is convenient to choose the basis functions

and

where the valence band basis t'unction~;:ire taker1 from the spherical harmoni c s YIo

-

IZ}, arid If,.

,

-.

+

1

.Y 2 iY), 6;(

I

Ecl. (A.32b) in A p p ~ % d A., i ~ for

-7

the p-statt: wave functions of' a hycii-:,)gt:n 2:itori;. 'The reasoti forthis choice is that the electron wave F~.l:~ctions arit ;l-lj!:e near the top of the valence band ancl .r-like ne;r th.c botiu-q~of' t172 i . ~ n d ~ i c t j

13;.

L,L>,

-A-n--12 Ex,

Lut tinger-Kohn model.

Condrlction band

.

Valence band Hea~jyhole

Light hole

(I):i, .

.-

.h -.

6

p-

1 ( /y + ~ Y ) J ) + V3 - I Z T ) = '

-..

136

'TI-IEORY OF'EI.,ECTRO~-.J!C:3-3/-1?dDS?, 0.468, the kalli~lmmole fraction is increased; therefore, t!le l;!tt.ice constant .is decreased and the

PI-IEC ik OF ELEC'I'RGNIC B!\NII STIIIJCTlJ t:ES 'IT4 SEM1CONDTJCTC)RS

: 50

Table 4.1. Material Parameters

G ilAs

Paraincters

a , (A) E, (eV)

5.6533

Y1

6.85

Y2 Y3

2.1 2.9 11.879

a, b (eV) =

- a,,

m$/n.zo

InP

1.424

C1, (10"dyn/cm2) ~ , , ( l ~ " d/cm2) ~n a

In As

(eV)

5.376 - 9.77 - 1.7

0.067

In,-,Ga,As will be under biaxial tension (Fig. 4.7). Here biaxial tension means that the lattice in the parallel ( x y ) plane will experience a tensile strain with a simultaneous compressive strain along t h e growth ( 2 ) direction. O n the other hand, if x < 0.468, we have a ( x ) > a,, and we wi1.l have the case of biaxial compression. At the zone center, k = 0, we have only PE and Q , appearing in the diagonal terms of the matrix (4.5.13) nonvanishing. Therefore, we obtain the band-edge energies of the heavy-hole and light-hole bands:

On the other hand, the conduction band-edge energy of the electron is given by

Note that both the conduction and valence band energies are defined to be 'positive for the upward direction of the energy. T h e net energy transitions will be

for the conduction to heavy-hole band, and

for the conduction to the light-hole band, where E, is the band gap of the unstrained semiconductor, and

is the hydrostatic deformation potential. Sometimes the hydrostatic and shear deformation energies, 6Eh, and 6E,,,, are defined, respectively, as

and

The effective band gaps are given by

For the Hamiltonian in Eq. (4.5.13) or i .....24) in Appendix C, the valenceband structure of a bulk semiconductor is determined by the algebraic equation det[ffij(k) - s , ~ E ]= 0

(4.5 -36)

where k is now interpreted as a real vector, and the envelope functions are taken as plane waves. For the 4 x 4 Hamiltonian, the solutions of Eq. (4.5.26) are simply

E

,,,.,(k)

=

-P, - P, - S ~ ~ ( Q , ~ ) { ( Q .+, Qk12+ (Rk12+ I S ~ I '

.*

(4.5.278)

for the heavy holes and light holes, respectively. Each of the solutions is cloilbly degenerate. Note that i t i-j in- porta ant to include the sign factor sgn(&,)( = + 1 for Q , > 0 and = .- ! for Q, L,,/2.

+

,

Exdrnple: An Irr - ,Gd-r,As//rr, - ,Gn,As, P,- , Qk~antulnWell Again, a convenient way to find the valence subband encl-gics :!ad eigenfunctions is to use

the block-cliagonalized E T a n ~ i l t o n i a ~(4.S.38) s anci (4.8.331, which sin~plifythe numerical cr~lculationsby i! sigr~ificantnntoafi:. As a nt~rncricaiexample. w e plot in Fig. 4.22 the energy dispersion z:Lri1es c:~lculated using 311 efficient

r . . ' . , . ' . . , . . . .

I

.

.

.

.

(b) ~=0.468

Figure 4-22. The valence subband dispersions of an In ,-,Ga,rAs grown on I n , -,C'ra,.As,P,-, quaternary barriers (with a band gap 1.3 ,urn) lattice matched to InP substrate. (a) x = 0.37 (biaxial cympression), ( b ) -1- = 0.4-65 (unstrainerij, (c) x = 0.55 (smail tension), and ( d ) x = 0.60 (large tension). Here the wave vcctor along the (21)0] dircection is normalized by 277/a, where a = 5.6533 A is the lattice constant of G:iAs. (After Ref. 37.)

k,

propagation-matrix approach in Ref. 37 for the valence subbands of an ln,L,Gn,~s quantum well grown on an In , A , v C;a,rAs,P,-, substrate with a band-gap wavelength 1.3-,~~n1 lattice matched to an I ~ substrate. P Th.e galliilln model fraction s is 9;irii.d. C:l$ .'i :-= 0.37 (biaxir~lconlpression), (b) ,r 0.468 (unstrnirlrd). ( c ) .u - 0.55 (snxlil tsr;sIc;n), and (rl) .r 0.60 (large tension). These band structr~i-ss z c ~.i.s::inl it; ir-ivcxtigate the strain effects o n the effective m:~.ss, th(2 c ! c ~ . , Y ;1.i:'~ L q.,~r.;-l i C 5 , ~ ~ p t i z again, I and ;ibsorptir~r!in .. cluantum wells. VY;e c_ii>,g:i!.sst!:;. ;si.i.;.t '.r applicx'iit~ns-f~lr'thi:r irl C h ~ ~ p t e9r sand 10. 3

-

-

1

b ~ ! i l

,

THEORY Or. ELEC'I'IIC, '4IC" B:\N 13S7'Rl-lC7r,JRE:.N;:; S7'R ,lC'r!J!':ES I N SEiM;2(,]N]]UCTC)I<S.'

Using the definition for D;;%n (4.3.1 1)-(4.3.13) show that the matrix representation of the Lut~ingel--Kohn Hamiltonian H L" is given by (43.14) and (4.3.1 5). Summarize the effects of the strain on the conduction band edge and the valence band edges from the results in Section 4.5. Calculate the strains (E,~,, E,,, E,,), PE, Q,, and the band-edge shifts for In,-,Ga,As quantum-well layers grown in IfiP substrate with (a) x = 0.37 and (b) x = 0.57. Calculate the band-edge effective masses using (4.5.33) and (4.5.39) for Problem 4.11.' Compare the values of the two structures. Show that the eigeilvalues for the matrix H(k by (4.5.36a)-(4.5 - 3 6 ~ ) .

=

0) in (4.5.34) a r e given

Calculate the band-edge energies with spin-orbit coupling effects using (4.5.36a)-(4.5.36~) and compare with those using (4.5.19) without the spin-orbit coup!ing effects for the I n , -,Ga,As/ InP system with (a) x = 0.37, (b) x = 0.47, and (c) x = 0.57. Show that the eigenvalues of the matrix (4.5.13) are given ,by (4.5.27a) and (4.5.27b). Derive the propagatiorl matrix (4.6.7) from the boundary conditions. Discuss in the infinite barrier model (i.e., Vo = a and VN+,= m) (a) how to modify the boundary conditions in the propagation-mi\!;-ix method in Section 4.6.1 and (b) how to find the eigenenergy E and its corresponding wave function. Summarize the prucedures for solving (a) an rz-type and (b) a p-type modulation-doped quantum well self-consistently in ~ e c t i o i s 4.6.2-4.6.4. Discuss qualitatively how Fig. 4.12 should be modified if the donor profile is within 20 A at the center of the quantum well N D ( r ) = ND for lrl r 20 A and N,(z)= 0 otherwise. Derive .(4.7.24a) and i4.7.24b) in the Kronig-Penney model.

(a) Discuss the procedure for determining the number of bands that can be considered a s "bound states" of the superlattice from Fig. 4.15. (b) What happens if the barrier height V, Palls within a miniband of Fig. 4-15? ,~.,AS Compare the rninib3ncl energics of the G ~ A S / A ~ ~ . ~ G Z ~snperlattice in Fig. 4.16 with tllcrse calc~.~iatsd for a single isolated quarltum well with the s;:inle well width r.tr ..= 100 A from Chapter 3, Section 3.2.

,

4.23

(a) Using the band-edge discorltinuity rulcs disc~issedin appendix D, design an I n , -,.Ga,As / l n P heterojunction such that A E, = 0.3 eV. Assume that strain occurs only in the I n , -,rGa,As layer, since the substrate I n P is thick. (h) Calculate AE,., E,iIn Ca,As), P,, Q,, and so on for part (a).

,--.,

4.21

Comment on the results from the subband energies in Figs. 4.1% and c.

4.25

Check the subband energies at the band edges ( k t = 0) in Fig. 4.19 with your calculated values. Comment on the discrepancies if there are any.

4.26

Derive the block-diagonalized Hamiltonian (4.8.16) using the axial approximation and the basis transformation (4.8.1.5).

4.27

(a) Find the eigenvalues- and the corresponding eigenvectors for the 2 x 2 Hamiltonian )I" in (4.8.16) in the planerwave representation:

a Cr -

( 1

-E

(2)

-

[ , ( . H I U= -

[P+ Q

(2)

lit

P-Q

R

1

(b) Repeat part (a) for

4.28

(a) Express,fhe wave function *'(kt, r) in (4.8.27), which is in bases 1 3 11) and 12) in the original basis I$, f ) , /$, f ) , - ?) and IS, - f). 3 3 1 (b) ~ x ~ r e the s s bases IT, v ) ( u = -2 -2 - 2 7 - f) in terms of bases [I), 12>, 1 3 , and 14).

4.29

Label'all the energies in Figs. 4.21cr-c for In, -,Ga,As/ wells with (a) x = 0.37, (bj x = 0.47, and (c) x = 0.57.

4.30

Cal.culate the first quantized subband energies for the electron, heavyhole, and light-hole subbands in Problem 4.29 using the method discussed in Section 3.2.

1.

la7

N.W. Asllcrof: a n d N.D. pi!cr::.!!~~,

New York, '1976. 2. F. Bussalli 2nd G , Pastori f':! l.r.:16.:.ji::, .:,i,1. . Solir!~,Pergnmc:n, Osfc,rc!, IJ k. 1 9 75. T

-7

1 . 1 i-'

.j':.iiif.,

..::::rc.

InP quantum

f-'/l,\;sics. HI:)!^. Rinehrirt

L'!,:c'!i'~~,llr Slllri..s

l~:?i/

bVinsto11,

C ) l l t i ~ l i /Trcuisitior!.~ it2

i

3. J . Bardeer.,, "An irnprovecl cillculation of the e ~ e x g i e sof metallic Li and Na," J. Chem. Plrys. 6, 367-371 (1933). 4. F. Seitz, Tile Modem Tlleol-))of Soiid.~,McGraw Hill, New York, 1940, p. 352. 5. W. Shockley, "Energy band structures in semiconductors," Phys. Re[?.78, 173-1 74 (1950). 6. G . Dressclhaus, A . F. Kip, and C. Kittel, "Cyclotron resonance of electrons and holes in silicon and germanium crystals," Plrys. Reo. 98, 368-384 (1955). 7.E. 0. Kane, "Band structure of indiuni antimonide," J. Phys. Chetrr. Solicls, 1, 249-261 (1 957). 8. E. 0. Kane, "The k - p method," Chapter 3, in R. K. Willardson and A. C. Beer, Eds., Sernicondr~ctorsurzd Setnirnetals, Vol. 1, Academic, New York, 1966. 9. J. M. Luttinger and W. Kohn, "Motion of electrons and holes i l l perturbed periodic fields," Pilys. Re[!. 97, 869-883 (1955). 10. J. M. Luttinger, "Quantum theoiy of cyclotron resonance in sen~iconductors: General theoly," P/rj,.s. Rer:. 102, 1030-1041 (1956). 1 1 . I. M. Tsidilkovski, Bunt1 J'tt-r~ct~ite of Sernioondr~ctot-.F,Pergamon, Oxford, UK, 1982. Senricond~rctot~s, Springes, Berlin, 12. B. R. Nag, Electron Tr.urlsport in Cotnpo~~rzd 1980. 13. C. R. Pidgeon, "Free carrier optical properties of semiconductors,'' Chapter 5 in M. Balkanski, Ed., Hnndbook orz Serrzicor-~d~~ctors, Vol. 2, North-Holland, Arnsterdam, 1980. 14. For some recent discussion on the effective-mass theory, see M. G. Burt, "The justification for applying the effective-mass approxima tion to microstructures," J. Phys.: Condens. hI{i!ter 4, 6651-6690 (1992); M. G. Burt, "An exact formu:::tion of the envelope function rnethod for the determination of electronic states in semiconductor microstri~ctures,"Setrricorrd. Sci. Techrzol. 3, 739-753 (1985); M. G. Burt, "The evaluation of the matrix element for interband optical transitions in quantum wells using envelope functiops," J. Phys.: Condens. Matter 5, 4091-4095 (1993). 15. J. R. Chelikowsky and M. L. Cohen, "Nonlocal pseudopotential calculations for the electronic structure of eleven diamond and zinc-blende semiconductors," Phys. Re[:. B 14, 556-582 (1976). 16. G . C. Osbourn, "In,Ga, ~ s - l n , ~ -,As a , strained-layer superlattices: A proposal for ~ ~ s e f unew l , electronic n~aterials,"Phys. Re[?. B 27, 5126-512s (1953). 17. G. C. Osbourn, "InAsSb strained-layer superlattices for long wavelength detector applications," J. V a c ~ d' ~ e. c h n o l .8 2 , 176-1 78 (1954). 18. E. Yablonovitch, "Band structure engineering of semiconductor lasers for optical ~le 6 , 1392- 1299 (1988). communications," J. I , i i l r t ~ ~ . a Techrzol. 19. T. P. PearsalI, Volume Editc;r, "Strained-layer superlattices: Physics," in R. K. WilIardson and A. C'. Bees, Eds., Sert~icotzd~~c.to~.s nnd Sevrritnelols, Vol. 32, Academic, New York, 1090. 20. P. S. Zory, Qrlontt~mJYel/ Ln:jc.rs, Acnden-{ic,P J a v York, 1993. 21. G. E. Pjkus and G . L.Bir, "F::fPects of clel'oi-nl::~ionon the hole energy spectrum : Sliltt: I . 1502-1517 (1960). of germanium and si lico~l. 50;'. p/~)~s.-So/r'd -.

-.

7

-

.

22. G. L. Bir and G . .:i Pliill:c!-i;\! (ir~:-i ,Stl-~!i;i~.-l!z~br;cc.cl 13'I'0cts i r ~Senzic:oncbrrc~o:;s, 23.

24.

25. 2(i.

27,

28. 29.

Wilcy, New York, 1071. R. P. F~~llmii11, 12. 13. Leigi-i~on,a!~cjM. Sancl:;. Tizr F ' ( : ! ; t l ; ~ ~ ~Lecirr~'es il or1 Plzysisics, Chapter '39, Addison-tVcsley, Rcac!in_e, MA. 1964. J. F, Nye, Pilysiccrl .PropCrlic*sof'Ctystnls, Oxford Universily Press, Oxford, UK, 1985. K. Seeger, Scmiuo?icblrctor P/iy.sic.s. Springer, Berlin, 3.9S2. J. C. Sense1 and G . Fehcr, "C'yclolron resollance experiments in uniaxially stressed silicon: Valerlcc band inverse mass parameters and deformation potentials," I'hys. Reu. 129, 11141-1062 ( I 963). H. Hasegawa, "Theory of cyc1utr.on resonance in strained silicon crystals," .Phys. Reu. 129, 1029-1040 (1963). A. Gavini and M. Cardona, "Modulated pjr,zorel'iectance in semiconductors," I't1y.s. Reu. 3 1 672-682 (1.97Uj. L. D. Laude, F. H . Pollak, and M. Cardona, "Effects of uniaxial stress on the indirect exciton spec.trurn o l silicon," Pfzys. Re!.).H 3 , 2623-2636 (197 !). K. Suzuki and J. C. Ilensc.1, "Quantum resonances in the valence bastds of germanium, I: Theoretical considerntions," Phys. Rev. B 9, 4184-4218 (1974). J. C. Hensel and K. Suzuki, "Quantum resonances in the valence bands of germanium, 11: Cyclotron resonances in uniaxially stressed crystals," Phys. R ~ I B I. 9, 4219-4257 (1974). M. Chandrasekhar and F. H. Pollak, "Effects of ~lniaxialstress on the electroreflectance spectrum of Ge arld GaRs," IYlys. Rei!. B 15, 2127-2144 (1977). D. E. Aspnes and h.1. C'rii-dona, "Strain dependence of' effective masses in tetrahedral senlicontl:::.tol-s," F'jrys. Recl. J3 17, 726-740 (1978). H. Mathieu, P. Merle, E. !. Arilezic!ne, B. Archilla, J. Camassel, and G. Poiblaiii.i, "Deformation potentials of the direc.t and indirect absorption edges of Gap," ~'!EJ',s.R ~ L :B. 19, 2209-2223 (1979). D. L. Smith and C. h.f;!i!hiot. '"Theory of semiconductor superlattice olectroilic r;:ructure," RPL'.R.IocI. .F'lly~.62, 273-234 (1990). D. I-. Srnith and C. Mailhint, "Strained-layer semiconductor superlattices." Crit. Rrr.. Solid St~rte14.f~rti$r.. Sci. 16, 13 I -160 (1990). S. L.. Chuang, "'Etliciznr hand-slructl.~recalculations of strained quantum wells using a two-by-two Hanliitc?niarl.'' Phys. Rc.r~. H 43, 9649-9h61 (1991). (Note that L i , . in this paper sklouid l;z inker; ;is - c i V to compare with data in the literature.) C . Y.P. (_7hao and S. 1-.Chuarlg. "Spin-orbit-coupling effects on the valence-band structure OF strainzd sei~:icor!dixcor quantum wells," Pitys. Rei'. R 45, 4110--4122 ( ic)92). .T. ,As. K ~ t l g ,Elc~.fr.orijttgr:c~ti:: ! . r , ' i i r ~ . Th:o:ji:., 2ci ed.. LVilcy, Ki'ew Yorli, 1990. I-'. Yell, ()j7rii.a! [.j:'ll;.p.s i,.! ,c!!-i:;.r:.: ;\.[t.::j'j!:, b-qiiey. I\iexvv Ypl.k, l\)S:$. E.J. Ro:.~nand S. L.. i3i-:i.l:1::; '-!..illi:.i~ iirr!! nor,iii:~arintcrs~ihbandelcctroatrsorpli()gs iil :i [ ~ ~ ( ) < ~ ~ ~ ] ; ~ [ i ; ) ~ : - ~ ] , ; ; :;;!::-::[-i~ ) , ~ < . ~ .$:L:.>'! . . $ 1 :'. . J . +,,'iiq.?l.]'!~Y.s. 69, 324?---325(](199 11. -I

30. 31.

32. 33.

34.

'55. Sf:).

.;7.

36.

39. ,413.

-I.:.

.

(.\I.,

42.

; d'r' - r')

and

where k

= 0 6 .From

the charge continuity equation, we have

T h e electric field in the time domain is given by

and in the frequency domain is

E(r)

=

itoA(r) - V@(r)

in

has been used. T h e term J V' . ( g ~ l d ' r '= # gJ dS' = 0 since the surface integral at infinity sl-lould vanish, assunling there is n o source at jniinity and g(r - r') = exp(i klr - rfl)/4nlr -- r f1 -+ 0 as rf apprc.,aches infinity. Here P is a unity matrix:

Defining a dyadic Green's function as

we have

and the magnetic field is given by

Suppose we consider an ecluivalent magnetic current source M and an equivalent magnqtic charge density p,: "

where

Ec~t~ivale,ntlv, from the duality principle, we (;an find

r' 0 ) . rt

-

Obsemation point

r=m

Source aperture

-

F r', ,where F is a unit vector along the Figure 5.4. Far-field approximation Ir - rll = r direction of r. T h e length Ir - rfl is approximately r subtracted by the distance of the projection of r' on F..

Thus, in the presence of both eIectric and magnetic sources, J and M, we find

5.4.2

E(r)

=

/[iw,ue(r, r') J ( f ) - O X c ( r , r')

H(r)

=

/[v

-

x E ( r , r') - J ( i+ ) i w e c ( r , r') -

~ ( r ' ) d3r' ] (5.4.15) ~ ( r ' )dGr' ]

(5.4.16)

Far-Field Ap y roximatiors

In many cases of radiation of the fields, for example, if the source distribution is over a finite region and the observation point r is far away from the source, we may approximate

where r^ is a unit vector along the r 'direction, as shown in Fig. 5.4. T h e distance r^ - r' is the projection of r' on r^ and the distance Jr- r'l is approximately the difference -between the length r and: i - r'. With this approximation, we further approximate

where k = kr^ is along the direction of observatiod. We keep the term F . r' only in the phase factor, since any small variation in r' comparable with wavelength causes a big change in the phase ,of e-ik"', but a negligible change in the denominator of (5.4.18). since r is very large in the far-field region. We consider radiotioa into the free spa&, k = li, = W ~ ThercG fore,

.

Figure 5.5. The coordinates foi- calculating the far-field piittern at position r ( r , B , + ) from an y') at the laser output facet plane z = 0. The angle 0 , is measured from aperture field E24(.r', the 2 axis along the x direction, and O,, is measurcd from the r axis toward the y direction. =;

-

Thus, using i we find

X

(I - W)-

M

=

i

+ & M + )= i

X ($!lf,

- ik?

x

/ d?r'

ik''

x M since 3 x i

M( r')}

=

0,

( 5 -4.21)

.4s an example, for a sen~iconductorlaser, shown in Fig. 5.5, if the aperture field at the output facet of the laser is given by ~,(r'), we may use an "ecluivaXen t source" by simply taking El]

and the magnetic current scwrce i:;; . . c:.)mpo~~enl: of E., and !ts Irn:ig~?v;iti?

;i

ct.lrr.enc .;f~.eetfrom the tangential

where 1^2 is a unit normal to the aperture surface and a factor of two accounts for the image source. We then integrate over the surface of the aperture to find the radiation field using Eq. (5.4.21.):

T h e radiated power density is

--

where q0 = J P O / f O ?

=

=

R

377

sin

6,

in free space, and

cos & 2

+ sin Bsin & 9 + cos 8 2

(5.4.26)

is a unit vector alopg the direction of observation r, which is the sanlc as the direction of wave propagation in the far field k. If we consider the aperture field to be due to a TE mode in the active region of .the semiconductor laser

then the equivalent magnetic current sheet is

Consider two cases. 1. 8 , nlensr~rerne~li.If the far-field pattern is measured along the x direction in the far-field plane, 4 = 0 (or 4 = T ) , and B = 0 , , we obtain i r'

=

x'.?

=

+ cos8.f

sinB,-t

+ y'y^

since z'

=

(5.4.29)

(5.4..30)

0 on the aperture

ikeikt. = +[ -

-

27rr-

d , ~ e - - i k s i nL~- ~ ' E , ~ y( 'x) ' , (5.4.31)

'

. - (I'

-L>

T h e optical power intensity pattern i s

P ( 6 .) a cos' 0 ,

1

- I!

,a!,

(4.';

-- :j

1 -

L !.

Lj

>.

1

,. V

IJ'

-.

.-

bin 9

.t

(

,

)

(5.4.32)

where the extra factor cosZ8 is necessary due :to the polarization of the aperture field. 2. Oil nzeasirrement. If the far-field pattern is measured along the y direction in the far-field plane, Q, = ~ / 2 and , 0 = Oil, we obtain P

=

sin 01,j7

+ cos B l l i

(5.4.33)

and the radiation power intensity pattern is

. -.

T h e above results for the far-field patterns are useful for studying the beam divergence for a single-eIement semicond~lctorlaser and coupled laser arrays. A simple scalar formulation can be found in Ref. 4, where numerical results for various diode lasers are shown.

PROBLEMS 5.1

(a) Take the Fourier tra~lsformsof A(r, t ) and +(r, t ) with respect to time in (5.1.21) and (5.1 '23,)and express A(r, o ) and +(r, w) in , ). terms of ~ ( r ' o, ) and ~ ( r ' w (b) Take the Fourier transforn~sof A(r, t ) and +(r, t ) with respect to both the spatiaI and time variables r a n d t and express A(k, w) and #(k, o ) in terms of J(k, w) and p(k, w).

5.2

Check the duality principle using (5.2.15).

5.3

Derive the Kramers-Kronig relations between the real and imaginary i ~ ( w ) similar , to parts of the complex refractive indes, E(w) = rz(w) Eqs. (E.14) and (E.15) in Appendix E.

5.4

Derive the reciprocity relation (F.17) in Appendix F.

5.5

(a) A plane wave is reflected between the free space and a bulk GaAs semiconductor with a refractive index assumed ,to be n = 3.5. Calculate the reflection and the tran.smission coefficients of the field, r and t , at normal inciclence. (?I) Kepeitt part ( a ) if' the wave is incident frorn the GaAs region onto the GaAs/air interface at nornlal incidencz. rc) For oblique incidsncc in part (h), lil;d the critical angle and t h e Brewster ang!e.

+

220

"

ELECTROMAGNETICS

5.5

For a plane wave incident from an InP region ( n = 3.16 and h , = 1.55 p m ) to the InP/air surface and reflected with an angle of incidence 0, = 15", calculate (a) the reflection and the trailsmissjon coefficients, r- and r , of the optical field for both TE and T M polarizations, and (b) the reflectivity R and the transmissivity T for both polarizations.

5.7

Calculate the Coos-Hanchen phase shifts for an angle of incidence 8, = 45" between an InP/air interface for both TE and T M polarizations ( n = 3.16 and h , = 1.55 prn).

5.8

(a) Calculate the reflection and transmission coefficients, r and t , for a plane wave normally incident on a slab of GaAs sample with a thickness d = 10 p m , refractive index n = 3.5 at a wavelength A , = 1 p m . (The background is free space.) ( b ) Find the power reflectivity and transmissivity in part (a).

5.9

Find the far-field radiation patterns P(0,) and P ( 0 , , ) in Fig. 5.5, assuming a uniform aperture -field EA(x, y) = E, for 1x1 I D and J y JI LV, and E,(x, y) = 0 otherwise.

5.10

If the aperture field is a Ganssian function in both the x and y - ( Y / ~ , ) ~in] Fig. 5.5, find directions, EA,(x,y ) = EDexp[-(s/x,)' the far-field radiation patterns P ( B and P(0,,).

1. J. A. Kong, Electro171ngrzetic PVczce Theory, 2d ed., Wiley, New York, 1990. 2. J. D. Jackson, Clnssicnl Electrodyn~zrnics,2d ed., Wiley, New York, 1975. 3. L. C. Shen and J. A. Kcmg, /lpplierl El~.crro~~z~ig~zetisr.n, Brooks/Cole Engineering

Division, Monterey, C.4, 1983. 4. H. C. Casey, Jr., and M . B. Panish, Hetel-osllz~ctz~re Lasers, Pczrr A: F~lndamerztal Principles, Academic, Orlando, FL, 1978.

Propagation of Eight

Light Propagation in Various Media The propagation of light is an interesting topic because many common phenomena such as refraction of light, polarization properties of light, and scattering of light can be observed every day. The Rayleigh scattering of light by water molecules has been used to explain why the sky is blue in the daytime and red in the evening. In this chapter, we discuss some basic properties of the propagation of electromagnetic waves in isotropic and uniaxial media [I-21. For light propagation in gyrotropic media and the magne tooptic effects, see Appendix G.

6.1 PLANE WAVE SOLUTlONS FOR MAXWELL'S EQUATIONS IN HOMOGENEOUS MEDIA We shall investigate solutions to Maxwell's equations in regions where J = p = 0. Maxwell's equations for a harmonic field become

We also assume that the media are homogeneous; therefore, we have plane wave solutions of the form exp(ik r). Since all complex field vectors E, H, B, and D have the same spatial dependence exp(ik. r), we obtain the basic relations for plane waves prop-agation in source-free homogeneous media:

We see that the wave vector ki is altx~yspez-pendiculrtr tu B and D from (6.1.2~)and (G.1.2d). But we cannut sity th::tt k is always perpendicular. t o H o r

E unless the media are isotropic. J.n isotropic media, D = E E and 'sB = p H . Therefore, 11 and I3 a r e parallel to E and H, respectively. Otherwise, the media are called anisotropic. 'Thus, the tirne-averaged Poynting vector, which is given by $ R ~ ( EX H'"),does not necessarily point in the direction of k in an anisotropic medium. In other words, the direction of power flow of a plane wave may not always be in the direction of the wave vector k. In the following sections, we discuss how characteristic polarizations and propagation constants in a given medium are obtained. A characteristic mode or normal mode is defined as a wave with a polarization, which does not change while propagating in the homogeneous medium. This polarization is called the characteristic polarization of the medium.

6.2

LIGHT PROPAG.ATION IN ISOTROPIC MEDIA

In isotropic media, the constitutive relations are simply

Thus, D is always parallel to E, and £3 is parallel to H. Any polarization direction of E can be a characteristic polarization of the isotropic medium. Equations (6.1.2a)-(6.1.2d) become

These are the equations satisfied by plane waves in isotropic media. One sees clearly that

1. k is perpendicular to both E and H. 2. k x E points in the direction of H, and k of -E.

X

H points in the direction

Thus, the three vectors E, H, and k are p:-rpendicular to each other. Let us take the cross pr-~!uctof (6.2.3~1)h : ~k from the left-hand side:

The left-hand side can be simplified using the vector identity

where we have made use of (6.2.36). We obtain

using (6.2.3b). The result is simply

The solutions to the above equation are either a trivial solution, E field, which is not of interest), or

=

0 (no

-_

which is the dispersion relation in an isotropic medium. We obtain the wave number k = wG. If we define the refractive index 1-2 = where p o = 4 v x I o - ~ H / ~ and , E~ = 8.854 X i 0 - " ~ / m = (1/36?i) x 1 0 - ' ~ / m , we have

d

w

,

l/dz

and c = = 3 X l o 8 m / s is the speed of light in free space. Let us such that define the unit vectors 2, 17, and

E

=

2E

H = ~ H k=Lk

(6.2.9)

Substituting (6.2.9) into (6.2.3a), we obtain

is in the direction of A. Since O r simply x ,i? is perpendiculir to O (6.2.10) leads to and

L

$ ~ 2 = / ^ l

Using k

= w

G,wz have

k, 2, and h are all unit vectors

k& = u,~i.i-l

(6.2.11)

L.!GHT PRoPAc~ATIONIN VARIOUS MEOIr\

Figure 6.1. In isotropic media, D = E E , B = p H . The three vectors E, H, and k, or D, B, and k forn~a right-handed rectangular coordinate system.

where q is called the characteristic impedance of the medium. We conclude, as shown in Fig. 6.1, that

1. The three unit vectors 2,

h, and

construct a right-hand rectangular

coordinate system. 2. The magnitude of the electric field is equal to the magnitude of the magnetic field multiplied by the characteristic impedance of the medium q . Propagation Constant and Optical Refractive Index in Semiconductors. In most semiconductors, ,u = p o , and the permittivity function is complex:

.. where the real and imaginary parts of the permittivity function satisfy the Kramers-Kronig relations (Appendix El. T h e compl& propagation constant k = k' i k " can be written in terms of the complex refractive index Z ( o ) = n(o) ~K(w),

+

+

I

where the imaginary part of the complex propagation constant is half of the absorption coefficient a :

wherz h is the wavelellgth in fret: space. For n plane wave propagating in the -I-z direction with

polariz;ltIon along .f, the electric field behaves as

follows:

= 1 2 0 n is the characteristic impedance of the free where 7, = 1/= space. Its time-averaged Poynting vector for the optical power density is

-

--

which decays exponentially as the wave propagates farther along the z direction with a decay constant determined by the absorption coefficient. At an optical energy below the band gap of most semiconductors, the absorption is usually small or negligible. Above the band gap, the optical absorption is important. When a plane wave is normally incident from the air to a semiconductor surface, the reflectivity of the power is

which takes into account the absorption effec*when the refractive index E is complex and n , of the air is 1. ~VumericalExample For InP material, the dispersive effects of the real and ) given in Appendix J as a function of the imaginary parts r t ( w ) and ~ ( w are the photon energy h w or the free-space wavelength A. At an energy Aw = 2.0 0.62 ,urn), which is close to that of a HeNe laser wavelength, we eV ( A have

-

The absorption coefiicient is

'The reflectivity for the reflected powcr t':- or^! tllc slxnicondr~ctoris

LiC:il?' PRC)PAGIY~iC)NIN VARIOUS MEDIA

Consider the case of a uniaxial medium described by

in the principal coordinate system, which means that the coordinate system has been chosen such that the permittivity matrix has been diagonalized. Two of the three diagonal values are equal in a uniaxial medium. The medium is positive uniaxial if E~ > E , and negative uniaxial if E , < E . Here, the z axis is callgd the optical axis6.3.1

Field Solutions

Since the medium described by (6.3.1) and (6.3.2) is invariant under any rotation around the z axis, we can always choose the plane containing the wave vector k and the z axis to be the x-z plane, i.e.,

without loss of generality, as shown in Fig. 6.2. In Table 6.1, we summarize the forms of vectors and dyadics and their corresponding matrix representations. Using (6.1.2a) and (6.1.2b1,

k=?k,+?&

Figure 6.2. -1-h~.wave vci.t E),

(k, - k,)d we obtain

=

Tr 37 (2n + 1 . ) ~ - -- ) . . . ) , ete. 7 2' 2

-

A -

( n an integer) (6.3.35)

'??-.

Llt'iHT PRGP:!,GA4TIC)N I N VAKIOTJS MEDIA

whicli b e c c n ~ e scircularly polarized at the output end x

=

d. Using

and defining

where A,, is the wavelength in free space ( w / c

=

2 7 ~ / A , ) ,we have

Thus the plate, which can transform an incident linearly polarized wave into a circularly polarized wave, is called a quarter-wave plate. Note that the value A , is neither the wavelength in the free space nor the wavelength in .the uniaxial medium; it corresponds to A. divided by the difference in the two refractive indices n , and no. In many crystals such as lithium niobate (LiNbO,) or KDP (KH,PO,), the refractive indices have the property n , > n o . In some crystals such as quartz (SiO,), n , < r z , . 7.he velocities of the two characteristic polarizations c / n , and c / n , are not equal. T h e axis along which the polarization propagates faster is called the fast axis and the other axis is called the slow axis.

PoIaroid. Tf we have a uniaxial medium given by

we see that for an incident wave propagating in the x direction, if the electric field is polarized in the JI direction, i t will propagate through with a propagation constant k , = wl&, which is real. However, if E is polarized in the z direction, i t will propagate with a complex propagation constant

=

[

- 1 '"d-

I---

1

CTz

-

=

(6

I

U-

i,ui (!J

(7-

tor

B ELr)

(6.3 -40)

Polaroid

Quarter-wave plate

Min-or

Figure 6.7. A n kxperimental setup with a polaroid, a quarter-wave plate and a mirror for complete absorption of the light incident from the left side.

and tlie wave will be attenuated significantly in the medium. Thus, if the thickness of the plate is large enough, an incident field with an arbitrary polarization will have its z component attenuated when passing through the plate. The transmitted field will be essentially polarized in the y direction only, which is linearly polarized. Example An experimental setup for an absorber of laser light uses a polaroid, a quarter-wave plate, and a mirror, as shown in Fig. 6.7. For an incident light with an optical field E,, randomly polarized and propagating along the z direction, the transmitted field E, after passing through the polaroid is linearly polarized along the direction of the passing axis, which makes 45" with the x and y axes:

where k , is the propagation constmt in free space. The bvave E , also makes a 45" angle with the fast (,v) and SIC)? (\-'I axes of the quarter--wave plate. The transmitted field at z = d of Lhe quarter-wave plate is

w l ~ e r e( w / c ) ( r z ,-

,q .)if

r

7,-,/L.7'1.1~ ~~ h

-

l ? ? ~ i .ti';lcr2 t r ! fr.c)m the

mil-TOY is

.

-.

L.lGl iT PK()PAG.I\I'IOI\: I N 'JARICIUS MEDIA

240

propagating i n the - 2 direction. Upon impinging on the quarter-wave plate at z = cl, the A- component of the electric field propagates with a propagation constant wn,./c, and the y component propagates with a propagation constant w n , / c again with an additional phase difference of ~ / 2 :

Therefore, E, is linearly polarized in the direction ( - 2 i9 ) which is along the absorbing axis of the polaroid. T h e final output light E,,, = 0 since E, is absorbed by the polaroid. This setup actually works as a light absorber, which can be used to absorb laser light to avoid stray light reflections in the laboratory.

PROBLEMS 6.1

For a laser light with a power 1 mW propagating in a GaAs serniconductor ( n = 3.5) waveguide with a cross section 1 0 p m x 1 p i n , find (a) the power density if we assume that the intensity is uniform within the waveguide cross section, and (b) the electric fieId strength and the magnetic field strength, assuming that the plane wave is uniform.

6.2

Consider a LiNbO, crystal with the ordinary and extraordinary refractive indices, r z , = 2.297, n , = 2.205. Write down the dispersion relations for the ordinary waves and the extraordinary waves.

6.3

For a LiNbO, crystal with the permittivity matrix given by

Find the two cl~aracteristicpolarizations yvith corresponding propagation constants for a wave propagating in the y direction. (b) If a plane wave propagates in + z direction, what are the possible characteristic polarizations'?

(a)

6.4

In Problem 6.3, if a plane wave propagates in a direction k = k(,t sin 8 + 2 cos 8 ) on the x-z plane, find the explicit expressions of the fielcls E, 31, I), and I3 for ( a ? the or.clinasj/ wnve and (b) the extraordinary wave. cl:~arter-waveplate for a LiNbO, assuming n o 2.397., ir -= 3.201; ;it a wavelength h = 0.633 p m . (b) Repeat part ( i ~ ) for a "(CP cryst;:! vvitl-1 I I , -- 1.5074. r l , = 1.4669 at

6.5 ( a ) falculatz the thickne;,.: Ai,,/'4 for

-

the san.!e w;~vt.length.

I,,

ri

6.6

If we replace the mirror by a polaroid in Fig. 6.7 wit11 the passing axis pcrpe~ldicularto that of the entrance polar-oid, find the optical transmitted field passing through this new polaroid.

6.7 Two polaroid analyzers are arranged with their passing axes perpendicular to each other and a quarter-wave plate is inserted between them. Initially, the fast axis is assumed to be aligned with the passing axis of the entrance polaroid. Find the optical transmission power intensity for an incidcnt randomly polarized light as a function of the rotation angle 0 between the fast axis and the passing axis of the entrance polaroid. Plot the transmission power vs. 8.

REFERENCES 1. J. A. Kong, Electrornng~retzcWave Theory, 2d. ed., Wiley, New Yor-k, 1990. 3,. M. Born and E. Wolf, Prilzciples of Optics, Pergamon, Oxford, UK, 1975.

Optical Waveguide Theory This chapter covers topics on optical dielectric waveguide theory, which will be useful for studying heterojunction semiconductor lasers. W e present the fundamental results in Section 7.1 for transverse electric (TE) and transverse magnetic (TM) nlodes in a symmetric dielectric waveguide including the propagation constant, the optical field pattern, the optical confinement Factor, and the cutoti conditions. We then discuss an asymmetric dielectric waveguide in Section 7.2. A ray optics approach is shown in Section 7.3, which explains the waveguide behavior. More practical structures, such as rectangular dielectric guides used in index-guidance diode lasers, are then discussed using an approximate Marcatili's method in Section 7.4 and the effective index method in Section 7.5. The results for a laser cavity with gain and Ioss in the media are derived to obtain the lasing conditions of a Fabry-Perot semiconductor laser cavity. Reference books on general waveguide theory can be found in Refs. 1-4 if the readers are interested in more extensive treatment of the subject.

7.1. SknaMETFUC DIELECTRIC SLAB WAVEGUIDES ,I

The dielectric slab waveguide theory is very important since it provides almost a11 of the basic principles for general dielectric waveguides and the guidelines for more complicated cross sections such as the optical fibers. Let us consider a slab waveguide as shown in Fig. 7.1, where the width cv is much larger than the thickness d , and the field dependence o n y is negligible (d/iJy = 0). From the wave equation

(G' VVT

+ w ' ~ . E ) E= O

(7.1.1)

shal.1 find the solutions for the fields everywhere.

VVe assume that the waveguide is symmetric, that is, the permittivity and thc permability are t. ancl p,. respectively, for I s ( 2 d / 2 , and E , and p , for 1 . ~ 1 < d / 2 . The orisin h:~s been chose:^ t o be at the center of the guide, since ihe waveguide is synl~nciric;tkerefore, wi= ht-t-ve even-mode and odd-mode solutions. We separate the c;olutions of the fields into two classes: TE po!:irization nnci T M po1ariz;:t;ion. Here, a wavegilide mode or a normal

I Figure 7.1..

GaAs

A simplified heterojunction diode 1ase.r structure for wavegiiicle analysis.

mode is defined as the wave solutio~lto Maxwell's equations with all of the boundary conditions satisfied; the transverse spatial. profile and its polarization remain unchanged while propagating down the waveguide. -.

7.1.1 Derivations of Electric Fields and Guidance Conditions for TE Polarization For TE polarization, the electric field only has an E , component:

TE Even Modes. We obtain the guidecl mode solutions by exanlining wave equation (7.1.2), which h:ts solutions of the form {exp(ik , z ) , expf - - i k , z ) ) x {exp(ik , x ) , exp(-- i k , x ) , or cos kL.x, sin k , x). Since the guide is trmslationaIly invariant along the z direction, we choose exp(ik,z) for a guided mode propagating in the + z direction. We then choose the standing-wave solution cos k,rx or sin k , x inside the guide and exp(-as) or exp(+crx) outside the waveguide since the wave is a guided mode solution. With the above obse~qations,the electric field for the eve,n rrlodcs can be written in the form

(b) Odd modes

(a) Even modes

Figure 7.2. (a) The electric field profiles of the TE even modes. (h) T h e electric field profiles of the TE odd modes.

as shown in Fig. 7.23, whe.re k x , k , , and

CY

satisfy

which are obtained by substituting (7.1.3) into the wave equation (7.1.2). Matching the boundary conditions in which E , and Hz are continuous at x = d / 2 and x = - d / 2 , we obtain

where H z = (l/icup)(dE,/dx) has been used. Note that the permeability is p , for 1x1 I d / 2 . Eliminating C , and C,, from (7.15a) and (7.1.5b), we find the eigenequation or the guidance condition:

.

The above guidance condition also shows that iri order for E , of the form (7.1.3) to be a guided rnode solution, the standing-wave pattern along the transverse .t- direction or- the amount of oscillations (determined by k , d ) has to match the decay consttint a oatside ille guide. An interesting limit is to consider that the decay rate LY is infiniteiy large, E,, will be zeru outside the m r , which is guide, and (7.1.6) gives ii.,.d/") ~ / 2 , 3 ~ / 2. . ,. or i;,ycl

-

-

the same as the guidance condition for a metaIlic.waveguide, of which the tangential electric field vanishes at the surfaces of the perfectly conducting waveguide.

TE Odd Modes. The electric field call be written in the form

as shown in Fig. 7.2b. Matching the tangential field components E,, and at A: = d / 2 or - d / 2, we obtain

H, -'

Thus

# .

7.1.2

Graphical Solution for the Guidance Conditions

To find the propagation constant k Z of the guided mode, we have to solve for .k,, k,, and a from (7.1.4a), (7.1.4b), (7.1.6), and (7.1.9). We rewrite the eigeneq~ationsin the following forms:

2

n

n2

TE even modes

(7.1.10a)

7 . l odd modes.

(7.1.10b)

111 = -

Pl

[k,x.)cot(k,r~)

Subtract (7.1.4b) from ( 7 . l . h ) to elintin:iti" k.- :.md i.n~.dtiply thc: result by d,/

Figure 7.3. A graphical solution for t h e eigeneq~lationsto determine a d / 2 and k , d / 2 for the 'even modes, TE,, TE,, etc., and the odd modes, TEI, TE,, and so on.

We have only two unknowns, k , and cr. The solutions for a and li,, can be obtained from a graphical approach by looking at the (a d / 2 ) vs. ( k , d / 2 ) plane [4, 51. Equation (7.1.11) appears as a circle with a radius

in Fig. 7.3 and the two eigenequations (7.1.10a) and (7.1.10b) arc also plotted on the same plane. Their interceptions give the solutions for ( a d / 2 ) and ( k , d / 2 ) . Here the refractive index inside the guide n, = \ / p , ~/ ,( p , ~ , ) and outside n = J p E / ( p U ~ Ohave ) been used and 4 , = , = w / c is the free-space wave number. Given the waveguide dimension d, w e find the propagation constant k = from either (7.1.4a) or (7.1.4b) after k , or cr is determined. -

I

7.1.3

Cutoff Condition

1

It is clear from the graph that the cutoff condition occurs at R = rn 7 ~ / 2for the TE,, mode

Thus the TE, mode has no cut08 frequency. (Note: Here we assume that the refractive indices are i n d 5 p ~ r l c i ~ noft the fl-equency. Xn semiconductors, the

frt:quency-dependent refractive indices should be used.) For

there are (nz + 1) guided TE modes in the dielectric slab. --For a single mode operation, the condition ( k , d / 2 ) f i 1 : - n2 < ~ / is2 required. This puts a limit on o, d , or n: - n'. For example, given the wavelength h , in free space and the waveguide dimension d ,

If n7- is almost eclual to 12, as is the case in an A1,,.Gal-,As/'GaAs/ A.l,Ga, -,As wztvcguide, we have

using

nl

- tr2 = 2rt1 A N . For example, at n , = 3.6, we have An

< 0.035 at

A, = d. Single-mode operation is achieved with a small difference in the

refractive index An < 0.035. 7.1.4

Low- and High-Frequency Limits

1. In the low-frequency limit, the waveguide nlodcs can be cut off. At H I ~ / 2 ,and a = 0, as can be seen fi-om the cutoff? ((k0d/2)J-1'= gr.;lphicul soiution. (The cutoff condition can be achieved by reducing the frequency, or d, or ni - n'.) We find /i,,~1/2 = nz 7 1 2 and from (7.1.4h)

C. Thereforz, approaches the propagiltion c(:)nstant

cc~r7stant cjf :!l~ guided rnode /cwi.i,'c ouc:;Ide the waveguide. This is expected since the decaying ccx-stant c? =. 0 ar c!.ltoff-, which jnear?s that the w:~veguiclernode does no! decay c>utsicle ii,lc guide. AIrrlost all the mode power propagates r>utsicle :he g~:ide;.:ilercfurc. the vcIc2city of tilc mode will equal the speed of [ighl o u ~ s l d etht: guide (Fig. 7.43).

sjnce a

fii?

pri:p;;gntio;!

of

"PHigh w

ko

Figure 7.4. The dispersion curves-6f the T E , modes. (b) The ficid 2rofile in the low-frequency limit. (c) The field profile in the high-frequency limit.

2. In the high-frequency limit, R + a.The graphical solution in Fig. 7.3 shows that ( k , d / 2 ) --. (nr + l ) r / 2 and ( a d / 2 ) --. for the TE, mode. Since m is a fixed mode number, we find that ( a d / 2 ) + R + m, as can also be seen from the graphical solution. Therefore,

p = wrr,/c. T h e propagation constant k , approaches that in or k = u the waveguide since the mode decays rapidly outside the guide ( a -, m). Most of the power is guided inside the waveguide, Fig. 7 . 4 ~ .

7.1.5

Propagation Constant k , and the Effective Index n,,

From Eqs. (7.1.17) and (7.1.181, we see that the propagation constant k , starts from w f i at cutoff and increases to w as the frequency goes to infinity. The dispersion cuives k , vs. w \ / p o e , are plotted in Fig. 7.4a. An effective index for the guided mode is defined as

%4

= 1 5 and the upper bound Again we have the lowcr bound O V , L L E u d K / k , , = n , at the low- a n d high-frequency limits, and the dispersion curves in Fig. 7.421 map to a set of effective i i ~ d e xcurves as shown in Fig. 7.5a.

Figure 7.5. (a) The effective index neff = k, / k o of the TE,,, modes vs. the free-space wave number w / c . (t,) The normalized propagation parameter vs. the nofrnalized parameter V number for the TE,, modes.

fi

Very often a normalized propagation parameter

is aIso plotted vs. the V number or normalized frequency

in Fig. 7.5b, showing a set of universal curves for the lowest three TE modes. Notice that the above discussions assume that n and n, are independent of the frequency. In semiconductors, the dispersion effect of the refractive index near the band edge can be significant. It is important to know the refractive index at the operation wavelength r z ( A ) . The effective index n,,(A) still falls betweer1 n ( h ) < n e f l ( h )< n , ( h ) when the material dispersion is taken into account. 7.1.6

Refractive Index of ' A I , G ~ _ ,AS , System

-! lit. refractive

.

index of rhis system and many III-V materials [6-181 below the direct band gap can be approximately given hj, the form 1-

where E ' ( w ) is the real part of 'the permittivity function and the imaginary part ~ " ( wis) negligible for optical energy below the band gap [12, J.3, 151: cl(w) &

o

r

=A(x) f(y) + 2

1312

E ( x ) -t A ( x )

f ( J~S,)

1

+

B(x)

Here E , ( x ) is the band-gap energy and A ( . r ) is the spin-orbit splitting energy. For Al,Ga,-,As ternaries below the band gap, the parameters as functions of the aluminum mole fraction x are

These results have been compared with experimental data with a very good ) E'((~)/E agreement, as shown i n Fig. 7.6a [ I l l for the real part ~ , ( w = below the band gap. T h e refractive index n ( c d ) below the band gap is plotted in Fig. 7.6b. In Fig. 7.6c, we show the refractive index above and below the band gap for AI,rGa,-,As with various aluminum mole fractions .r. For comparison, the rekactive index of a high-purity GaAs is shown as t h e top dashed curve and that of a p-type doped GaAs is shown as the top solid curve. We can see the features near the band gap of the AI,Ga,-,As alloys. As, In Ga.,A,P, - Y , More data and theoretical fits on GaAs, N,Ga XnP, and other material systems are documented in Refs. 9-19. It is noted that the refractive index is gznerally dispersiCe near or above the band gap with a fast variation near various interband transition energies. It also depends slightly on the doping conce1;tration of the semiconductors, the temperature, and thi: strain. Above the bani! gap, both the real and the imaginary parts of the pel-m.ittivity func!ion ar'e important, and Eqs. (E.22a) and (E.22b) in Appendix E ::;i-~clclld be used to relate ~ ' ( o , )and ~ " ( oto) the complex refractive inde..c n -t i t c .

,-,,

,-.,

~

(a)

. I 1.0

I

I

20

1.5

PHOTON ENERGY T----r---'--

C cV

)

m-

0.3 3.9 3.7

.-

3.3

LLA--.I--A

3.1

0.8

1.0

1.2 1.4 1!.6 1.13 Photon Energy ( e V j

2.0

2.2

3.8

3.7

IC

,

3s

Z

-

2p

3.5

3.4

3.3

3.2

1.2

1.3

1.4

1.5

1.6

1.7

1.8

ENERGY, hv (aV) Figure 7.6.

(Cuntirl~ied)

7.1.7 Normalization Constant for the Optical Mode "The undetermined coeficien t C, is de terlnined by the normalization condi:. tion in which the total glided power is assumed to be 1. That is,

Since for TE modes,

where the permsa[.iliijl

is

/.L,

. . ir;::!cle

tlze guide

i ~ i l dy

outside the guide. We

'7

'O

-d 5 x 5 0

C, e'*z'

COS(- k , , d

+ &)e"""+ d ,

(7.2.17)

x 5 -d

where the constant C , can be found from the normalization condition, noting that E # z E ~ The . guidance condition is obtained from (7.2.8) after replacing p i by s i . We find

for the TM,,, mode. We summarize the guidance conditions in terms of the refractive indices 11, ?I1, rz2 and use p = p , = p , for dielectric materials using (7.2.8) and (-7-2.18):

RIlcd= tan-'

a -

kl,t kl,d

7.3

=

tan-'

a2 + tan--' + m.rr

-

nfa, --

n2hx

n:n-,x

nfa

(TE,,, mode)

(7.2.19~1)

k2,r

+ tan-'

+ nzr

(TM,,, mode) (7.2.19h)

RAY OPTICS APPROACH TO THE VYAWGUIDE PWOBLElMS

An efficient method to find the eigenequation for the dielectric slab waveguide probIem is to use the ray optic,s picture. We know from Section 5.3 that when a plane wave is incident on a planar dielectric boundary with an angle of:incidence 8 larger tharl the critical angle, t h ~reflection : coefficient has Cioos-1-iiinchen phase shift 2 $ ! , . For a mode to be guicled, the total iieid af'icr reflecting back and forth i[frum point .A to B) hy t h e ~ V J Gb~3ur!d;~rie:;s11o:dd 1-rpca'c itself (Fig. 7.LZ):

Figure 7.11.

A ray optics picture for a guided mode in a slab waveguide.

where the phase 2 k , , accounts for the round-trip optical path delay along the x direction. T h e above condition is also called the transverse resonance condition. In terms of-the Goos-Hanchen phase shifts, we obtain

where for TE modes

Therefore, cve find

The total phase delay (noting that the Goos-Hanchen phase shifts are negative, - 2 # 1 2 - 34,,, or advance phase shifts) should be an even multiple of 7 . Equation (7.3.4) gives the guidance condition for the TE,,, mode:

If p L = p and E -, and

= r,

i-e.. the slab wavegl;idc is symmetric; we have dl,

- dl,,

which leads to tan tan[k')

+,,

=

-cot

ILL 1 = --

+,,

if nz is even

4,

P k l xP =

- -I

(7.3 -7)

if rn is odd

.

which are the same as (7.1.10a) and (7.1.10b), as expected. I

7.4

RECTANGULAR DIELECTRIC WAVEGUIDES

The rectangular dielectric waveguide theory is very useful since most wavegliides have finite dimensions in both the x and y directions. There are two possible classes of modes. The analyses are approximate here, assuming krl 2 d, as shown in Fig. 7.12. The method we use here is sometimes calIed Marcatili's method for rectangular dielectric waveguides [20, 211. These modes are

1- HE,, modes or E & + I , ( ~ + ,modes: , H, and E,, are the dominacomponents. 2. EH,, modes or E&+l X q + modes: E , and H9 are the dominant components. I n general, we have d j d z = ik, and Maxwell's equations in the following -forn~swill be used as the starting equations in ,the later analysis for both

3 ?l'IC,SL WI',VEGU IDF THEC R'i

36.1

HE,,, a n d EH,,, modes:

Here we have assumed that the permittivity is a constant in each region. 7.4.1

HE,, Modes (or Ef,

+

,, modes)

Suppose we start with the HE,, modes. T h e large components are H, and E,.. T h e other components Ex, E z , H,., and Hi are assumed to be small. In principle, we can assume that H , = 0, and express Ex, E,, E,, and Hz in terms of H, using Maxwell's equations. These are called the HE,, modes o r E;, modes. Alternatively, we can also assume Ex = 0, and express E,, H,v;' H,, and Hz i n terms of E,. Since these are all approximate solutions, they have to be compared with the exact solutions [22, 231. W e find that bbth approximations give essentially the same eigenequations with minor differences. We call these modes HE,, modes, where p labels the nuniber of nodes for a wave function in the x direction and q for the y direction. T h e solution procedure is as follows.

1. Solve the wave equation for

2. Use (7.4.4):

H.,everywhere:

4. Since FI), = 0, we obtain

-1 a -Hz i w r Jy

E = -

"

=2

-1 a a -- -Hx wek, Jx Jy

Once H, is found, all other components (except that Hy = 0) can be expressed in terms of H, using Eqs. (7.4.6)-(7.4.9). The expression for H, can be written approximately as ' ~ , c o s ( k , r + Q,)cos(kyy C,cos(k2x

H,

=

+ 4,)

+ #,)e-"~(~-"'

c , -a.j(~r-.d)cos(k,,y ~ + 4,) .

e'*r'
0) or it has gain

(F"

< O), and the

where the real and imagina'i-y parts of the complex refractive index, n and K , have been tabulated and plotted in Appendix J for GaAs and TnP semiconductors. For a wave propagating in the + z direction, the field soIution is (consider E = FEY) ! E,.

=

~

~

e

~

~

'

T h e field dccays.as i t propagates in the .+-zdirection for a I::ssy r ~ e d i u mand grcms i f the medium has a gt3in n:cchanisu-I such as in the active region of a laser. For a waveguide that may contain gains ir'r t h e active region and losses in the passive region. the propa:,jatioi~c:::nst;;n! i s gerter~rillycoinpiex. h typic~zl

Figure 7.25.

A lossy (gain) waveguide structure.

E'

c id'

analysis is based o n the perturbation theory or a variational method. Consider a dielectric slab waveguide that has complex perrnittivi ties everywhere, as shown in Fig. 7.15:

E X )

=

(

F; F'

+ icr; + is"

inside outside

Inside the guide,

where k ; = w J s = ken,, and g = - ~ ; E ; / E ; E'; < 0 . Outside the waveguide we have

is the gain coefficient, if

where n = k r & " / ~is' the decaying corkant of the absorption coefficient for the optical intensity. Suppose we try to solve the TE mode

Figure 7.16. A semiconductor Iaser cavity with n net gain y in the active region and an absorption coefficient a outside the active region. The cavity length is L and the reflectivity is R at both ends.

R

R

using the binomial expansion assuming that the imaginary part is small compared with the real p:rrt, and k:) is close to k' and k ; . When applying (7.6.14) to a laser cavity, us shown in Fig. 7.16 we have the wavc bouncing back and forth in a round trip of distance 2 L :

where r- is the reflection coefficient of the electric field. T h e reflectivity R is related to r by R = r l z . Ignoring the small imaginary part of r , which is a good approximation for most cases, we have e i 2 k ~ u ) ~ e r . y ~ - (R1 = - ~ l) a ~ Thus

2 k : ' ) ~= 2 m n

nl

=

an integer

(7.6.17)

which determines the lorlgitudinal modz spectrum of the semiconductor laser, and

which determines the threshold gain condition. H e r e Tg is the modal gain of' the dielectric waveguide. Strictly speaking, if the modal dispersion is important, k',O' of the guided mode solution should be used. For most semiconductor waveguides with small indzr differences. we can assume kto) = ( ~ / C ) I L [ and (7.6.17j can also be approximated by

,

The frequency spectrum is C ftTl

= t?Z --

2n,L

and the Fabry-Perot frequency spacing is

Alternatively, we write in terms of the free-space wavelength A

and

when the dispersion effect ( d n , / d h # 0) is taken into account. For two nearby Fabry-Perot modes, Am = 1 and A A given in (7.6.23) gives the wavelength spacing of the Fabry-Perot modes at A:

In Fig. 7.17, we show the amplified spontaneous emission spectrum of a InGaAsP/ InGaAsP strained quantum-well laser at 300 K. The cavity length .C

-

Figure 7.17. Amplified sgcmt;ini-.c>iisrmissio~isprctrL!?;l crT a "37-)-,1,~ ssc~iconciuctorlaser cavity !:I room temperature below threshold. 'I?;;-. iiljeclicsn ctirrent is I R mA. (Above the laser threshc-,ld current I,!, .= 13.5 mA, I,~:;iri; ~ t ~ ~ i t s . , ;

L is 370 p m . T h e threshold current is I,,, = 13.5 niA at this temperature. Using T I , = 3.395 and d r ~ , / d h= -0.264 (prn)-" near h = 1.49 p n l for the barrier i n , -,Ga, As, P, -,(A, = 1.28 pin, y = 0.55 and x = 0.25, see App e n d ~K) ~ [lo, 161, we obtain the wavelength spacing A h = 7.9 A, which agrees with the spacing shown in the figure.

7.1

Consider a Ga,vIn,-,As waveguide with a thickness d confined between two InP regions, where the gallium mole fraction x is 0.47, as shown in Fig. 7.15. (a) If the free-space wavelength h o is 1.65 p m , what is the photon energy (in eV)? (b) Find the maximum thickness d o (in p m ) so that only the TE, inode is guided at the wavelength h o given in part (a). Assume that the waveguide loss is negligible at this wavelength. (c) If the thickness d of the waveguide is 0.3 p m , find the following parameters approxinlately (in l / p m ) for the TEo mode using the graphical approach in the text: (i) the decay constant cu in the InP regions, (ii) k , , and (iii) k , in the waveguide. (d) Calculate the optical confinement factor for the T E o mode in part (c).

Figure 7.18.

UP

n = 3.16

7.2

Repeat Problenls 7.l(b) and (c) for the T M polarization. A r e the answers the same as those for the TE polarizationc?

7.3

Calculate the Goos-Hiinchen phase shifts for the TE, and TM, guided modes in Problem 7.l(c).

7.4

Show that for a symmetric dielectric waveguide with the V number

approaching zcro. the propagation c.:>nstant k - of the TE,, mode can be written as

:,

1' R (.-I E L .I.' Tvi'

s

'>7G

I , '

Find the expression for C,. (Hint: Start with the guidance condition and the dispersion relation for V --+ 0.1 7.5

Consider a symmetrical slab waveguide with a thickness d (Fig. 7.19a) and a symmetric quantum well with a well width d (Fig. 7.19b). Cotlsider the guided TE even modes for the optical waveguide and the general even bound states for the heterojunction quantum-well structure. Summarize in a table an exact one-to-one correspondence of the wave functions, material parameters, eigenequations (guidance conditions), propagation constants, and decay constants, for the two physical problems above.

P* f2

+d Figure 7.19.

Consider a symmetric optic,al slab waveguide wit11 a refractive index inside the guide 3.5 and a refractive index outside the guide 3.2, the waveguide dimension is 2pm, and assume that the magnetic permeability is the same everywhere. (a) For a light with a wavelengtli 1.5 ,urn in free space, find the number of guided TE modes in this waveguide. (b) Using the ray optics approach, find the angle 8,, of the guided TE,,, mode bouncing back and forth between the two boundaries ,of the waveguide (Fig. 7.20).

Figure 7.20.

( c ) (Independent of parts

a : ~ dh).) Find th:: cl.~toft'wavelength A , , OF the TE,,, mode. Ev:tlu;.~te the decay constant n outside the waveguide ar1t.i 1:;-ii. propagatici: constant Xr _ at the cutoif condition. . (:I!

7.7

(a! Plot the refractive index n vs. the optical energy Aw using Eqs. (7.L.22) to(7.1.24)for s = 0, 0.1, 0.2, and 0.3. (b) Design a GaAs/Al,Ga,-,As waveguide using the above results such that only the fundamental T E o mode can be guided at a wavelength A, near the GaAs bandgap wavelength at room temperature. Ignore the absorption effects for simplicity. Specify the possible parameters such as the alumir~ummole fraction x and the waveguide thickness d .

7.8

(a) Derive the normaIization constant C , for the electric field of the

T E mode in an asymmetric dielectric slab waveguide. (b) Repeat (a) for the magnetic field of the TM mode. 7.9

(a) Find the analytical expressions for k,, k,, and a in a symmetric waveguide for the TE,, mode at the cutoff condition. (b) Repeat (a) for the TM,, mode.

7.10

(a) Find the analytical expressions for k,r, k ; , a , and a , in an asymmetric waveguide for the TE, mode at the cutoff condition. (b) Repeat (a) for the TM, mode.

7.11

For an Ino~,,Gao,,,As/InP dielectric waveguide with a cross section ( 2 p m x 1 pm), find the propagation constant of the lowest HE,, (or E:,) mode. You may use the graph in Fig. 7.13, assuming n l = 3.56 and n = 3.16 at a wavelength A, = 1.65 p m . What is the effective index of this m o d e ?

7.12

Repeat Problem 7.11 using the effective index method. Check the accuracy bf the result compared with that of the numerical solution from Fig. 7.13.

7.13

List all of the assumptions made in order to derive the threshold condition

7.14

Show that if the reflectivities from the two end facets of a Fabry-Perot cavity are different, R , # R,, the threshold condition is given by

7.15

Consider the laser str~lctureshow11 in Fig. 7-31with losses described by the absorption coctficisnt a , = 10 c m - ! In the substrates and a = 5 cm-I in the active regior,. The ref-lectil/ity K is 0.3 at both ends and

+

>

500 prn

Figure 7.21.'

the waveguide length is 500 p m . Assume, for simplicity, that the propagation constant of the guided mode is approximately w n / c , where n = 3.5. (a) What is the threshold gain g,,, of the structure if the optical confinement factor of the guided mode is 0.9? (b) What is the longitudinal mode spectrum of this laser structure? Calculate the frequency spacing A f and the wavelength spacing Ah of two nearby longitudinal nlodes at h o 0.8 p m and ignore the dispersion effect for simplicity.

-

1. D. Marcuse, Theory of Dielectric Optic PVnr~egz~ides, Academic, New York, 1974. 2. A. W. Snyder and J. D. Love, Opticizl JVnvegt~irle Theory, Chapman & Hall, London, 1953. 3. A. Yariv, Optical Eleutrorrics, 3d ed., Holt-Rinehart & Winston, New York, 1985. 4. H. C. Casey, Jr., and M. B. Panish, Heterost~zrct~lre Lnsers, Part A: F~mdnrnc~ztal Principles, Academic, Orlando, FL, 1978. 5. J. A. Kong, Electronragnetic Wnr9eTheory, Wiley, New York, 1991). 6. M. A. Afromowitz, "Refractive index of Gal-,A1 ,As," Solid S t ~ ~ rConzrn~in. e 15, 59-63 (1974). '7. J. P. v a n der ZieI, M. Ilegems, and K.M. Mikulyak, "Optical birefringence of thin GaAs-AlAs multilayer films," Ayyl. Phys. Lett. 28, 735-737 (1976). 8. J'. P. van der Ziel and A. C. Gossard, "Absorption, refractive index, and birefringcnce of AlAs-GaAs monolayers," J . ifpyl. Y/zys. 48, 3018--3023 (1977). 9. J:S. Blakernore, "Semiconducting and other major properties of gallium arsenide," J . Appl. Plzys. 53, K123-R181 (1983). 10. S. Arlaohi, "Refractive incliccs of 111--Vpmp:mnds:Key properties of InGaAsP i-clevar;t to device design," j . AppI. F'/~ys. 53. 5563-5869 (1982). i I . S. A.d;ict~i and K. C)e, ''X;ltel-c;~lstrain nr::! phtsloelastic efkcts in Cra, Al,As/ GaAs and i n , -,Gal As ,:P, -., / InP r:ryst:!Is," .I. App!. P!p.s. 54, (3620-6627 (1933). 12. S. Atlachi, "GaAs, i\iA:,. ;ii.it.! C;a,-,i-i!,rij.s: hlateriai p:!smleters for Lise !i; .. research ancl devic:: :1pplic;ltii?iis, .I. Appl. f'iz:k1s. 53, R1--K39 (198.5). -,[

-.

13. S. Adachi, "Optical properties of A! , G a l --,As alloys," Phys. Rer!. B 35, 12345- 12352 (198s). 34. H. C. Casey, Js., D. D. Sell, and M. B. Panish, "Refractive index of Al,Ga,-,As between 1.2 and 1.S eV," Aypl. Pl~ys.Lett 24, 63-65 (1974). 15. M. Amiotti and G . Landgren, "Ellipsometric determination o f thickness and films on InP," refractive index at 1.3, 1.55 and 1.7 p m for In(,-,,Ga,As,P,,-,, J. Appl. Phys. 73, 2965-2971 (1993). 16. S. Adachi, Physical Propei.fies of III-V Seiniconductor Cotnpozincls, Wiley, New York, 1992. 17. S. Adachi, Proycrties o f Indium Phosphi~ie,INSPEC, T h e Institute of Electrical Engineers, London, 1991. 18. K. H. Hellwege, Ed., La~ldoll-Boi-iisteirrN~iinericnlData and Functional Relationships it1 Science arzd Technology, New Series, Group 111 17a, Springer, Berlin, 1982; Groups 111-V 2221, Springer, Berlin, 1986. 19. 0. Madelung, Ed., Sei?iico~zdi~ctors, Grolip I V Elemeizts a r ~ d111-V C o i ? ~ l ~ o ~ i nind s , P. Poerschke, Ed., Dnta in Scierrce a n d Technology, Springer, Berlin, 1991. 20. E. A.--J. Marcatili, "Dielectric rectangular wavegiride and directional coupler for integrated optics," Bell Syst. Tech. J . 43, 2071-2102 (1969). 21.. E. A. J. Marcatili, "Slab-coupled waveguides," Bell S y s f . Tech. J . 53, 645-674 (1974). 22. J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguide," Bell Syst. Tech. J . 48, 2133-2160 (1969). 23. W. C. Chew, "Analysis of optical and millimeter wave dielectric waveguide," J. Elecfrom~lgrzeticWares Appl. 3, 359-377 (1989). 24. C. S. Lee, S. W. Lee, and S. L. Chuang, "Plot of modal field distribution in rectangular and circular waveguides," IEEE Traizs. hIicrorvarle Theory Tech. MTT-33, 271-274 (1985). 25. W. Streifer, R. D. Burnham and D. R. Scilres, "Analysis of diode lasers with lateral spatial variations in thickness," Appl. Pl~ys.Lett. 37, 121-123 (19SO). 26. J . Buus, "The e f f e c t i ~ c index method and its application to semiconductor lasers," I E E E J. Qci~llrrimElcctrori. QE-IS, LUS3-10S9 (1982).

Waveguide Couplers and Coupled-Mode Theory In this chapter, we discuss how a light can be coupled into and out of a waveguide, and how it can be coupled between two parallel waveguides or coupled within the same waveguide with a distributed feedback coupling. We present the coupled-mode theory for wave propagation in parallel waveguides. An analogy of this coupling between two propagation modes in dielectric waveguides is a coupled pendulum system or a coupled LC resonator system. The coupling of two resonators leads to energy transfer between two resonators at a frequency determined by half of the beat frequency of the two system modes: the in-phase mode and the out-of-phase mode. For coupling of propagation modes between two parallel waveguides, ;t beat length can also be defined, which is twice the minimum distance required for the exchange of the guided power between the waveguides. Many interesting physical systems exhibit the phenomena of the coupling of modes, which can be understood from a simple model of two coupled pendulums.

8.1 WAVEGUIDE COUPLERS 11-31

-We discriss transverse couplers, prism couplers, and grating couplers and point out the significance of the phase-matching condition in this section.

8.1.1

Transverse Couplers

Ilirect Focusing. As shotvll in Fig. 8.1, a lens is used to focus and shape the beam profile so that it is matched to the modal distrihutio~~ it1 the waveguide. 'The coupling e.fficie11cy depends on the ovsrlap integral between the field profile of the incident wave and t t ~ cl i ~ ~ d ;profile il of the guided mode in the thin film. This configuratii:,n can achieve early !.GO% efficiency. However, it may not be suitable for gent.r:~I intsgratcd optics applications because of the ilorxplnnr~.rconfiguration.

t t

(b)

Active layer

Figure 8.1.

~~g

(a) Direct focusing. (b) End-butt coupling.

End-butt Coupling. As shown in Fig. 8.lb, the efficiency of a n end-butt coupling can be controlled by the gap width t , the waveguide dimension d and refractive indices of the waveguide and the substrate, n, and n,, respectively.

8.1.2

Prism Couplers

For a waveguide mode, the propagation constant P along t h e z direction has to satisfy the conditions, k,n, < P < k,n, and k,n, < P < k,n,, where lio = 2.rr/ho is the wave number and h o is the wavelength in free space. For an incident plane wave from region 0, its horizontal wave number is k , = kon,sin 8, < lion, for any real incident angle Bi. "Thus, .the wave cannot be phase matched to the guided mode since k,no < P, Fig. 8.2b. The excitation of the guided mode is very weak. However,, using a prism above the waveguide with a small gap (assuming n, > no and n, > n,), Fig. 8.2a, the phase-matching condition can be satisfied: k,n,sin 8, = P , Fig. 8 . 2 ~ .This angle of incidence 8, will be larger t h a n t h e critical angle for the interface between the prism and region 0,

B;

71 4)

=

sin- ' :2,

since ,Cl > k,:l,,. 'Tl;eref;>re, ihe i ~ . ~ n s m i t t c1dq 5 ~ ~ ~inv t . the gap region is an ~ ille guicled mode. In evanescent type and can be resonantly C O L ; ~ ! Z ~to generaI, for a particular 'I'E,,, n~4)ciewith 1; prc:\xiyalion constant P,,,, w e can

Figure 8.2. (a) A prism coupler. (b) T h e phase-matching diagram for a guided fnode with a propagation constant P > k,no. (c) T h e phase-matching diagram for an incident wave fro111 the prism region and coupling to the guided mode with k D n psin 8, = P .

choose

8, such that

Other important parameters are the incident beam profile, coupling length, and gap dimension. 8.1.3

Grating Couplers

Consider first a plane wave incident on a periodic grating which is assumed to be a perfect conductor with the surface profile described by x = h ( z ) with a period A , as shown in Fig. 8.3. E~ = 9~~

- ikxO.x+ i l l Z O i

k ,., = k cos 8,

E.

?orfect ~ont!ucl(~r or n dielectric rncdiurn (p,, E ;)

1

!-

k

_,,= k sin Oi

EL!

-4

E-ignrc 8.3. ;\ p l a ~ l rvtave ELwith the polarizaticir~:1!03.g the y clir.c:cti~nis incidel~t(.,il a g ~ a t i n g YUI-LICC wi!h :I profile .v =-. / I ( , : ) a n d a periud A.

2. :6

%'A 'VEG U; :3Ei COU:" J ? R S f,,ND CCJPI,E,D. MCDE THEORY

The reflected wave is E, = $E,(s, r). Since the boundaiy condition requires that the tangential electric field vanish at the conductor surface, the reflected wave on the surface of the grating should satisfy

. is a where we have made use of the Fourier series expansion, since e-ik.vu"(z) periodic function with a period A. In the region above the grating, ER(ll:

For the initial excitation at z = O of a two-coupled waveguide, a(O) b(O) = 0, we obtain from (8.3.31) and (8.3.32)

i

o(C) = c o s $ t b(t)

=

ikb,,

-sin

(I,

=

1,

(8.4.7a)

fie

eiC"

(8.4.7b)

where

We can show from the exact relation (5.4.4)

The solutions (S.4.7a) and (8.4.7b) can be written as

The output power P', ir-I w;~vcguidei r \,\/her1 vv.:lveguidc b terminates

;it

z

=

t

is obtained using

where the expansion in (5.4.12a) or (8.4.13a) is in terms of individual waveguide modes, and in (8.4.12b) or (8.4.13b) is in terms of all the guided and radiation modes of waveguide a alone since they form a complete set. Here n = 1 refers to the fundamental guided mode Ela) and H i a ) used in (8.4.12a) and (8.4.13a). Cross-multiplying (8.4.12) by HI") and integrating over the cross section, one obtains [23, 291 -

L L ~ == )

-.

,;(e ) + C,,b(e

)

(S.4.14a)

Similarly, one finds

where t h e overlap integrals, C,, and C,,, are defined in (5.3.10). The guided power due to the first mode p, in waveguide n is

A similar procedure for the output power in waveguide b when waveguide n is terminated at r = t leads to

These results are very similar to those in Ref. 24 except that the parameters are defined in terms of the more accurate parameters y,,, y,, k , , , arid k,,.

3.4.2 P4umerical Resu3Ls for. Tlrvo Strongly 43oupled Waveguides 5.14 we show nurnericd results fur two coupled Ti-diffused Limo, channel waveguides modzied a s two zui~pled slab waveguides (which is possible gsing the effectivt: index method [38])with the I-efractive index in 1.11 Fig.

ASYNCHRONISM

-4 1 I 0.0 0.002 R E F R A C T I V E I N D E X OIFFERENCE

I 1

- 0.004

- 0.002

(a)

-

_ On

I

0.004

a/-..

.

-----..

OVERLAP INTEGRALS

I

- 0.004

0.i65 I 0.0 0.002 R E F R A C T I V E INDEX DIFFERENCE I

'

- 0.002

I

,An

0.004

av

(b!

Figure 8.14. (a).' async.hronisnl ;j is plottcil V C L S L ~ S the rzfl-active index diference of tivo-cuuplecl waveguidrs, A,! = ,l,; - ~ i , , .which is pri;po:.rion~li to t h e applied voltage I*'.(b) The overlap integrals C,,, (dashed l i n e ) . (:i.,, ('~!:::;t~ccll i ~ i : : ) . 21td C" -- (C7,;, -I- C',,,,)/Z (solicl Line) are shown and a r rrltru~st ~ icli.:rric-a!.

COUPLING COEFFICIENTS

1

I I0.000

- 0.004

- 0.002

0.0

I

O.OQ2

I

-On

0.004

gv

REFRACTIVE INDEX DIFFERENCE

(c>

OUTPUT POLVERS

REFRACTIVE INDEX DIFFERENCE

(dl

K,hk,,,,

Figure 3.14. ( C o ~ z r i ~ l ~(c? ~ e d'The ) cucipling ~i,c:tlicicnis A