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'''

[l
56 Ba

1

1

[KT] 4d1('

Note: [Ne] = 1 ~ ~ 2 [Ar] = [Ne] 3s'3ph [Kr] = [Ar] 3d'04s'4p6

[Xe] 6s' -

80 Hg [Xe] 4f I" 5dI06s2

[Xe] = [Kr] 4d'05s25p6

~

~

2

~

~

B

1 I

figure 1.1. (a) A zinc-blende structure such as those of G a A s and I n P semiconductors. (b) T h e

zinc-blende structure in (a) consists of two interpenetrating face-centered cubic lattices 9 + i), where a is the lattice constant of the semiseparated by a constant vector ( a / 4 ) ( i conductor.

+

(a) Bonding diagram

-..

(c) Bonding diagram

(b) Band diagram

(d) Band diagram Energy

Empty Photon

...................... ...................... I ...................... Full Figure 1.2. ( a ) A planar- bonding diagram for a G a A s lattice. Each bond consists of two valence electrons shared by a gallium and an arsenic atom. (b! T h e energy-band diagram in real space shows the valence-band edge E , below which all states are occupied, and the conduction band edge E, above which all stales are empty. T h e separation E, - EL, is the band gap E,. (c) A bonding di:lgrarrt showing a broken bond due to the absorption of a photon with an energy above t h e band gap. ,A free electron-hvlz pair is created. Kotr that the photogenerated electron is free to move around and chc hc;Ie is also f:c:c to flop artjunil at d i t k r e n t honds between the Ga and As atoms. ( d ) The e~iergy-b:tnd cli;lgrai:: s l v u i n g the triergy levels cif the electron and the holc.

constant and v is the frequency of the photon,

:

where c is the speed of light in free space and h is wavelength in microns (pm). T h e absorption of a photon may break a valence bond and create an electron-hole pair, shown in Fig. 1.2c, where an empty position in the bond is represented by a hole. The same concept in the energy band diagram is illustrated in Fig. 1.2d, where the free electron propagating in the crystal is represented by a dot in the conduction band. It is equivalent to acquiring an energy larger than the band gap of the semiconductor, and the kinetic energy of the electron is that amount above the conduction-band edge. T h e reverse process can also occur if an electron in the conduction band recombines with a hoIe in the valence band; this excess energy may emerge as a photon, and the process is caIled spontaneous emission. In the presence of a photon propagating in ,the semiconductor with electrons in the conduction band and holes in the valence band, the photon may stimulate the downward transition of the electron from the conduction band to the valence band and emit another photon, which is called a stimulated emission process. Above the conduction-band edge or below the valence-band edge, we have to know the energy vs. momentum relation for the electrons or holes. These relations provide important information about the number of available states in the conduction band and in the valence band. W e can imagine that by measuring the optical absorption spectrum as we tune the optical wavelength, we can somewhat map out the number of states per energy interval. This concept of joint density of states, which is discussed further in the following chapters, plays an important role in the optical absorption and gain processes in semiconductors. T h e recent pr6gress in modern crystal growth techniques [5] such as the molecular beam epitaxy (MBE) and the metal-organic chemical vapor deposition (MOCVD), has demonstrated that it is possible t o grow serniconductors of different atomic compositions on top of another semiconductor substrate Mtith monolayer precision. This opens up extremely exciting possibilities of the so-called "band-gap engineering." For example, aluminum arsenide (AlAs) has a similar lattice constant as gallium arsenide ( G a ~ s ) We . can grow a few atomic layers of AlAs on top of a gallium arsenide substrate, then grow alternate layers of GaAs and MAS.W e can also grow a ternary compound such as AI,KGa,-,As (where the aluminum mole fraction x can be between O and 1) on a GaAs substrate and form a heterojunction (Fig., l.3a). Interesting applications have been found using heterojunction structures. For example, when thc wide-gap A1 ,Cia, -,As is doped by donors, the free electrons from the ionized doril:*?~renil to fall to the conduction bancl of the GaAs region becri.~:seof the !ohver patentia1 eriergy on that side; the band diagram is shown ill Fig. 1.31;. (This band bending is investigated in Chripter 2.) An applied field i : ~;! c l I r e c t i l ~ npar2llel t o the junction interface

GaAs

AlxGa ,-xAs

heterojunction formed with different band gaps. T h e Figure 1.3. (a) A GaAs/Al,Ga,-,As band-edge discontinuities in the conduction band and the valence band a r e A E , = 67%A E, and AE,. = 33%AEG. (b) With n-type doping in the wide-gap AI,Gal-,As region, the electrcins ionized from the donors fall into t h e heterojunction surface layer on t h e G a A s side where the energy is smaller and an internal electric field pointing from the ionized (positive) donors in the A1,Gal-,As region toward the electrons with negative charges cl-eateT5a band bending, which looks like a triangular potential well to confine the electrons.

will create a conduction current. Since these electrons conduct in a channel on the GaAs region, which is undoped, the amount of impurity scatterings can be reduced. Therefore, the electron mobility can be enhanced. Based on this concept, the high-electron-mobility transistor (HEMT) has been realized. For optoelectronic device applications, hetero,i.rnction structures [6].play important roles. For example, when semiconductol lasers were invented, they had to be cooled down to cryogenic temperature (77 K), and the lasers could lase only in a pulsed mode. These lasers had large threshold current densities, which mearis that a large amount of current has to be injected before the lasers could start lasing. With the introduction of the heterojunction semiconductor IaserS, the concept of carrier and photon confinements makes room temperature cw operation possible, because the electrons and holes, once injected by the electrodes on both sides of tile wide-band gap Y-N regions (Fig. 1.4), will be confined in the central GaAs region where the band gap is smaller, resulting in a smaller potential energy for the electrons in the conduction band as well as a smaller potential energy for holes in the valence band. Note that the energy for the holes is measured downward, which is opposite to that of the electrons. For the photons, it turns out that the optical refraotive index of the narrow band-gap material (GaAs) is larger than that of the wide-band gap material (A1,cGal-,,As). Thereforz, the photons can be confined in the active regior! as well. This double confinement of both ied process niore efficient carriers and photons ~nztkesthe ~ t i ~ ~ - l u l aemission and leads to the room-tarnperatrire operation of laser diodes. The control of the mole fractions of different atoms also makes the band-gap engineering cxtrernely ex.~itiilg.For opticaI comm~inicationsystems,

INTRODUCTION

----.

E" -

Photon emission

v GaAs E i Figure 1.4. A double-heterojunction serniconductor laser structure, where the central GaAs region provides both t h e carrier confinement and optical confinement because of the conduction ant! valence-band profiles and the refractive index ~ r u f i l e This . double confinement enhances stim~~latecl emissiolls and the optical n ~ o d a lgain.

Refractive index profile

ns

I

I

I I

Position z

-..

it has been found that minimum attenuation [7] in the fibers occurs at 1.33 and 1.55 p m . It is therefore natural to design sources such as light-emitting diodes and laser diodes, semiconductor modulators, and photodetectors operating at these desired wavelengths. For example, by controlling the mole fraction of gallium and indium in an In,-,Ga,As material, a wide tunable range of band gap is possible since TnAs has a 0.354 eV band gap and GaAs 0.47, the In,Ga,-,As has a 1.424 e V band gap at room temperature. At x alloy has a band gap of 0.75 eV and is lattice-matched to .the InP substrate, because the lattice constant of the ternary alloy has a iinear dependence on the mole fraction:

-

where u(AJ3) is the lattice constant of the binary compound A B and a(BC) is that of the compound BC. This linear interpolation formula works very well for the lattice constant, but not for. the band gap. FOI: the band-gap dependence, a quadratic dependence on the mole fraction x is usually required (see Appendix K for some important material systems). For &,Gal-,As ternary compounds with 0 5 x < 0.4, the following linear formuia is commonly used at room temperature:

Most ternary compounds require a quadratic. term. From the above formu.la, we can calculate the cond~.ictic>nand vall:r!cc: Garid-edge discontinuities between a GaAs and a r ~A1 $ 3 , _, As heterojul~ciion using A E,. = 67%A E j! and A EL,= 33%AE,. where A E, = 1.?~17.1 (cV). When very thin layers of heterojunction structures are g : - ~ w i l with u layer thickness thinner t h a n the

(a) F i e l d d

(b) Field>O

/'

Figure 1.5.

(a) A semiconductor quantum well

without an applied electric field bias showing the quantized subbands and the corresponding wave functions. (b) With an applied electric field the tilted potential has the quantized energy levels shifted by the field and the wave functions are skewed from the previous even or odd symmetric wave ft~nctions.

coherent length of the conduction band electrons, quantum size effects occur. These include the quantization of the subband energies with corresponding wave f~inctions(Fig. 1.5a). T h e success in the growth of quantum-well structures makes a study of the introductory quantum physics realizable in these man-made semiconductor materials. Due to the quasi-two-dimensional confinement of electrons and holes in the quantum wells, many electronic and optical properties of these structures differ significantly from those of the bulk materials. Many interesting quantum mechanical phenomena using quantum-well structures and their applications have been predicted and confirmed experimentally [8]. For a simple quantum-well potential, we have the partic- in a box (or well) model. These quantized energy levels appear in the optical absorption and gain spectra with exciting applications to electroabsorption modulators, quantumwell lasers, and photodetectors, because enhanced absorption occurs when the optical energy is close to the difference between the conduction and hole subband levels, as shown in Fig. 1.5a. The density of states in the quasi-twodimensional structure is also different from that of bulk semiconductor. A significant discovery is that room temperature observation of these quantum mechanical phenomena can be observed. 'When an electric field bias is applied through the quantum-well region using a diode structure, the potential profiles are tilted and the positions of the quantized subbands are shifted (Fig. 1.5b). Therefore, the optical absorption spectrum can be changed by an electric field bias. This makes practical the applications of electroabsorption niodulators using these quantum-well structures. Experimental work on l ( ~ wthreshold current quantum-well lasers [9] has been reported for different m:iterial systems, such as GaAs/AlGaAs, InGaAsPI' InP, and InGaAs /' InGr-ASP..kclvat~ tages of the quantum-well lasers, such as a higher temperature stability, an improved linewidth enhancement factor, and wavelength tgnabiljiy, have also been demonstrated. 71kese- devices are based on t h e band structriri: enpineering concept using, fur exarriple,

,

Photon emission

N-A1,Ga I-xAs Figure 1.6. The energy-band diagram of a separated-confintment quantum-well laser structure. The active GaAs layer, which has a dimension around I00 A, provides the carrier confinement and is sandwiched between two AlYGa,-,As layers, where the aluminum mole fraction y is As cladding regions. The AI,Ga, -,As Iayers smaller than those ( x ) of the outermost A1 ,rGa, -, are of t h e order of submicrons (or an optical wavelength) and provide the optical confinement. The mole fraction y can also be graded such that it varies with the position along the crystal growth direction.

-. a separate-confinement heterostructure quantum-well structure to enhance the carrier and the optical confinements (Fig. 1.G). The effect of the uniaxial stress perpendicular to the junction on the threshold current of GaAs double-heterostructure lasers was studied experimentally in the 1970s. The idea of using strained quantum wells [9, 101 by growing senliconductors with different lattices constants for tunable wavelength photodetectors and semiconductors was expIored in the 1980s. Strained-layer quantum-well lasers have been investigat for low threshold current operation, polarization switching, and bistability applications. Important advantages using the strained-layer superlattice or quantum wells include the reduction of the threshold current density due to the raising of the heavy-hole band relative to the light-hole band, the elimination of the intei-vale~lceband absorption, and the reduction of the Auger recombination. Due to the selection. rule for optical transitions, the polarization-dependent gains are also changed by the stress, since the optical gain is mainly T M polarized for the transition between the electron and the light-hole bands and TE polarized for t h e transition between the electron and the heavy-hole bands. Many of these details for valence subband electronic p r o ~ e r t i e sand polarization selection rules in quantum-well dcvices are explained in this book.

1.2

OVERVIEW

This book is divided i ~ ~ five t o parts: I, F;ili:~damentaIs; 11, Propagation; Ill, Generation; IV, hlodulatior,; and V, r):icoti~l?.oF Light. \Ve start with thc fundame t~talso n serniconductol- elcc:ronics, cl~i211turn.mechanics, solid state

Schrijdingcr Equation (Effective Mass Theory) (Electronic band structures, energy levels, wave functions, dielectric function. absorption and gain s p c u a )

f Maxwell's Equations

Semiconductor Electronics Equations

(Optical electric and magnetic fields, guided modes and propagarion constants)

(Carrier densities, potential. bias voltage, quasi-Fermi levels and current densities)

Optoelectronic Device Characteristics (Current vs. voltage curve, rcsponsivity, quantum efficiency, and semiconductor laser output)

Figure 1.7. tics.

'

Fundamental equations and their applications to optoelectror~icdevice characteris-

physics, and electronlagnetics, with the emphasis on their applications to optoelectronic devices. In Fig. 1.7 we illustrate the important fundamental equations and their applicatio~is.In the presence of inje.. .ion of electrons and holes using a voltage bias or an optical source, the semiconductor materials may change their absorptive properties to become gain media due to the carrier population effects. This implies that the optical dielectric function can be chanied. This change can be modeled with the knowledge of the electronic band structures, which require the solutions of the Schrodinger equ,ation or the so-called effective-mass equation for the given bulk or quantum-well semiconductors. Using Mawell's equations, we obtain the optical electric and magnetic fields, assuming that we know the dielectric @nction of the semiconductors. The electronic band structur-e is. aIso dependent on the static electric bias voltage, which determines the electron and hole current densities. The semiconductor electronic equations governing the electron and hole and their corresponding current densities have to be solved. -concentrations Tne device operarion ctlaracteris~ics,such as the current-voltage relation in p-n junction diode structnr~:, the esternal quantum efficiency for the (:onversion of electric to ogtic;ll power i*:l ;! sen-!icc>ncluctorlaser, 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:i<,s. C a h r i c l g e UnivcrsitjPress, Cambridge, 1-71<. !0::9. --?

?

26. W. T. g'sanp, Volume Ed.. Lig17t~vci~'~ C0nzt7zit1zi~atiornT ~ c h ~ ~ oVOI. l o ~22, ~ ,Parts A-E in Ser?li~oizllric.to~* C I ~ I C / S e r r l i n ~ c ~ ~K. l s: ,;1 Willarclson and A. C. Beer, Eds., Academic, New York, 1985. 27. R. Dingle, Applicntioru of ~ t l r r l t i q ~ ~ uWells, ~ l t ~ ~Selcctir:e ~n Doping, arid S~iperlrztctncl Seminletrrls, R. K. Willardson and A . C. Beer, tices, Vol. 24 i n Semicon~h.lcforEds., Academic, Now York, 1985. 28. J. Wilson and J. F. R. Hawkes, Optoelectronics: Atz Intr.ocli.iction, Prentice Hall, Englewood Cliffs, NJ, 1983. 29. R. G. Hunsperger, Integrated Opfics: 77reor-y a n d Technology, Springer, Berlin, 1984. 30. K. J. Ebeling, Integmltxi Optoelectrotlics, Springer, Berlin, 1993. 31. K. Chnng, Ed., Ha~zdbookof II.Iicrowave arid Optical Coinponenfs, Vols. 3 and 4, Wiley, N e w York, 1991.

Semiconductor Lasers 32. G . -H. R. Thompson, Physics of Setrzicondz~ctorI,u.ser Der!ices. Wiley, New York, 1980. Lnsers. Parts A and B, 33. H. C. Casey, Jr., and M. B. Panish, Hete?~osrri~ct~ir-t. Academic, Orlando, FL, 1978. 34. G. P. Agrawal and N. K. Dutta, Long-W~~celettgtlzSe?nicorrdrictor Lasers, Van Nostrand Reinhold, New York, 1956. 35. P. S. Zory, Jr., Ed., Qlicznt~lmPVefl Lnsers, Academic, San Diego, 1993. 36. G. A. Evans and J. M. Hammer, Eds., Surfi~ceEmitting Semicorldlictor Lrisers and Arrays, Academic, San Diego, 1993. 37. R. K. Williardson and A. C. Beer, Eds., Lasers, Jilnctiorzs, Tr~~t~spot.t, Vol. 14 in Senzico~zdrtctorsarzd Srrrrirnetals, Academic, New York, 1979. 35. W. W. Chow, S. LL7. Koch, and M. Sargent 111, Sernicond~rctor-LnswPhysics, Springer, Berlin, 1994. 39. J. K. Butler, Ed., erni icon duct or It~jectionLasers, I E E E Press, New York, 1980. 40. J. J. Coleman, Ed., Selected Pnpcrs or7 Sernicorzdrictor Diode Lcrsers, SPEE Milestone Series, Vol. MS50. SYIE Optical Engineering Press, Bellingham, WA, 1992.

Optical Waveguides and Nlodulators 41. H. Nishihara, M. Haruna, and T. Suhara, Optical Irrtegrtr[cd C'ircriits, McGrslw-Hill. New York. 1989. 43. T. Tamir, I~~re,yr.ated 0-ptics, Springer, Bcr!in, 1979. 43. T. T a ~ n i r .Griirlu~I bP'c':r~.e~?prc~t~ft~crrorlic.r, 2d ecl., Springer, Berlin, 1990. 44. D. Marcuse, T/rrory off?ie!rct:-ic Opiict~l!fi;:~'eg~~icLe.s, A c a c I ~ r n i ~NCW , York, 1974. 45. A. W. Snyder a n d J. D. LZ~:)ve, Oi,~itc~;/ !i,'irr.eg~:i:ie Tlleo~y,Chapman LC Mall, London, 1953. 46. /I-'\. B.'Buckm:\ri, G:/;llc~i-i-l.321!,,; ?!cv/vriic',~. S:tiln:!er:; College, Ncw Yurk, 1992. 47. A. Y;\ri*f ar~clP. E'c.11, il;)fi;.[c!l,ii.l~,cs. i;? C;.\,.:.l;~!.r., LVilcy. NCLV York, 1984.

45. J. I>. Vincent, F~~rtdcrnrenta1.s qJ' Irzfr(rre~% Ll~fectorr)I>c>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<:TORR ELECTL ONlCS

22

2.1.1

PJlam~ell'sEgnatinns in MKS Units

VxE=--1B

Faraday's law

dt

V x H = J + -

i)D

Amp?re7slaw

(2.1.2)

V-D=p

Gauss's law

(2.1 -3)

V-B=O

Gauss's law

(2.1.4)

at

where E is the electric field (V/m), H is the magnetic field ( A / m ) , D is the electric displacement flux density ( c / m 2 ) , and B is the magnetic flux density ( v - s / m b r webers/m2). T h e two source terms, the charge density p (C/m3) and the current density J ( A / ~ ~ )are , related by the continuity equation

where no net generation or recombination of elect]-ons is assumed. In the study of electromagnetics, one usually assumes that the source terms p and J are given quantities. It is noted that (2.1.4) is derivable from (2.1.1) by taking the divergence of (2.1.1) and noting that V (V X E) = 0 for any vector E. Similarly, (2.1.3) is derivable from (2.1.2) using (2.1.5). Thus, we have only two independent vector equations, (2.1.1) and (2.1.2), or six sca1;-!r equations, since each vector has three components. However, there are E, H, D, and B, 12 scalar unknown components; thus we need six more scalar equations. These are the so-called constitutive relations which describe the properties of a rrledium. In isotropic media, they are given by D

=

EE

B

=

pH

In anisotropic media, they may be given by -

D = ~ - E B = P - H where

is the permittivity tensor and

For electromagnetic fields

:it

is the permeability tensor:

optical frscluencies, p

5

0 and J

=

0.

2.2.2 Boundary Conditions By applying the first two Maxtvell's equations over a sinall rectangular surface with a width 6 (dashed line in Fig. 2.la) across the interface of a boundary and using Stokes7 theorem,

the following boundary conditions can be derived by letting the width 6 approach zero [I]:

ii

X

( E l - E,)

fi x ( H ,

- H,)

=

O

=

J,

(2.1.10) -.\

(2.1.11)

,

where J, ( = limJ,,, a ,J6) is the surface current density (A/m). Note that the unit normal vector fi points from medium 2 to medium 1. Similarly, if we apply Gauss's 1:lws (2.1.3) and (2.1.4) and integrate over a small volume (Fig. 2.1b) with a thickness 6 and let 6 approach zero, e.g.,

we obtain the following boundary conditions: .-

i ' ( D , - D,) = p , ii' ( B , - Bz) = 0

,,

where p, ( = lim ,,,, ,p a ) is the surface charge density ( c / m 2 ) . For an interface across, two dielectric media, where no surface current or charge

:ii.i t.hr: intc:rl;,c~ ~ t tkvu ' rnedi:): Figure 2.1. Geometr?, f't~rddrr.i.;Irlg th:? i ) i ; i i r ~ , J i ; : - y ~ ~ , ! ~ c ~ ! ~ ! c;rrl.oS:; (a) a rectartgulnr surface: is ct?;;losecl by tllz iiiilll.)t:r (.' [cl:~shztl line); a n d ( h ) a small volun-rc with iI thickness 6.

density can be supported, J,

=

0 and p,

=

0, we have

For an interface between a dielectric medium and a perfect conductor,

since the fields E,, H , ? D,, and B, inside the perfect conductor vanish. The surface charge density and the current density are supported by the perfect conductor surface. 2.1.3

Quasi-electrostatic Fields

-

For devices with a dc or low-frequency bias, since the time variation is very 0), we usually have slow ( d / d t

..

and H = 0, B r 0 for the electronic devices for which no external magnetic fields are appIied. En this case, the solution of the electric field can be put in the form of the gradient of a n electrostatic potential (25:

and

in an isotropic medium. Ecluatio~l(2.1.19) is Poisson's equation. When the frequency becomes higher, f o r examplc, in a microwave transistor, one may includc the displacenlcnt c u r~e n t delis~ty c ~ ( ~ E ) / din t the total current c~!rrt'rltd e n ~ i t yJ,,,,: density in addition to the cancl~tcl~or~

.-.

2.2

SEMICONDUCTOR ELECTRONICS EQU,!YI'%ONS

In this section, we present the basic senlicorlductor electronics ecluations, which are very useful in the modeling of semiconductor devices [4-71. These erluations are actualIy based o n the Maxwell's equations and the charge continuity equations.

2.2.1

Poisson's Equation

As shown in the previous section, Poisson's equation in the semiconductor is given b y (2.1.19):

where 4 is the electrostatic potential, and p is the charge density given by p =q(p - n

+ C,)

c,=N,+-N, C is the magnitude of a unit charge, p is the hole Here q = 1.6 x concentration, n is the electron concentration, N,f is the ionized donor concentration, and Ni is the ionized acceptor concentration.

2.2.2 Continuity Equation Since Ampkre's. law gives

. where the conduction current density is

and J, and J, are the hole and electron current densities, respectively, we have

Assuming that C,, =

N;- Ni is independent of time, we have a

+ q -d(l y

V - (J, + J,,)

- n) =

0

(2.2.8)

Thus, we may separate the above equation into two parts for electrons and holes:

V - J,,

-

a dr

q-n

+qK

=

(2.2.9)

where R is the net recombination rate ( c m V 3 s - 9 of electron-hole pairs. Sometimes it is convenient to write the generation rates (G, and GI,) and recombination rates ( R , and X , , ) explicitly:

R

=

(2.2.11)

R,, - GI,

for electrons and

for holes. Thus, w e have the current continuity equations for the carriers: 1

an -

dt

2.2.3

=

G,

--

R,,

+ -rlC

J,,

Carrier Transport Equations

T h e carrier transport equations (assuming Boltzmann digtrib~rtionsfor carriers) c a n b e written as

where E = -V$ is the electric field, p , and p, a r e the electron and hole nlobiIity, and D,! a n d U,,a r e [he electron and .hole diffusion coefficient, respectively. We may express the electric field i n terms of the electrostatic potential in the carrier trilrlspc)ri equation. We then have J,),J,,_6 , p , n , o r nine scalar components as unltnotvns. We also have (2.2.1), (2.2.13), (2.2.14]), (2.2.15), and (2.2.16);o r nine scalar eclu~tticjns.W e may also eliminate some af the ~ ~ n k n o wfui:ctions n YLI.L'[.~ ;!s J,, ai;(l J,, and reduce the number of

equations to three:

JP 3t

-;=:

4 V

1

- .Rp -- - V

4

-

. [-qp,pV+

( F V C ~=) - - q ( p - n

-~D~VJI]

+ C,)

(2.2..1.8) (2.2.19)

with three unknowns n , p , and 4. In principle, these three unknowns can be solved using the above three equations once we specify the bounda.ry conditions for a given device geometIy. 2.2.4

Auxiliary Relations

Often it is convenient to introduce two auxiliary relations with cwu more functions, F,(r) and F,(r), the quasi-Ferrni levels for the electrons and holes, respectively:

n(r)

= rl,

exp

(2.2.20 j

p(r)

= rl,

exp

(2.2.2 1j

where the intrinsic carrier concentration n i depends on the band edge concentration parameters N, and N , , the band gap, and the temperature

and the intrinsic energy level is

E , ( r ) . = -qsi,jr) -t- E,.

(2.2.23)

Mere E,. is a reference constant energ!,. We may substitute (2.2.20) and (2.2.21) into (2.2.17)-(2.2.19). and use only three unknowns, +(r): +,,(r), and +,(r?, which have the same dimensions ( V ) :

..,

t ~ v oauxiliary relations a r e t'irr- !:onclegzneratc sernico.rlr:luctors, for wtlich the hlnswell-Bol tznzann stilt is ti.t.s ;:re applicable. To Eaki" into a s c w n t the effect of degeneracy, one m a y rncjilify (2.2.20j and (2.221) simply by using the 1 he

Fermi--Dil-ac statistics together with the elcc tron a n d hole del~~ity-of-state frlnctions p,(E) and p,,(E):

and

where

is theJermi-Dirac

distribution for electrons and

is the Fermi-Dirac distribution for the holes. Density of States. 'The density of states for electrons, P,(E), is derived as follows. The number of electrons per unit volume is given by

where the factor of 2 takes care of both spins of the electrons. For the electron states above the conduction band, we may assume that the electrons are in a box with a volume L , L L L - with wave numbers satisfying 27~ 27 k .t =nz-, k,=nP, L x

L,,

23k , = e,

L-

nz, 1 1 ,

e

-

integers

Thus, the number. of available states in a small cubc d k , d k,, d k the k space is d3k divided by the a m o ~ l ~ l t

=

d 3 k in

for each state

c c c /-----d3k ; ;

I;,

A,, k ,

.

(2r)3/~

Thus 2

v

d3k k,

47i-k d k

4Ti3

k , X-:

===

1-k 2 d k 2

(2.2.33)

Ti-

If the parabolic band nlodel is used,

where E, is the conduction band edge, we obtain ,k ' d k

,=a

where p , ( E ) is called the density of states for the electrons in the conduction band 3/ 2

(E

-

E.)

'

for E > E,

(2.2.36)

and p , ( E ) is zero if E < E,. A similar expression holds for the density of states of the holes in the valence band, 3 / '7 \

( E L - E ) I/'

for E < E,.

(2.2.37)

and p,,(E) is zero for E > E l , , where E,. is the valence-band edge. In the nondegenerate limit, it can be sl~ownthat (2.2.26) and (2.2.27) reduce to (2.2.20) and (2.2.21). Using the Fermi-lt3irac: intr:gi:.il tdelinecl I 3 - j

31'

];;JS1<.'

SEhllL!:,TY)REL15CTROKICS

C i a m n ~ afunction, we m a y rewrite (2.2.26) and (2.2.27) as

where 1' is

and

where

and r ( 3 / 2 )

=

&/ 2 has been

used.

Approximate Formula for the Fermi-Dirac Integral. An approximate analytic form for the Fermi-Dirac integral f ; ( ~ valid ) for - m < 7 < is given by [5, 8-10]

where, for j

=

f , one

uses either [ S ]

Figure 2.2. A plot of the Fermi-Dirac integral F,,2(77) ,q < - I and 77 x- 1.

VS.

7 and its asymptotic limits for

The ~naximumerrors of (2.2.43) are only 0.5 percent. We also see that

These asymptotic limits are shown as dashed lines and comparecl with the exact numerical integration in Fig. 2.2. Determination of the Fermi Level E F . For a bulk seniicond~~ctor under thermal equilibrium, the Fermi level EF ( = F, = F,) is determined by the cllarge neutrality condition:

N;^

where is the ionized acceptor concentration and N,f donor concentration. W-e have

,

is the ionized

where g is the ground-state dcl.r,!,::?cr:roy factor for acceptor levels. g . , equals 4 because in Si, G e , rind Cl;ai';> ;:..ii:h acceptor I~cvclcar1 accept or:e hole c>E either spin and tile ;lcceptur iavc:! i:; d t ~ t ~ h ciegen.=s;lt.e ly as a res~tltol the two

ciegcnerate valence bands nf k

=

0,and

-

where gD is the ground state degeneracy of the donor impurity level and equals 2. For example, assuming that NA 0 and N, = 5 x 101'/cm3 for a GaAs sample at 300 K, Fig. 2 3 , we can plot n , p, and N ' from (2.2.39), (2.2.40) and (2.2.47) vs. energy to determine the intersection point between the curve for NL+ p vs. energy and the curve of n vs. energy [4]. T h e horizontal reading of the intersection point gives the Ferrni energy E,.. In Fig. 2.3b7 we repeat the same procedure for a larger donor concentration N, = 5 x 10's/col'. We can see that the Fermi level EF moves closer to the conduction band edge E,. Notice that at thermal equilibrium, F Z = ,~ n:.-~or extrinsic semiconductors, we have two cases: 1. n-type, N,+- N;>> n,, therefore, no =

NL- Ni and

p,

1

=

N;

-

Ni

2. p-type, N,- - N,i >> n , , therefore, pO = Ni- NL and n o

1.1 =

N;

f

-

N,+

(2.2.45b)

So far, we assume that p is given by the heavy holes only, since the density of states of the heavy holes is usually much larger than that of the light holes. If we take into account the contribution due to light holes, the total hole concentration should be

where

Both terms l ~ a v sto 1-x ~ ~ s e ci ni clc.!crn:~rting

charge ~l~uti.aliL!/ condi tivn.

Lit.1~

Ferrni level E, using the

,

(a)

Energy (eV)

T

T

'

.

r

'

r

.

'

I

.

.

'

.

I

'

.

.

'

L

'

'

-

4

~

G a A s (300 K) 1020

-

?--4, b

7

- - +-.--.-----.--.-\

10'5

- ND

101"

-

..

+

.

I

Ev 105 -0.5 c

n

n

.

!

-

0.0

0.5

1.5

T .O

Energy (eV)

2.0

..

(a) A graphir:;it approach to determine t h e ~ e r r n iIcvel EF from rhr charge neutrality condition 1.1 = N,T+ p. Both n and N2-t p vs. t h e horizontal variable E F are plotted and the intersection of the two curves give the Fcrlni level of the system N, = 5 X ~ ~ ' ~ / c (b) rn~. Same as (a) except that ND = 5 X l ~ ' ' / c r n ~ . Figure 2.3.

An inversion formula is sometimes convenient if we know the carrier concen (ration r z . Define

The maximum error [1.0, 111 of the above furmilla is 0.006. Further improvement b y two orders of rnagnitctde is possible [12]. A n alternative approach for taking into account the degenerate semiconductor is to replace n i in (2.2.20) and (2.2.21) b y an effective intrinsic carrier concentration n,, such that

where AE takes into consideration the effective band-gap narrowing, which has to be taken from electrical measurements. 2.2.5

Boundary Conditions [61

O n the surface of ideal ohmic contacts

The quasi-Ferrni potentials satisfy

at the ohmic contacts. Here 4,, n,, and y , are the values of the corresponding variables for space-charge neutrality and at equilibrium. On an interface with surface recombination, such as that along a Si-SiO, interface in a MOSFET, one may have I*

where R , is the surface recombination rate. When interface charges p, exist such as those of the effective oxide charges on the Si-SiO, interface, one has

where B points from region 2 to region 1. On a semiconductor-insu~i~tor interface xhere no surface I-ecombination exists, one has

-

2.1 x l o 6 the ii~trinsicc:iirie; ;oncentration,! we have the electron concentration

N,

B ?ti

and '1

i

= - =

4.41 x

cn~-*<

rr,,

Nrl

We can also calculate the band-edge concentratio11 parameters using 0.0665 m, and tnz = 0.50 rrzO and T = 300 K.:

a;, = 2-51x

10"

crn

r r z , 300

HZ: =

-"

'The Fermi level can be obtained from the inversion formula (2.2.51 using

and we find

-

That is the Fermi level is 35.7 rxeV below the cond~ictionband edgc. If we use 77 = In rl - 1.458, we obtain El.. EL.= - 37.7 rneV, The error is only 3 meV.

IE this section, we describe ;i pkic~~:.>fi><:i<~~(.>>::;<:,i: iipprt.iach tc:~ tllc geni;'r;;tion and recombination pi.c?cesr-;es if-i ; ~ . * ~ ~ : i ~ i : ) i , i : : I iA ~ < quantui-n t~r~. rrlei:l-i;~~ic;i! approach can also be taF;et; :~sir;g I?:.:; E~ZIC-:!.:~?. . L I < . ; ~ ~ ? C:~~:rt~irb::ii.o~i t:l-ie~~.p,, .

1

BASIC SEMICONDUCTOR ELECTRONICS

36

Fermi's golden rule. The latter approach will be discussed when we study the optical absorption or gain in semiconductors in Chapter 9. In general, these generation-recombination processes can be classified as radiative or nonradiative. The radiative transitions involve the creation or annihilation of photons. The nonradiative transitions do not involve photons; they may involve the interaction with phonons or the exchange of energy and momentum with another electron or hole. The fundamental mechanisms can all be described using Fermi's golden rule with the energy and momentum conservation satisfied by these processes. The processes may also be discussed in terms of band-to-bound state transitions and band-to-band transitions. 2.3.1

Intrinsic Band-to-Band Generation-Recombination Processes

For band-to-band transitions [13-15) as shown in Fig. 2.4, the recombination rate of electrons and holes should be proportional to the product of the electron and hole concentrations (2.3.1) and the generation rate may be written as

Gn

=

Gp = e

(2.3.2)

where c is a capture coefficient and e is an emission rate. The net recombination rate is R=R n -Gn =R p -Gp =cn'P-e

(2.3.3)

If there is no external perturbation such as electric or optical injection of carriers, the net recombination rate should vanish at thermal equilibrium:

o=


(h l Recombination of electron-hole pair

Rn;Rp;cnp

Gn=Gp=e

iJ!ure 2.4.

(2.3.4)

cno'Po - e

Ec---'f---

Ec --+----

E v ---+--

Ev

--*---

The energy band diagram s for (a) generation and (b ) recombination of an

ketron-hole p.<:Iir.

where r z , and p,, are the clcctror. and hcle concentrations at thermal equilibrium ,,lop,, = nj. The n e t rccumh!naricn rate can be written as

If the carrier concentrations rz and p deviate from their thermal equilibrium values by Srz and Sp, respectively,

we find that the recombination, rate under the condition of low-level injection, 6n, 6p ( n o + p,), is given by

where S n = b p , since electrons and holes are created in pairs for interbnnd transitions, and the lifetime

has been used. 2.3.2 Extrinsic Shoclkley-Read-Hall 6 R . H ) Generation-Recombinat:on Processes 116-191 . c r-

[here are basicallyLfour processes; Ei8, 191 as shown in Fig. 2.5. These processes'may all be caused by the absorption or emission of phonons. 1 . Electron ccrpttlre. The reqombination rare far the electrons is proportional to the density of el'ectrons t i , and the concentration of the traps

(a) E]ecwo;l captilrc ).

K,,=cnnPI,(1 -ft)

.

.,. -..,..,>;:

-A-. -1,. Z,CC ....1 311 , II?;:,.,

,

4

!id) Hole emission R : ~ = c , ~ N ~ ~G,=e ~p Nt(1-ft)

,(: i

cJI:=-i;2.. p,: gf:

,-

:

I

.,

,.\

Figllrc 2.5. 'T'lle erlcrgy l;~;!iliJ tli;lgt,:~~~::, G ;: ~k:: .>!-::.l:!.:i(.: processes: (:I) crlzl;tron ~ a p t ~ l : e( ,f j ) i ! ; ' ~ : ~ i : iii:.'l\..i';i;, ~ t

.

- i..t ..I,!--1F:ill a',, ' :i i,::;lt'

<e:lc!-ation-recomhinatiiln i:&lpt?irz,arlii (d) hole ernisl;iuil.

N,,rn~lltipliedby the probability that the trap is empty ( 1 f, is the occupation probability of the trap, R,,

=

- f;),

where

(2.3.9)

c l l n W ( 1 - f;)

where c,, is the capture coefficie~ltfor the electrons. 2. Electl-orz en7ission. T h e generation rate of the electrons due to this process is

-

where e,, is the emission coefficient, and Nff, n , is the density of the traps that are occupied by the electrons. 3. Hole cnptr~re. T h e recombination rate of the holes is given by the capture of holes by occupied traps; the number is given by N,f,:

4 . Hole enzissiorz. T h e generation rate for this process is proportional to the density of the traps that are empty (i.e., occupied by holes):

where e, is the emission coefficient for the holes. Based on the principle of detail balancing, there should be zero net generation-recombination of electrons and holes, respectively, at thermal equiiibrium. We have

R,,

-

GI, = crltzoN,(l- f r o )

-

e n N l f r , , =0

(2.3.13)

where the subscripi 0 denotes the thermal equilibiiurn values. Thus, we obtain a relation between c,, and el,, c , and e,.

and 0 , f r a m the t~boveequations, and = . The r c ! t i l , ~may be replaced by an 1 , p = ~cmiconductors. effective concdntr.ation i : , ; , fa!- Ll~ge."ncr:ite

where f o r convenience -;~,eCjctillc

The net recornbination rates for electron and lmles are

I

We note that a combination of electron and hole captures (processes 1 and 3) destroys an electron-hole pair, while a combination of electron and hole emissions (processes 2 axid 4) creates an electron hole pair. At equilibrium, these two rates are balanced; the fraction of occupied traps f, is then given by

The net recombination rate is

where

which are the hole and electrori lifetimes, respectively. We note that the above capture and emission processes can be due to optical illumination at the proper photon energy in addition to thermal processes. Under the low-level injection condition, the net recornbination rate (2.3.21) can be written in the same. form as -(2.3.7):

:!ASIC 5EMIC:ONDUCTOR ELECTRONICS

41)

2.3.3

Auger Generation -Recornbina ti011 Processes

For band-to-bound state transitions, the following processes [IS, 191 are possible (Fig. 2.6). 1. Electron captures with the released energy taken by an eIectron or a hole. T h e recombination rate is given by

R,

=

(c,?n + c , P p ) n N , ( l - f,)

(2.3.25)

where we note that the capture coefficient c,, in SRH recombinations has been replaced by cnn + c f p , since the bvo possible processes for electron captures are proportional to the concentration of the electron or hole that gains the energy released S y the electron capture. 2. Electron emissiorls with the required energy supplied by an energetic electron or hole:

(a) Electron captures

(c) Hole captures ~~=(c;n+c;p)p~~f~

A

(b) Electron emissions

(d) HoIe emissions

~ ~ = ( e ~ n i - e ~ p-f,) )N~(l

t

-1'hc energy b:~liil cli~~?r':ti??~lo: t h z 5;1rlci-t:)-bouncI stntt: A u g e ~generation/ recombin;ltion processes: (2,) ;[rctrc,rl c;!pr?lrr, (1)) slt.c:rOn enlission, (c) hole capture, and (d) hole emission.

Figt.rre 2.6.

3. Hole captur-es with the releasecl energy talcen by an electron or a hole:

4. Hole emissions with the required energy supplied by a n energetic electron or hole:

G,

=

(ein

+ e;p)N,(l

-

f,)

( 2 -3.28)

If all these processes of the impurity-assisted transitions due to thermal, optical, and Auger-impact ionization mechanisms exist, we use

The net recombination rate is given again by (2.3.20)

For band-to-band Auger-impact ionization processes, we have the four possible processes (Fig. 2.7):

1. Electron capture. An electron in the conduction band recombines with a hole in the valence band and releases its energy to a nearby electron. (a) Electron capture (b) Elecuon emission (c) Hole capture (d) Hole emission

Figure 2.7. Tht. e l l r r g y h;!rld c l i ; ~ ~ i a i ; : ri,.i;. ; tll;: :>>~ci-io-band tlugcl. grnri-~ttion/rccoml;ination . . processes: (3)~ l e ~ t c:ip[ure, r ~ n I'hj ~ i r : < : : . i ~ i - ~ c'r:iiSric>n, j c i !~..lirc:;~f~turq.:,and i d ) I ~ o i semission.

,- 7

I his process destroys an electron-hvic pair. T h e reconlbination rate is

2. Elect~.o/z emission. An electron in the valence band jumps to the conduction band because of the impact ionization of an incident energetic electron in the conduction band, which breaks a bond. This process creates an electron-hole pair. The generation rate is G,, = e nA U n

(2.3.35)

3. Hole capture. An electron in the conduction band recombines with a hole in the vaIence band with the released energy taken up by a nearby hole. This process destroys an electron-hole pair. T h e recombination rate is

4. Hole enzissiorz. An electron in the valence band jumps to the conduction band (or the breaking of a bond to create an electron-hoIe pair) due to the impact of an energetic hole in the valence band. T h e generation rate is

At thermal equilibrium, no net generation/reconlbination exists. Thus process 1 and its reverse, process 2, baIance each other, as do processes 3 and ,4. . Therefore,

Similarly,

.,ALr

;\U

= c,,

'

rl;

(2.3.30)

T h e t o t a l net Auger recombir~ationrate is the sum o f the net rate5 of electrons and holes, s i i l c ~ each procchs creates or destroys one 2, electron-hole pair, Thcrct'i)~

2.4

2.3.4

EXAMPLES AND i9P PL.JI-:!;I'iOF!S

'i"3 TIP-t ~.-~~L.Ec'I'RC:NIC: DEW(-:ES

.

43

Impart Ionization Generation-Reco~al~inationProcess [61

?'his process is very much liltc the reverse Auger processes discussed above. However, the hot electron impact ionization processes usually depend on the incident current densities instead of the carrier concentrations. Microscopically, the processes are identical to the Auger-generation processes 2 and 4, which create an electron-hole pair due to an incident energetic electron or hole. These rates are usually given by

and

.

I

where a, and 3 ,, are the ionization coefficients for electrons and holes, respectively. a, is the number of electroll-hole pairs generated per unit distance due to an incident electron. j3, is the number of electron-hole pairs created per unit distance due to an incident hole. The total net recombination rate is

Usually the ionization coefficients are related to the electric field E in t . ! ~ ionization region b y the empirical fbrmuIas

where E,,, and E,, are the critical fields, and usually 1 5 y 5 2. These impact iorlization coefficients are used in the study of avalanche photodiodes in Chapter 14.

I!I this section, wc ~ ~ ) n . ~ j i;j d c.~c;K.. r ~ i ~ n p e.;iarnplcs fc to illustrate t.he optical ge neration-recombin;:~tifin pl-~+:;~r::;;,s;.-: ;ii-~dtl-i;:Ir cfifcctl; cr; photoc!etectors. :.ire ~Si:,ci:ss!;.ct ie Chapter :L4. More details about ph~todeiec!:{cjri:

Optical jnjectio~l

-1-lI Figure

homogeneous semiconductor under a uniform optical illumination from the side.

In the presence of optical injection, the continuity equations are

a r.1 -

; Ir

=

+ .

G,, - R,,

1 3 --J,,(x)

ax-

If a semiconductor is doped p-type with a n acceptor concentration iV,, we know that pO = N 4 and

2.4.1. :

nf no -- -

%

Uniform Optical Injection

Suppose the optical generation rate is uniform across the semiconductor,

which is independent of the position (Fig. 2.8) and the recombination rate is R,, = 617/7,,. We then have

T h e new carrier densitic.:; nre

where the t:xces;s c2r:isr zonc.entrations: sr-i.; ;.(:!-la1 8 p = 611 lor the interband uptic,aI generi~tionprocess, S ~ ~ I L ' wz C . a;s\iii12 !hat electrons and 11oles arc

created in pairs. Furthermore, for a constant light intensity,

we find the steady-state solution

Therefore, the amount of the excess carrier concentration is simply the optical generation rate G, multiplied by the carrier lifetime. In the presence of an electric field E along the x direction, the conductio~lcurrent density is

and the total current density is J = J,,

+ Jp = a.E = u-ve

with the conductivity

where Y is the applied voltage, and !is the length of the semiconductor. Therefore, at thermal equilibrium when there is no optical illumination,

is the dark conductivity and .*

ACT =

q ( ~ , , S n+ P , ~ P )

is called the photodonductivity. We find

- cr(wn + w,)Go~,? which is proportionaI to the generation rate and the carrier lifetime. The photocurrent is

and A is the cross-sectional area of the sernicond-crctor-. For most seioicondtlctors, 'the electrorl mobility is z-nxcl? !;irger t h a 2 tile hi>!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.,

!<.

-

F,\S'C SEMlCOr~DUC'TOriELECTRONICS

52

Here V, is the total potential drop across the junction, whereas Vo, is the portion of the voltage drop on the p side and VoN is the portion of the voltage drop on the N side. The contact potential is evaluated using the bulk values of the Fermi levels F, and FN measured from the valence or conduction band edges E,, and E c N , respectively, before contact (see Fig. 2.10a):

and

BE, = 0.67(EG,

-

for the G ~ A S - A , ~. G vA ~ s, system (2.5.14)

E,,)

-.

Before contact

level

< .

(b) After contact

Figure 2.113. contact.

Energy h a n d diugrafll

lLjr

il

IJ-~V l i ~ ' t r ' [ ' ~ j ! ~ l l i t i (il) o i ~ bef(31.e conlact: and ( b ) after

2.5

ST'M!COI'IDUC'T'OR

r - A ' AND

HF,~'E~P.OJ~~NCTTCN>;

il-:"

For nondegenerate semiconductors,

Fp - E,, = - k g T ln

;i,) -

-

-

where NcN and Nu, are evaluated from (2.2.41) and (2.2.42) for the N and p regions, respectively. Since the electron concentration on the N side, N ND, and the hole concentration on the p side, p N,, the contact potential Vo can be evaluated from the above equations when the doping concentrations Nu and ND are known. Using the two conditions (2.5.9) and (2.5.12) and the total width of the depletion region x , from

we find immediately that

-

Thus we may relate x , to Vo directly:

The band edge E,(x) from the p side to the N side is given by (choosing E L . ( - m ) = 0 as the reference potential ene?gy) E,,(x)

x

AE,. -

=

i

-4#(x) -AE,,+q+(x)

+ X,) ' N ~x:,

2;

p side

Nside

-..

(a) A p-N junction geometry

(b) Charge distribution

4 P(x)

(c) Electric field

t

.(XI

(d) Electrostatic potential

f

Figure 2.11. Illustrations of (a) a p-N junction geometry, (b) the charge distribution, ( c ) t h e e l e c ~ r i cfield, and (d) t h e electrostatic potential based on the depletion approximation.

V" -x P

--

-'XN

=-x

The conduction band edge E , ( x ) is above E,(.Y)by a n amount E S Pon the p side and by an amount EGN on the N side. E,.(.r) is always parallel to E , . ( x ) :

Illustrations for the charge density, the electric field, and the electrostatic potential are shown i n Fig. 2. Ill. T h e final energy bapd diagram after contact is pIotted in Fig. 2.10(b).

,

-

U.3) het.erojtinction is formed at Example A p - G a A s / M - N , , G a -.i - 4 s C-v thermal equilibrium witho:.lt a n exterr~al bia:; at room temperature. T h e doping concentration is Id, = 1 x 10'' c r ? ~ in- ~tile p side ant1 N, = 2 x 10''

~ r n in - ~the N side. Assume that the density-of-states hole effective mass for A1,vGa,-,As is

which accounts for both the heavy-hole and light-hole density of states. Other parameters are

where x is the mole fraction of aluminum. (a) We obtain for x = 0.3

The band-edge discontinuities are

AE,

=

1.247 x 0.3

=

0.374 eV

AEc=0.67AE,=250.6meV (b)

=

374 meV AE,=0.33AEg=123.4meV

We calculate the quasi-Fermi levels F, and FN for the bulk semiconductors for the given N, and N,, separately. p-GaAs region

(c)

The contact potential is calculated using (2.5.13)

(dl

The depletion widths are

The energy band cliagranl is plotted in Fig. 2.12.

1

Biased p-N Junction. With an applied voltage V, across the diode (Fig. 2.13a), the potential barrier is reduced by V,, if V4 is positive. Note that our convention for the polarity of the bias voltage V., is that the positive electrode is connected to the p side of the diode. ,Thus the depletion width x,,,is reduced. Explicitly,

-0.06

-0.03

0.00

0.03

0.06

0.09

0.12

Position x (pm) Figure 2.12. Band diagram of a p - G a A s / N - A l o . 3 G a o . , A s heterojunction with N,, in the p region and N, = 2 x lo1' ~ r n in - ~the N region.

=

1 x 10Is

The potential drop on the p side of the depletion region is +(O) shown in Fig. 2.13b

which is reduced from V,, by an amount Vp. Similarly, the voltage drop across the N side of the depletion region is

Figure 2.13. ( a ) A p-N jl.ifictioi: wit11 a hiit>. I.:, a n d (h) t l i l correspoilrJing elcctr.i.)static potcntiril (,h(xi (solid curve for V4 3 0 ;and dnsheci c:r tq;e f o r I.'I ===0 ) .

which is reduced from V , , by , a n arnounl bc equal to the bias voltage V,:

VN.T h e sum of

y, and

J',, has to

Again the charge neutrality condition

and

together with

lead to

Thus the depletion region is reduced since s,,is proportional to (V, - v,)''~. If the diode is reverse-biased, then V, is negative and x,, becomes wider. The potential function +(x) in (2.5.22)determines the band bendings E , ( x ) and E,.(x)again through (2.5.19)and (2.5.21). T h e resultant band diagrams for a forward-biased and a reverse-biased diode are shown in Figs. 2.14a and b, respectively. Quasi-Fermi Levels and Minority Carries Iaections. T h e quasi-Fermi levels Fp(x) and F,(X) are separated b y the bias voltage V , multiplied by the magnitude of the electron charge (1 = 1.6 X 10-'I' C .

his fact can be seen clearly by comparing the band diagrams of the zero bias and forward bias cases in Figs. 2.10b and 2.14a, respectively. T o find the variations of the quasi-Fcrini levels 6,t.r) and Fv:,(x)as functions of the position, we have to find the carrier conce~trationseverywhere in the diode. Ass~lminga low-level injection conclition, which means that the amount of injected excess carriers d ~ l et o the voltaye bias is always much less than the majority carrier concentratic.~nin each regiori, the minority carrier concentration is much more strongly affectecl than the niajority carrier concentration in each region. As a icsult of this, there are two different quasi-Fermi levels, F i d F[](.rj, f o i the elecrrorl and the hole concentrations, respectively,

(a) Forward bias

(b) Reverse bias

Figure 2.14. Energy band diagrams for a p-N heterojunction under (a) fonvard'bias, and (b) reverse bias based on the clepletion 'approximation.

near the j~lnctionregion. Far away from the junction region, the quasi-Fermi level should approach its thermal equilibrium value in the bulk region, respectively, that is, FN(x -+ +m) --+ FN (bulk value on the N side) and F,(x + -a) + Fp (bulk value on the p side), since the ,carrier concentrations are hardly affected away from the junction. It is usually assumed that the quasi-Fermi Ievels stay ,as constants across the depletion region, that is, F,(x) = FNfor -x, 5 x < :+m and F,(x) = F, for -a < s < x N , as shown in Fig. 2.14. This is equivalent to the statement that the carrier distribution in eilergy lor the same spe'cies of carrier across the depletion region stays the sanle in the depletic~nregion. As, a result of this assumption, we find that, for t h e BoItznla~lr!distribution, on the p side

i]i?..Sii ST7.i\djc~i.~TjiJi;Ti?R ELECTRONICS

60

On the

1V

side

Since at thermal equilibrium (i.e., no current injection, V, FN(x), we have

=

0), F p ( x ) =

where n i p is the intrinsic carrier concentration on the p side. Thus, if V, # 0, for -x, < n < 0,

At the edge of the depletion region, x fore,

=

- x p , p,( -x,)

= ppo =

N',. There-

Here the subscript 0 in p,, and n,, refers to their thermal equilibrium values. T h e minority carrier near the edge of the depletion region, n , ( - x , ) , differs by a factor exp(iiV' / k B T ) from its thermal equilibrium value n,, due to the carrier injection. Similarly, at the edge of the depletjon region on the N side, x = x,, the minority carrier concentration is

where PNO= rI:N/Nl) and n,, is the intrinsic carrier concentration on the N side. If we compare the current density equations for the p side of the diode, assuming that rz,, and p,, are independent of x (dn/dx = d(Sn)/dx, ap/d.t. = d(6p)/dx),

qp,pE', since 17 p on the p side of the we see that in general, gp,,rzE diode. 1 j , and j, are of the same ordsr of magnitude a n d the charge neutrality condition, 6 r ? ( x i --- & p ( - r ) , applies in the quasi-neutral region (i.e., s < -x,), where E is very small, we expect :hat the drift current is much lcss than the difFusio~lc!irrent tar the eiectrons. Thus, the minority

current density j,, is only doi~iinaledby the diffusion current density (this is not true for the majority current density as wilI be seen later):

Using the charge continuity equation for the electrons,

We find, under steady-state conditions, d(Sn)/at

=

0,

-.

The solution with the boundary condition, 6 n ( x

= -m)

=

0, has the form

where

is the diffusion length for electrons in the p region. T h e and L,, = injection condition from (2.5.35) has been used. The total carrier concentrations on the y side for x < -.up are .+

and

where 6 p ( x ) = Sn(x), I,,;,, = h;,, and rl],,! == t f 2i t , , , / N c l . Simila~-ly,in the quasi-ncutral rcgion on ctle rV side o f the ciiode, s > x,, the minority (hole) current der;.si ty is u'ppraxin~;~ttsiy

and

which lead to a ~ ( , = ~ ~) P ( ~ ~ ) ~ - ( ~ - ~ N ) / L P

(2.5.47)

where ~P(xN= ) P ( x N ) - f'No -p ( L 3 -

-

1

)

(2.5 -48)

and L , = J D ~ is T the ~ diffusion length of the holes on the N side. T h e total carrier concentrations on the IV side of the diode, x > x,, a r e

From the carrier concentration for x < - x , and x > x , , the quasi-Fermi levels can be obtained from (2.5.31b) a n d (2.5.32a):

where kBT ln(rz,,,/N,,)

=

F,

-

E,, on the p side has been used. Similarly,

where kBT In( P , b / N V N ) = E m - FN has been used. From (2.5.51) and (2.5.52) we have I

and F,,,(,Y -+ -a) + F,, F,(x + +=) -+ F,. 1 / k D T ) 72 1, w e For the forward bias case: I.:,> 0, assuming that exp(ql., 1 1;lve

F,\.(

-Y) =

Fv -t

(

4-

.r,)) k,jT 1-,,

f r j i . .L I ---x P

and

( xf

I

XI,

la,,

Fp(x)

=

Fp

+

( X -

xN)IiBT

for x

and

2 x,

x - X, <-s

-

L,

(2.5.54b) These results are plotted in Fig. 2.14a. For the reverse bias, VA < 0, assuming that exp(qVA/ k B T ) << 1, we have from (2.5.50,

-

- FP

-

k , T e(" +.'.'/L.

for x

+ x p <<

-L,, (2.5 -55b)

-. Similarly, for F,(x) on the N side,

F , ( x ) = F,-

-

- FN +

k B T l n [ l - e-'X-'~'/Lp]

for x > x ,

k B T e("--r~)/Lp

for x - x ,

(2.5.56a)

= L,

(2.5.56b)

The quasi-Fermi levels are plotted in Fig. 2.14b. Current Densities and I-V Characteristics. The current densities are obtained for the minority carriers first:

O n the p side

O n the N side

64

BA

I<'S E~:?~c.:c;~~DLJCTOR ELECTRONICS

Assuming that there is no generation o r recombination current in the space charge region, that is, j,, and j, are constant dver the space charge region, the total current density is thus the sum ol the two current densities:

The total current is I

= jA

with A the cross-sectional area of the diode:

Canier concentrations

region I

I I

4 Current denybes I

I

j = Total current density

Figure 2-15. (a') Tbc carrier coocen1r:~liunsand (b? tlr:? cllvrc~ltdensities as lunctions of position s in a fol-wnrd biased p-N hcti-r-uj~lnctic.)ndiode sing 1 1 . 1 ~c l c p l ~ t i u in~p p r o x i , ~ ~ n t i o ~ ,

Figure 2.16. (a) The carrier concentrations and (b) the c u r ~ k n tdensities as functions of x for a reverse biased p-N heterojunction diode using the depletion approximation.

From the total current density j , which is a constant across the diode, the majority carrier current densities j,(x) a i d j,(x) can be obtained from j

=j

-j

)

j N ( x ) = l -j p ( x )

on the p side on:the N s i d e

x 5 -x,

(2.5.61)

x 2 x,

( 2 -5.62)

The complete current distributions are plotted in Fig. 2.15 for the forward bias case and in Fig. 2.16 for the reverse bias case. 2.5.2

Semiconductor n-P Neterojuncticln

If the narrow gap semiconductor is doped rz-type and the wide-gap semiconductor is doped P-typel we may have a band diagram such as that shown in .Fig. 2.17a before contact. '$hen the hetcrojtlnction is formed, a space charge region will exist d:re to the cliffusion or redistribution of free carriers at

B :\::!C S!
(a) Before contact Vacuum level

After contact

-----

Figure 2.17. contact.

-+

-

/

fl

- - -------

- - - - - Vacuum le vel -4 $(XI

+ -

Energy band diagrams for an n - P heterojunction (a) hefore contact and (b) after

thermal equilibrium. T h e excess electrons on .the t z side near the junction may spill over the P side and the holes on the P side near the junctio; may spill over the n side. Under the depletion approximation, we have a net charge distribution such as shown in Figs. 2.18a and b. The procedure to find the band diagram is similar to that for the p-N heterojunction in the p r e v i o ~ ~section. s We summarize the major results here. T h e charge density p ( x ) is

The eiectric fietd E(s) is (Fig. 3,.LSc)

2.5

S E M I C . ~ N I ) I J C T O Kp-A' :lP.lT) 1.-P F~E'I~EROJI.~N('TIONS

(a) An n-P heterojunction

(b) Charge distribution P(X)

(c)

Electric field

-

X

-xn

X~

(d) Electrostatic potentiaI

1.cm

.

Xp

W X

Figure 2.18. (a) An rz-P heterojunction, (b) the charge distribution, ( c j the i$,ltctric field distribution, and (dl the electrostatic potential using the depletion approxirnation.

Von r

.

where the bounda~ycondition at x

The electrostatic potential is

=

0 gives

I

as shown in Fig. 2.1Sd, where

Again the total depletion width is

and the contact potential Vo is

The band-edge function E , ( x ) or E , . ( x ) can be obtained from

E,.(x) =

-9Nx)

x < o

-qd)(,r) - A E ,

x

-

q+(x):

>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'~,,,

<

( 2 ,v.v , - .Y') 2 ~ / ',

O<.r~x, ,X Xp 1

(2.5.72)

and

E,(x)

E , . ( x ) f E,,, =

Example An sz-GaAs/P-A1, temperature and a zero bias. cmp3 in the n region and parameters were given in the 1. n side (GaAs)

x
E , . ( x ) + EGp x > O

,Ga ,.,As heterojunction is formed at room The doping concentrations are N, = 4 x 1016 NA = 2 X 10" ~ r n -in~ the P region. The previous example of a p-N heterojunction.

N, = 2.5 x 1019

=

4.30 x l0I7 cmA3

3/ 2

Nv

side ( ~ l , , G a , , A s ) x 10" cnlP3

P

P

=

NA

=

=

2.5 X 10"

Nv exp ( E v ~ , F p )

2. The contact potential is calculated using (2.5.70):

3. The depletion widths are

-4

-6 Figure 2.19.

-2 0 Position x (pm)

4

Band diagram of an unbiased ~ I - G ~ A ~ / P - A ~ ~ . ~heterojunction G ~ , , . , A S with N,, = 11 region and A!, = 2 X 10" rii the P region.

&&-' in the

4 X lot6

2

'

4. T h e energy band diagram of the n-GaAs/P-Al,,Gao.,As

plotted in Fig. 2.19.

junction is

I

Biased n-P Heterojunction. T h e band diagrains for the rz-P heterojunction

under folward ( V 4 > 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
T

A step-by-step illustration for the energy band diagram of an tt-N (isotope) heterojunction (a) before contacl aild (b) after c o n u c t showing five steps to obtain the steady-state energy band dia,oran-1. Figure 2.21.

of the Fermi levels. The charge distribution is

Note that far away i;om t!ls junctior,

/:!;.

.-,

-=)

--?

?(,.

The electron field

I.6

FiF~r~<~.!~JU~>iCC1'~'IC,.~PI?~; +QJI) !~!L*1..4i,...;EM~c.:~I
-1..:

satisfies

Since E ( - m ) = 0, i.e., the el-ectric field is zero far away from the junction, we integrate the above equation from -CQ to 0 - and obtain

where

is the surface electron concentration. SimilarIy,

~71.'

-

[ " ( X I )

-

- -x

N,]d x ' x < o (2 -6.7)

rz

i-

~ N D ( x- x N )

-

0 <'x < x,

EN

.Judging from the charge distribution, the electric field and the potential profile, one'sees that the band diagram in Fig. 2.21 (step 5 ) is very similar to that of the 12-N heterojunction in Fjg. 2.lOb. IJsing the Boltzrnann distribnlion

where E,,(

-

EL.(c

c )

is a constant, in (76.2) and

we obtain a differential equation for E , ( x ) :

Using

we can rewrite ( 2 . 6 1 1 ) as

Integrating fronl x at .r = 0-1

= -m

to 0 - , w e obtain an expression for t h e electric field

+ [ E L , ( 0 - )- Ect,]~ - E . . , / ~ L J ~ ) ~ F ~ (1.6.14) ~/*DT

Equation (2.6.14) can be furtlzer simplifiecl to

,:.AT HI",TEX DJTJKC'I-IC;,NF.A N D ~'~ETAT,-:EM]C{ ,NI31JC?'OR

JUNC'-TIObrS

75

Using the fact that the contact potential is given by

and

we obtain at x

=

0

Therefore,

Thus from. (2.6.15) and (2.6.20)

the only unknown VO,lis obtained by solving (2.6.21). After finding V,,, we obtain V,,, and x , from (2.6.19). 'As an example, we show in Fig. 2.22 the energy band diagrams for a P-n-N heterojunction. The n region is ass~lmedto be wide enough -sucl~ that the two space charge regions near the two junctiorls do not merge. If they do, the flat portion in the center 11-type semiconductor will not exist, and the band .diagram is further distorted.

C:onsicier a metal with :i w ~ r kf i : i ~ - r ~ t i(I.',,, ~ ) ~ lbrought i[i ~:{:~ntact with a.n n.-type . . semiconductor with ar! eler:trc!r? afl~lnlty,y. For- ;if: ideal cor,tact, the barrier

(a) B e f o r e c o n t a c t -

_

_

_

P

_

_

-

_

-

-

L

-

-

-

-

-

_

_

n -

-

-

-

-

-

-

-

-

-

-

N

(b) A f t e r contact

Figllre 2.22. Energy band diagrams for a P-rr-N heterojunction structure (a) before contact and (b) after contact. Assurne thermal equilibrium.

height is given b y (see Fig. 2.23),,

i

and a negative charge dist'ribution builds u p at the metal surface with a space charge region fortned in the semiconductor region. The charge ciistributian in t h e semiconductor region is given b y

(a) Before contact

&f - - Vacuum level ---f-----Ec 3 @m

Fn

L

-

(b) After contact Band diagram Ec

Charge distribution

EIecGc field

tT

> X

I;"; E(O+)

Potential 'nction

X

Figure 2.23. (a) The energy band diagram of a metal-semiconductor junction be€ore contact. (b:) The band diagram, charge distribkition, electric field, and potentid function after contact.

where E,,,

vihere I;

=

=

E,(=). We h w e

4(x + a),with

:hs i-cfrrence potential $(O?

=

0, a n d

B,A,SIC S E L ~ ~ I C ' C C N WCTOR ELECYTKONLCS

75

Multiplying (2.6.24) b y E ( x ) on both sides

and integrating from 0,

to

+m,

we obtain

The contact potential is given by Ti, = [a,,, - ( E , - F n ) ] / q . The factor exp(-qVo/kBT) in the last expression in (2.6.28) is usually negligible for > 0. When the metal serniconductor is forward biased with a voltage V and the barrier is reduced to C,' - V, we have

v,

If the electron concentration width x,,

iz(s)

is assumed to be depleted in a region of

Then

and

An iterative approach to include the effect of the electron is to use E(O+) frdm (2.6.29), and obtain

PR0T)ILEMS

PROBLEMS 2.1

Derive the boundary conditions using the integral forin of Ma,nvell's equations for a given charge density p, and current density J,. Check the dimensions of all quantities appearing in the boundary conditions.

2.2

Consider an ideal case in which the electrons are distributed in a two-dimensional space with a surface carrier density n, (l/cmz). (a) Derive a two-dimensional density of states and plot it vs. the energy E . (b) Find the relation between n, and the Ferrni level at a given temperature IT. (c) Repeat part (b) for T = 0.

2.3

Assume that the electrons are distributed in a "quantum-wire" geometry, i.e., an idealized one-dimensional space with a given "line" carrier .density n , (l/cm). (a) Derive a one-dimensional density of s t a t ~ sand plot it vs. the energy E. (b) Find the relation between n , and the Ferrni level at a given temperature T . (c) Repeat (b) for T = 0.

-

2.4 Use the inversion formula (2.2.511, and plot q vs. u. Compare this curve with the cuxve in Fig. 2.2. What is their relation? 2.5

2.6

2.7

Derive Eq. (2.3.24) for the average carrier lifetime

T,.

Explain the band-to-band Auger recombination processes in k space by drawing the band structure including conduction (C), heavy-hole (H), ' light-hole (L), and spin-orbit split-off (S) bands [Ref. 271. Suppose the carrier generation rate in Fig. 2.9 is

(a) Find the excess carrier concentration 6n( y ). (b) Plot G,,(y) and 6 r z ( y ) vs. j1 for S 10 p m and L ,

-

2.8

=

4 pm.

Plot the .band diagram for a GaAs/'_410,Ga,~.,As y-N junction. The GaAs region is doped with a n acceptor concentration N,, = 1 X 10"/ ern" The AI,,Ga,.,As region is doped with a donor concentration iVD = L x ll)'8/cm-'. Assui~tezero bins. A E , = 0.67 AE, and A E , 0.33 A E,, where A E , == 1.247.r (c'V) is the hand-gap difference between

-

13

Al,rGu, . A s a n d G;iAs. E v a l u a t a t h e Fel-mi levels, the contact potential

V,,a n d the d c p l e t i o ~width. ~ 2 .

Repeat Problem 2.8 f o r a reverse bins of - 2 V.

1. 2. 3. 4. 5.

J. A. Kong, Elect!-ornrlg17etic Wkue Theory, 2d ed., Wilcy, New York, 1990. J. D . Jackson, Classical Elect~.od~,narnics, 2d ed., Wiiey, New York, 1975. M. Born and E. Wolf, Pri~zc.iplesof Optics, Pergamon, Oxford, UK, 1975. Devices, Wiley, New York, 1951. S . M. Sze, Physics ofSet?zicotz~iiicfor S. Selberherr, A~znlysisrirld Si~nrllatiol-1o f Sctnicondrlctor Deoices, Springer, New York, 1984. 6. W. L. Enyl, H. K. Dirks, and B. Meinerzhagen, "Device modeling," Proc. I E E E 71, 10-33 (1983). 7. A . I-I. Marshak and C. ~ 1 . V a nVliet, "Electrical current and carrier density in degenerate materials with nonuniLorm band structure," Proc. I E E E 72, 148-164 (1954). 8. X. Aymerich-Humett, F. Serra-Mestres, and J. Millan, "An analytical approximation for the Fermi-Dirac integral t;J/2(.x)," Solid-Stale Electron. 24, 951-952 ( 1951). 9. D . Bednarczyk and J. Bednarczyk, "The approximation of the Fermi-Dirac inlegral FV2(77)," Phys. Lett. 64.4, 409-410 (197s). 10. J . S. Blnkemore, "Approximations for Fermi-Dirae integrals, especiaIly the Function F,/,(77) used to describe electron densily in a semiconductor," Solici-Stare E ~ ~ L . I I .25, O I ~1067-1076 . (1 982). 11. N. G . Nilssorl, "An accurate approxin~alionof the generalized Einstein relation for degenerate semiconductors," Phj s. Stat~rsSollcli: A 19, K75-K7S (1973). 13. T. Y. Chang and A . Iznbelle, "Full range analytic approximations for Fermi in terms of FlI2,"J. Appl. Phys. 65, energy and Fermi-Dirac integral F2 162-2164 (1959). 17. R. B. Adler, A . C. Snlith, and R. L. Loneini, Ii7:rodiictiori to Senlicondiictor Ph~lsics, Scrnicondi~ctor Electro~~ics E~liicatioil Corrzm~t/(.r,Vol. I, Wiley, New York, 1964. 11. R. F. Pierret, Semicoudiictor Fri~rciame~ztals, in R. F . Pierret a n d G. W. Ncudeck, Eds., Modrrlrrr S e ~ i r s011 Solitl Stole Dei,ict.s, Vol. 1 , Addison-Wesley, Reading, MA, 1.983. 15. S. W ang, F~~ncItl/~lc~rrtnlss o f St.i?rico/zcllrc~li,,.T!:~or.v~ l r l d D e ~ ~ i cPIzysi('.~, e Pre n tice Hall, Englewood Cliffs, NJ, 1 9 ~ 0 . 16. W. Shockley ancl W. T. Re'ld, Jr-., "St;~tistic.it)t' ttlc recornbin;~tionsof holes and electl-ons," Plzys. R(ir7. 87, 835-8-12 ( lij52i.

,,,

18. C. T. Sah, "The equivalent circuit model in solid-slate electronics, Part I: T h e single energy level defect cer-lters," Proc. I E K E 55.'654-671 (1.967); "Part 11: The multiple energy level impurity centers," I'roc. lEEE 55, 672-684 (1467); "Part 111: Conduction and displacenlctlt currents," Solid-State Eiectri~tz. 13, 1547-1575 (1970); "Equivalent circuit models in semiconductor transport for thermal, optical, Auger-impact, and tunnelling recombination-generation-trapping processes," Phvs. S t n t ~ i sSolidi: A , 7, 54 1.-559 (197 1). 19. C. T. Sah, F~indanzentn1.sof Solid-State Electronics, World Scientific, Singapore, 1991. 20. El. C. Casey, Jr., and M. B. Panish, ~~~~~~~~~~~~~~~~e Lasers, Purr A: F~indamental Principles; Part B: Materials and Operating Characleristics, Academic, Orlando, FL, 1978. 21. R. L. Anderson, "Germanium-gallium arsenide heterojunctions," IBM J. July, 283-287 (1960). 22. K. I,. Anderson, "Experiments on Ge-GaAs heterojunctions," Solid-State Electron. 5, 341-351 (1962). 23. W. R. Frensley and H. Kroemer, "Predicti* of semiconductor heterojunction discontinuities from hulk band structures," J. Vac. Sci. Techtlol. 13, 813-815 (1.976). Sci. Techtzol. 14, 24. W. A. Harrison, "Elementary theory of heterojunclions." J. VLZC. '1016-1021 (1977). 25. H. IGoemer, "Heterostructure devices: A device physicist looks at interfaces," Silrf. Sci. 132, 543-576 (1983). 26. R. F. Pierret, "Extension of the approximate two-dimensional electron gas forrnuiation," IEEE Trans. Electron Deuices ED-32, 1279-1287 (1985). 27. A. Sugimara, "Band-to-band Auger effect in long wavelength multinary 111-V alloy semiconductc;r lasers," IEEE J. Q ~ ~ ~ n t zElectron. srn QE-18, 352-363 (1982).

Basic Quantum Mechanics In this chapter, we present some basic quantum mechanics, which in later chapters will help explain the physics of optoelectronic processes and devices., In quantum mechanics, there are three important potentials, which have exact analytical solutions and are presented in most undergraduate texts [I, 21 on modern physics o r introductory quantum mechanics. 1. A simple sqrlnre rvell porerltinl. This potential is presented in Section 3.2. We take into account the fact that the effective masses inside and outside the quantum we1.l are different for a semiconductor quantum- well. More discussions on semiconductor quantum-well structures are presented in Chapter 4. 2. A harrnorzic oscillcrtnr. T h e solutions arc Hermite-Gaussian functions [I.-31 and are discussed in Section 3.3. A simiIar situation exists for the eIectric field of a waveguide with a parabolic permittivity or gain profile, since the solutions to the wave equation with a parabolic dependence on the position are also the Hermite-Gaussian functions. 3. A hydrogen ntonz. T h e solutions for both the b o ~ l n dstates ( E < 0) and the unbound (continuum) states ( E > 0) are presented. Most textbooks cliscuss the bound states only, because they are important for hydrogen atoms. We use the hydrogen atom model to describe a n exciton, which is formed by an electroll-hole pair. T h e optical absorption process involving the CouIomb interadion between the electron and hole is called the excitonic absorption. The excitonic absorption spectrum is strongly dependen.t on the energies and wavc functions o f the exciton bound and continuum states. In bulk semiconductors, t h e el-ectron-hole interaction is a Coulomb potential in a three-dimensional space. In quantum-well semiconductors, electrons and holes are confined to a quasi-twc-diinensional structure; thus, their binding energy is diffkrent from the three-dimensional results. In Section 3.4 we summarize the analytical soltltions for the hydrogen mode1 in both threedirncnsional ,and two-ctimensional space fur both the bound and unboiirld states [4-81,. The cle tailed derivntior.ls are prescn ted in Appendix A. These results are 'used in Cl~apttrr i.3 w h e w vie (discuss excitonic absorption ancl cluantuni-confined Stark elYzct.:i. Since most potentials e:tc'c p t tliz ai?(:)vr: three clo not have exact, :tnalyticnI so!utiuns, we h:~ve to use pert~rrbatiol!rr!i:thod~ to fincl their eigenfunctions

and eigenenergies. We present a conventional time-independent perturbation tl~eoryin Section 3.5, then a slightly modified perturbation theory, called the Lowdin's method [9] in Section 3.6. The Liiwdin's method is applied to derivc the semiconductor band structures, such as the Luttinger-Kohn Hamiltonian in Chapter 4. In Section 3.7, we present the time-dependent perturbation theory and derive Fermi's golden rule, which is important to our ~~nderstanding of tk.emission and absorption of photons.

The Schrijdinger equation in a nonrelativistic quantum mechanical description of a single particle is

where the Hamiltonian H is

the potential energy function V(r,t ) is real, zf is the Plailck coilstant I-1 divided by 2r,and rn is the mass of the particle. For a free particle in an uilbounded space, V ( r ,t ) = 0, the solution is simply a plane wave:

where the energy is

and. V is the volume of the space. The probability density is defined as

r ) = fi:K(r,t ) d ~ ( r ,

(3.1.S)

a n d the probability ctlrrent density as

Here

(id3r i s tilt: probahiiit:; of fil-liling rt-1:: particle in a volume d 3 r near. the pr:sitiol~ r a t time i. The: wave function 1s normalized such thal f :I:

d 3 = 1, that is. thc probability of finding the particle in the whole space is unity. It is strnightfo~wardto show that /;[II Sp,rcc

which is the continuity equation or the conservation of probability density. It is analogous to the charge continuity equation in electromagnetics. The expectation value of any physical quantity is given by

where 0 is a n operator for the physical quantity and the volume of integration is over the whole space. In the above real-space representation, that is, @ = $(r, t ) , the corrcsponclence for the operators is Position

rOp= r

Momentum

p,,

fl

= -

i

V

If V(r, t ) is independent of t , the solution @ ( r ,t ) can always be obtained using the separation of variables:

and

which is the so-called time-independent Schradinger equation. The solution may be in terms of quantized energy levels E, with corresponding wave functions @n(r),or a continuoils spectrum E with corresponding wave functions $&-). In generaI, any solution of the Schrodinger equation may be constructed from the superposition af these stationary solutions: :

where l~i,,~' gives the probabiil!~that th:: particle will be in t h e n t h stationary 1 stiidying a rime-dependent potential state $,,(r, f ) with an enersy I;,',;. 5 problem, very often ;I pzrt~irbationnppiotlch is used if the time-dependent perturbing potential is st!l:ill ~c;n~pax.ei.lwith the unperturbed Hamiltonian, and the aboya expl-insicn i n t i r m s OF i!?e stationary states $,,(I-)or r/,,(r),

1

are solutions to the unperturbed problem, is very usefu!. In this case, 0 , (and a,) will be functions of time sirice the perturbation is time-dependent, and 1 u,(t)12 will give the time-dependent probability that the particle is in state n of the unperturbed problem. For a two-particle system, we use a wave function +(r,, r,, t ) to describe the particle 1 at r , , and particle 2 at r,, al time t . Here we assume that the two particles are not identical, such as those of a hydrogen atom. The Hamiltmian is ' I

1

where

and Vi refers to the gradient operator with respect to ri (i = 1,,2). This case will be solved in Section 3.4. If we write the Fourier transforms of $(r) and V(r) ir, the single-particle Schrodinger equation (3.1.1I), $(k)

=

/$(r) e-".'

d'r

(3.1.16)

we then obtain the momentum-space representation of the ~chrodingex. equation

which becomes an integral equatior,. This integral equation is discussed in Chapter 13.

3.2

THE SQUAREVWLL

We considel- a square ('or rectang~lrir)q u ~ t\.lin ~ n welX wit!? a tarrier hcight .C\. In the 011e-clirnension:i! case, the tirne-,inclcpendent Scfli-%linger equaticrn is

56

3.2.1

'_>.i'\,S I(.'

QUANTUM idEC!.I./\NITS

Infn.ite Barrier lVZodel

First we assume tha-t V,, is infinitely high; thus, the wave function vanishes at the boundaries. T h e solution to the infinite barrier model satisfying the boundary conditions 4,,(O)= +,(I,) = 0 is

+,,( Z )

=

/

sin

(7.)

n

=

1 , 2 , 3 ,....

with corresponding energy and wave number

i-e., the energy En and the wave nutnber k 7 in the z direction are quantized. T h e wave function has been normalized j ~ l ~ , , ( z ) 1d2z = 1. If the origin of the coordinate z = 0 is chosen at the center of the well, V ( + z ) = V ( - z ) , the solution can always be put in terms of even or odd functions by parity consideration:

n even n odd

which can be obtained from (3.2.2) by replacing z by z + L / 2 and discarding any minus signs, since we have the freedom to choose the phase of the wave function. The parity consideration is very useful since the symmetry properties of the system are employed; thus, the wave functions have associated symmetry properties. I n general, if V ( z ) = V ( - z ) , for any solution + ( z ) of the Schrodinger equation (3.2.1.), + ( - z ) will also b e a solution with the same eigenenergy, as can be seen by changing I ro - z in (3.2.1). If we form linear combinations of + ( z ) and (b(-z), they will also be solutions. Specifically,

where

a,(.z j is an

cs.L.1: j';l~,:ii~::;

;:rid

+ , , ( z )j;

311

odd function.

The complete sc)lution for tl-it; poiential well V ( z ) in a three-dimensional space is solved from

The normalized wave function is

with a corresponding energy

where #,,(z) is the same as in (3.2.2), and A is the cross-sectional area in the x-y plane. This energy dispersion diagram E vs. k , or k , is plotted in Fig. 3.1b for = 1 and 2. In semiconductors, rn should be replaced b y the effective mass rnzFof the carriers. For a GaAs /A1,Ga, -,As quantum-well stnicture, suppose the CiaAs well width is 100 A and t.he aluminum mole fraction x in the barrier region is large such that an infinite barrier model is applicable. We have the effective mass of the electrons r z z z = 0.0665~n0 in the well region. The :ampb

>.

( 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

<

2 L

2

Using boundary conditions in which the wave function 4 and its first derivative divided by the effective mass (l/nz)(d+/dz) are continuous at the

' i y 3 . A quantum well with ;1 width L :!:id a tinit:: barrier height Vo. T h e energy ievels fur r! =. .I and rr 2 and their cori-ep o r i d i n g wiive functions are show!?.

-

I

BASIC QUAN7'UM MECI3AN!C 3

92

intei-face between the barrier and the well, that is,

and

we obtain

Eliminating C , and C3, we obtain the eigenequation or the quantization condition

m,k L a=tan k -

2

m w

The eigenenergy E can be found from the above equation by substituting k and a from (3.2.19~~) and (3.2.19b) into (3.2.22). SimilarIy for odd wave functions, we have solutions of the form

d(z)

=

C 2 sin kz

The boundary conditions (3.2.20) give C, r.u C , --.-

ti?,,

=

-

C.', sin -,

--

k

rn ,,,

i

L: 7 7

C'",C()$

L 2

3.2 TT-3E SQU AXE k;IEL:-

Thus, the eigenequation is given by m,k

a==---

" 2 n,

L cot k 2

which determines the eigeneilergy E for the odd wave function, again using k and n in (3.2.lOa) and (3.2.19b). In general, the soiutions for the quantized eigenenergies can be obtained by finding ( n L / 2 ) and (kL/2) directly from a graphical approach since from (3.2.19a) and (3.2.19b)

and even solution

cy-

L

2

= -

nib L -k-cot TTZ, 2

L k2

odd solution

The above equations car1 be solved by plotting them on the a L / 2 vs. liL. ? plane, as shown in Fig. 3.4. If m , = HZ,, Eq. (3.2.26) is a circle with a

Figlire 3.4. A g r ~ ~ p h i c asolt,tti<:!~ l for th;: :lrc:~:!ing curlstail! :Y ;~:lif w;lve i?tirlrller li of a fifiite quitniuni wrl!. Hers v,c di.!ilit; I ; ' = I- {;,',, /;;:,, 1" acceua! iOi. t h r ditTcreiit rfIective, masses i n tile vieil a n d in the bari'ier.

radius

Jm( L / ~ { I ) If . m,,. + rn,, we define a '

-

n Jm,/m,,and rewrite

O n the a f L / 2 vs. k L / 2 plane, Eq. (3.2.26) is still a circle with the same ( L / 2 h ) . T h e solutions to a' and k are obtained from the radius intersection points between the circle (3.2.26) and either the tangent curve (3.2.28a) or the cotangent curve (3.2.28b1, as shown in Fig. 3.4. It is obvious that if the radius of the circle is in the range

-4

there are N solutions. After solving the eigenequations for a and k , we obtain the quantized energy E.-' The bound-state solution ( E < V,) after normalization l",l+(z)/* d z = 1 is

+(z)

=

c

L cos k- e - n ( l z l - L / 2 ) 2 cos kr

lzl

>

1.4 <

L -

2

(3.2.30)

L -

2

where

+

( l / a ) is the If m , , = m b , we have C = \l2/( L + 2 / a ) . The length L well width plus twice the penetration depth l / a on each side of the well. The normalization factor is very similar to that of the infinite well (3.2.2) except that we have an effective well width L, = L + (2/a). For odd solutions

3.2 1TIE SQUAKF: WEL7.

where

Again C reduces to the expression J ~ / ( +L( 2 / n ) ) when m, = m,. Alternatively, an effective well width LC,, can be defined by using the ground-state eigenenergy E obtained from the solution of the eigenequation (3.2.28a) and setting it equal to the ground-state energy ( n = 1) of an infinite well:

The appropriate wave functions are then

where the origin is set at the left boundary of the infinite well. Consider a GaAs/Al,Ga, -,As quantum well shown in Fig. 3.5. Assume the following parameters Exan.. ie

( 0 Ix

< 0.45, room temperature)

A E , ( x ) = I ,247.x (cV), !I&.= 0.67AE,, and A E , the free electron mass.

=

0.33AE,, where

nlu

is

a. Consider the aluminum mole fraction x = 0.3 in thc barrier regions ( X = 0 in the well region) and the well width L , = 100 A. How many bound states are there in the c o n d ~ ~ c t i oband? n How many bound heavy-hole and light-hole subbands are there? b. Find the lowest bound-state energies for the conduction subband ( C l ) and the heavy-hole (HH1) and the light-hole (LH1) subbands for a 100-A GaAs quantum well in part (a). What are the C l - H H I and C l - L H l transition energies? c. Assume that w e define an effective well width L,, using an infinite barrier model such that its ground-state energy is the same as the energy of the first conduction subband E,, in (b). What is Leff?If we repeat the same procedure for the HH1 and the LH1 subbands, what are L,,.,?

-.

SOLUT~ONS. We tabulate the physical parameters as follows:

Well Barrier ( x = 0.3) J .,,. = 100

0.0665m0 0.0916mo

2,A E ,

=

C).34nl0 0.466tno

0.094m0 0.107mo

0 . 6 7 A E g = 0.2506 e V , AE,

=

1.424 e V 1.798 e V 0.1235 e V .

a. The number of bound states N is determined by ( N - l)?i-/2 I f i r ~ : ~ o ( L / 2 A ) 5 N7i/2, V , = AE, for electrons and A EL,for holes, respectively. :

ill[&)=

For electrons: .

-

*

/

For h e a j l holes:

77-

3.30 < N 2

=

5-25 -

7

N

=

3 bound states

i

2

b. T h e eigenenergy E is found by searching for the root in

N = 4 bound states

the first subhand energy is

Similarly, E,,,, gies are

=

7.4 meV and ELF,,= 211.6 meV. Tlle transition ener-

c . The energy for an infinite barrier model is E, = ~r~h~/(2rn*l:+.,)

For C1, L,,,

=

/

) 2m:Ec,-

For HHI., Left=

For LH1, L,,

3.3

sr2h2

=

=

136 A

r2h2

= 122A ~ ~ : ~ E H H I

=

139.6 A

T I E HAFtMONIC OSCILLATOR

If an electron is in a parabolic potential of the form V(z)= ( K / ~ ) . z as ~, shown in Fig. 3.6, the time-indepsndent Schrodinger equation in one dimension is

J?igu?.e 3.6.

A. pa~.:tbolic c~I.lnnrr~rn we11 wit!! i t s i!:~i??~iizc
BASIC Q U A N ' T U ~MECHANICS ~

and change the variable from z to

5

Equation (3.3.1) becomes

The solutions are the Hermite-Gaussian functions

where f!,:([)are the Hermite polynomials satisfying the differential equation I101

and n is related to the energy E by

The Hermite polynomials can also be obtaihed from the generating function

The first few Hermite polynomials are

Another elegant way [3] to find the solutions for the harmonic oscillator is to use the matrix approach by defining the annihilation operator

and the creation operator

Note the relation

rp - pz which

L

=

ifi.

:be proved using p = ( f i / i ) ( d / d z ) and

. we find for any Function ( ~ ( z ) Therefore,

I?

f

rnw I?=--

2h z2

+

pZ 2rnf1w

+

i -(zp

2A

- pz)'

The Hamiltonian (3.3.1) can be rewritten in terms of @+a:

Since the last term fzccb/'2 is a sclnst;i.nt, t h e pr.ob!erri of soIving the Schrijdinger

equation

is the same as finding the eigenvector and the eigenvalue of the operator n'a. Defining

and noting that the Poisson bracket

[ a ,a + ]

= a n + - a +n

i = -{(JIZ

2A

- zp)

,-

(zp

-

pz)]

=

1 (3.3.19)

from (3.3.12) and (3.3.13), we find

Na

= n'nn

=

( a n + - l ) n = aN - a

Nn+= aCaa+= n t ( n t n

For any eigenstate of PI or N, say

+ I)

= a+(N

+,, = In > , we

(3.3.20)

+ 1)

(3.3.21)

have

since the wave function ( $) = a l n ) has a norm ( $ I $ ) which is always nonnegative. Let n be the eigenvalue of the operator N with the corresponding eigenvector In), ~ l n =) nln). We have

Thus a ( n )is an eigenstate of the operator N with eigenvaIue n - 1. Assume that the eigenfunctions are normalized for all r l :

Then 11 =

( ~ i l a + ~ l r= z j ((nrrt)(rrn)j

(3.3.25)

Since ( n j n j) is an eigenstrite of the opeiator IV with eigenval~ien - 1 fi-oxn (3.3.73) and its rnagnjtuc!e squarcd is rz f:-c.,.rn (3.3.251, we have

I

where the normalization condition ( Y E - 1/12 - 1) peating.the above procedure, we have

=

1 has been used. Re-

We find that the lowest state will be 10) with its eigenvalue n = 0 because of the nonnegative condition (3.3.22). Similarly, using (3.3.21), we find

In general, the nth eigenstate can be obtained from the ground state 10) by the creation operator

with Ro),

n

=

0 , 1 , 2 , 3, . . .

(3.3.30)

The ground-state wave function can also be obtained using the fact that a10)

Therefore,

=

0

.r

The solution to the above first-order differerltial equation is

which has been nc)rrnalized. I:~c~I-LI:$i!i.z) =: ( ~ ( 0 in ) (3.3.32), ail t h e other eigenfunctiorls can bi: CI-t:a;eil sei~uenei~tlly asing the creation. operalor in

(3.3.28)+

3.4

II's13[3E0GIEN A'FOPYI(3D AND 220 EXClTON EOTJND AND CONTINUUM STATES) [4-51

In this section, we summarize the major results of the energies and wave functions for the hydrogen atom model with both bound (E < 0) and continuum ( E > 0 ) state solutions. T h e hydrogen atom is a two-particle system for the positive nucleus (with a mass m,) at a position r , and an electron (with mass rn,) at a position r , . A general solution is to transform from r, and r , coordinates to the center-of-mass coordinates R and the difference coordinates r: R

+ uz2r2

m,rl =

m i + nz,

and

T h e complete solution is of the form

where V is the. volume of the space and +(r) satisfies

.

where m,.is the reduced mass, l / n z ,

=

l/rn,

+ l/m,.

3.4.1 3 0 Solutions

In three-dimensional space the eigenfunctions can be expressed as

, =

( I-) 0

R E ()

,) Y)

bound states ( E < 0) continuum states ( E > 0)

( 3-4.5a)

(3-4.5b)

where the radial functions R,,,(r) and R,,(I-) are shown in Table 3.1. T h e , ) are tabulated in Appmtli,.~.A. For bound-state spherical harmonics ~ , , , ( 6 o solutions, t h e energy let;eIs are quantized as

-s 3

Pi-

+

Cs

Pi rn

and the Rydberg energy for the hydrogen atom is

K

= Y

rn, e 4

h'

1

- -2 ( 4 ~ ~ ) ~ hZm, '

and the Bohr radius a , is

The wave function at the origin is

For continuunl state solutions, the energy E is a continuous variable and the wave function at the origin is given by

where E = ti 2 k 2/2172,.. The expression in the square bracket of (3.4.10) is called the Sommerfeld enhancement factor for a 3D hydrogen atom. 3.4.2

2D Solutions

The solutions for the two-dimensional hydrogen atom problem are given by CI

R *(r)

, v277 )

bound states ( E < 0)

=

corltinuum states ( E > 0) The eigenenergies for the bound states are quantized

'We have

It is noted that thc binding energy for the Is states IE,( is four times of that in the three-dimensionaI case. This enhancement of the binding energy is very useful in understanding the excitonic effects in selniconductor quantum wells and the observation of the excitonic optical absorption spectra, which we investigate in Chapter 13. T h e wave function at the origin is

For continuum states, the energy E is a continuous variable and the wave function at the origin is

where E = h2k2/2r?z,. T h e expression inside the square bracket is the 2D Sommerfeld enhancement factor. T h e 2D and 3D solutions for e x c i t o ~ ~are s summarized in TabIe 3.1.

3.5 3.5.1

TIME-INDEPENDENT PERTURBATION THEORY [I01 Perturbation Method

In most practi~lilphysical systems, the Schrodinger equations do not have exact or analytical soIutions. It is always convenient to find the solutions using the perturbation method when the problem of interest can be separated into two parts: O n e part consists of an "unperturbed". Hamiltonian with known solutions, Ht"),and the other pal-t consists of a smalI perturbing potential, H':

The unpertuibed wave functions known:

+jr)and eigenvalues Eiu) are assumed to be

~(0'&(0' = ~ ( 0 ) . tz

n

@)

T o find the solutions fur the total Hamiltonian

it

is convenient to introd~:i:~a per;:~rb;iti~)!?1;arn;:Icter ,\ (set

,\ =

later):

3.5 TIME-INDEPENDENT T'EF.rI'URI3 AT :C!N T! JEO R I'

107

We look for the solutions of the form

Substituting the above expressions for H, E, and equation, we find, to each order in A,

*

into the Schrodinger

( 3 -5 -6)

Zeroth order

H(O)+(')

First order

~ ( o ) + (+ l )~ ' ~ ( = 0 ~ 1( O ) q , ( l )

= E(O)*(O)

+~(1)+(0)

(3.5 -7) H ( ' ) + ( ~+ ) H'+(') = E ( ' ) + ( ~+) E(')+(')+ E(~)+(')(3.5.8)

Second order

Zeroth-Order Solutions. It is clearly seen that the zeroth-order solutions are the unperturbed solutions:

First-Order Solutions. The first-order wave function +(') may be expanded in terms of a linear combination of the unperturbed solutions:

Thus

-

( ~ ( 0 ) ~:))$;l)

=

E(i)#(O)- H,+;O) n

.+

MultipIying the above equation by

we obtain

and

where

+:E1"

(3.5.11)

and integrating over space, using

BAS LC' QUANTUM MECHANICS

J 08

T o the fi.rst ordcr in perturbation, we nced to normalize the wave function

Thus we have aLIA = 0 to the first order in perturbation. The result is therefore

E,,

= E~O'

+ H;,

Second-Order Solutions. Similarly, the second-order wave function be expanded in terms of the zero-order solutions:

$(')

We find

and

T o the second-order correction, the wave function may be normalized:

can

'

3.5

TIM:;-INDE PENDENT PER. J'~JRBA'f'l'3N TfTECIRY

Neglecting terms of third and higher orders, we obtain

The nortnalized wave function t,!~, and its eigenenergy E,,, to the second order in perturbation, are given by

E,,

=

EL0) + HA,

+ C tn

IH;,I~

+ n EL0' - E::)

Example

a. When an :finite quantun~well has an applied electric field F in the z direction (see Fig. 3.7), the Hamiltonian can be written, as '

Figure 3.7. Rn infinite i i ~ i i t n t u mwell v/ith~!:t 11r-i ;:c!:lieci firiil (1cF1 L ~ ~ L I ~ Cant1 ) with ill1 ; ~ p p l i a l field (right 1-igure) where t t p!>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<~-r? Jj'

we obtain, to the secorld order in .!I1,

has been used. where EAO) = If, however, the states in class A are degenerate, the diagonal elemel~ts are exactly or almost the same:

with differences of the first and higher orders. We have

Solving the eigcnequation using ~r,:

above hy letting

the eigenvalues for E and eigenvectors for cr,, are thus obtained, and the wave functions are given by

where a, for n E class R are obtained from (3.6.6). T h e above wave function can be normalized directly by dividing the above expression by ( ~ l L < ~ n + ~ o ) l ~ n ~ r t ~ > ~ ) ) or ,

3.7

TIME-DEPENDENT BERTtrRBATION THEORY [lo]

Consider the Schrodinpx equation

where t h e Warniltonir~r-l /I' consists of :r! i.ri~perturbed part H,,, which is timz-independcn t, and L: s m i ~ l iperrurlxk: iill-1 f!'(r. i ) , which ctepends on time:

Thc solution to the unperturbed part is assumed :known:

The time-dependent perturbation is assumed to have the form

T o find the solution +(r, t ) to the time-dependent Schrodinger ecluation, we expand the wave function in terms of the unperturbed solutions:

where lun(t)12 gives the probability that the electron is in the state n at time r. Substituting the expansion for into the Schrodinger equation and using (3.7.3), we obtain

+

fl

d"n(f) dt

4n(r)e-iE:,Jt/h

=

i

H

--

t )a t )( r ) e

i

f (3.7.7)

TI

Taking the innei groduct with the wave function +:(r), we find orthonormal property \ d3r#z(r)#,(r) =

and using the

where

ancl the r:iatri:< elenlents are defined r:s N / t: I?!

=

/ ~. ~ ; l , ~ ~ , r .3 ,~ ~ ~ * < r j q ~ ~ t ~ r ) c ~ , ~ r ;/I

,

Introducing tl.7~

parameter X (set X

=

1 later)

and letting

we obtain

Thus, the zeroth-order solutions are constant. Let the electron b e a t state i, initialIy:

We have the zeroth-order solution:

Therefore, the electron stays at state i in the absence of any perturbation. The first-order solution is obtained from

Suppose \ve are interested in a final solved directly by i n t e g r a t i o ~ ~ :

st::te

iil =

f'; ths above equation can be

If we consider the photon energy to he near resonance, either w w -- - - w , . ; ~we find

- o f i or

where the cross term has been dropped since it is small compared with either of the above two terms. When the interaction time is long enough, using

we find

The transition rate is given by

27~

=

..

--IHfi ' ' - ? ( % - E, - h w ) R

+

2r it

S ( E f - Ei+ h o ) (3.7.22)

where the property S ( f l w ) = S ( o ) / f z has been used. The first term corresponds to the absorption of a photon by an electron, since El = Ei h w , while the second corresponds to the emission of a photon, since El = EiW w. These processes are illustrated in Fig. 3.8 for a two-level system. In summary, we have Ferrni's golden rule: For a timk-harmonic perturbation, turned on at t = 0,

+

(a) Absorption

(b) Erni,ssion

Figure 3.8. Diagrar-ns showing the two prace.;s:?c i!: i<.7.21).(::) a[,sr,rvtic,n :ind, (b) emission of a piloton i i a~ two-level system.

The transition rate of an electron from aa initial state i with energy Ei to a final state with energd E , is give11 by

PROBLEMS 3.1

Consider an In,.,, Ga ,,.,,As / InP quantum well. Assume the following parameters at 300 K: In

,-,, Ga ,.,,As

region

InP region

where m , is the free electron mass. The band-edge discontinuity is A E, = 0.40A E,, and A E, = 0.60A E,. (a) Consider a well width L , = 100 A. How many bound states are there in the conduction band? How many bound heavy-hole and light-hole subbands are there? (b) Find the lowest bound-state energies for the conduction subband (Cl) and !be heavy-hole (HH1) and the light-hole (LH1) subbands in part (a). What are the Cl-HH1 and Cl-LHl transition energies? (c) Assume that we define an effective well width LeIf using an infinite barrier model such that its ground-state energy is the same as the energy of the first conduction subband E,, in (b). What is L,,? If we repeat the same procedure for the HH1 subband, what is L,,'? 3.2

In the infinite barrier mgdel, the dispersion relation for the electron subband is given by I

\lk

(a) Plot the dispersion relations E vs. I;, = + k ; for rl = 1, 2, and 3 on the same ctlilr-t. Plot the corresponding wave f~inctions +1(~)c , ~ ~ ( z and ) , 43!:zjwith k , = 0. (b) Derive the electron density of states p,( E ) vs. the energy .E and plot p,(E) vs. E. (c) Assume that there are t z , clzsti-or~sper ini it area in the quantum wells at T = 0. Finrl an exprzssior~f ( ~ r .the 'l'errrli-level position in terms of n,.

3.3

Calculate the F'ermi level for the electrons in a G ~ A s / A ~ , . ~ G ~ , , , A ~ quantum well described in the example in Section 3.2 with a surface electron coilcentration n, = 1 X l0l2/cm2 at T = 0 K. How many subbands are occupied by electrons

3.4

(a) Find an analytical expression relating the surface electron concentration ns(l/cm2) to the Fermi level at a finite temperature T in a quantum well based on the infinite barrier model.

(b) What is the carrier concentration at the ith subband? 3.5

If a finite barrier model is used, what is the electron density of states function p,(E) in Problem 3.2. Plot p,(E) vs. the energy E.

3.6

Consider a graded quantum-well structure with a parabolic band-edge profile along the growth ( z ) axis in real space, as shown in Fig. 3.9.

Figure 3.9.

-.

-

(a) What are the general dispersion relations of the cbnduction subbands E E,,(k,, k ,)?

(b) What is the density o-f states of the electron's in the cond.uction band p,(E)'? Pldt p,(E) vs. the energy E. 3.7

Show that the wave function in (3.3.32) is the solution of the differential equation (3.3.31) for the ground state of the llarmonic oscillator problen-1.

3.8

An exciton consisting of an electl-on with a.n effective mass r?zz and a holc with an effective mass J;.:,+;k c2n be de'scribed using a hydrogerl model. (a? Calculate the Ky dberg ~ ~ ~ R ,.c f!?ri -a ~ GaAs ~ ~semiconductor. ~ Assunle that E = 1 . 2 . 5 ~ ~I110 ; ; i:thc:r pilrnrrietcrs are give11 in t h e '7 rlumerical exalnple in Ssctlor~.!.-. (b) Calculate the: L s !ri,ur~i! s::;re ufiergy fur tirliis if the electron-.hole . . ~LIIY

I:<

r ~ s ~ . ~ - i ! i!,: ; f ;:!~ ~~I"I[{:: ! ta,;.,::.;j.impr?si:?!-::!! C;!:J;~L:C.

3.9

( a ) Piot the continuum-state wave function at the origin I$E,,(r

=

0)(2

vs. the energy E for the three-dinlensional hydrogen model. (b) Plot the continuum-state wave fu~lctionat the origin I$,,(r = O)I 2 vs. the energy E for the two-dimensional hydrogen model. 3.10

From the perturbation results in the example of Section 3.5, calculate the energy shifts at an applied field of 100 kV/crn using the effective well widths L,,, = (a) 136 A for electrons in-C1 subband, (b) 122 A for holes in HH1 subband, and (c) 139.6 A for holes in LH1 subband in a G a A s / A l o ,Gao,,As quantum well.

3.11

Discuss the differences between t.he Ldwdin7s perturbation method in Section 3.6 and the conventional perturbation method in Section 3.5.

3.12

Derive Ferrni's golden rule and ciiscuss the regime of validity [lo] for applying this rule.

1. A. Goswami, Q ~ ~ n n t i Mechalzics, ~rn Wm. C. Brown, Dubuque, 1A, 1992.

2. R . P. Feynman, R . B. Leighton, and M. Sands, Tlle Feyrlrrzon Lectr~resor2 Physics, Qllnnt~irnMecl.~nnics,vol. 3, Addison-Wesley, Readi1.19, MA, 1965. 3. R. P. Feynman, S;;vtisticnl i~leclzanics:A Set of' L e c t ~ ~ r e Addison-Wesley, s, Redwood City, CA, 19,:L, Chapter 6. 4. For both bound and contin~lumstate sol~rtionsin the three-dimensional case see L. D. Landau and E. M. Lifshitz, Q~lnntlrrn Mechunics, 3rd ed., Pergamon, Oxford, UK 1977, p. 117, and H. A. Bethe and E. E. Salpeter, Q ~ i a n t r ~ r n Mech~rl~ics of Onc- n~lrlT~~~o-EIectroil Atoms, Springer, Berlin, 1957.

5 . For an n-dimensioniii space, rz 2 2, see M. Bander a n d C. Itzykson, "Group theory and the hydrogen atom (1) and (II)," Re['. Mod. Plzys. 3 8 , 130-345 (1966), and 35, 346-355 (2966). 6. C. Y. P. Chao and S. L. Chuang, "Analytical and nun~erical solutions for a two-dimensional excjton in rnonlentilrn space," Ph~v.Rer-. B 33, 6530-6543 (1991). 7. E. Menzbacher, Qucrlltrlm Mechnnics, 2d ed., Wiley, New York, 1970.

8. PI. Haug and S. W. Kocli, Q i ~ ( l l l t l l T/lco)y ~l~ of !ILL' Optical 1 ~ 1 Electrot~ic d Propcrtizs of Scn.licondircto~.s,Worlcl Scientific? Singapore, 1990. 9. P. Lodin, "A note on the q ! ~ ~ n t ~ : ! : ~ - m e c h ~ ? nper-turtlation ii-~il theory," J . Chern. Plzys. 19: 1396-1401 !195 1). 10. A. Yariv, QILLIIIIIIIII k'!~(.~?.~r!ii..s, 3cl i : ~ l . .'>!5!i:ey. New York. 1989. 1 1 . A. Lukes, G . A. Ringv,ooci, ::nil l3. S L I ~ I . ' I P"i? ~ O p;irticlc , in a OX in i l ~ epresence of an elcotric field a n d i\;.)pljc;ttic.,n~tz) disorcicrcd systems," Pizy,sicci , 3 4 4 , 421-1124 ( 1976).

12. F, M. Fernandez and E. A. Clastro, "Hypervirial-perturbational treatment of a particle in a box i n thc presence of an electric field," Plzysicri IIIA, 334-342 ( 1982).

13. M. Matsuura and T. Karnizato, "Subbands and excitons in a cluantum well in an electric field," Phys. fiei~.B 33, 8385-8389 (1986). 14. G. Bastard, E. E. Mendez, I,. L. Chang, and L. Esaki, "Variational calculations on a quantum well in an elec-lric field," Phys. Reu. B 25, 3241-3245 (1933). 15. D. Ahn and S. L. Chuang, "Variational calculations of subbands in a quantum well with uniform electric field: Gram-Schmidt orthogonalization approach," Appl. Phys. Lett. 49, 1450-1452 (1986). 16. S. Nojima, "Electric field depexldence of the exciton binding energy in GaAs/ Al,Ga, -,As quantum wells," Phys. Reu. B 37, 9087-908s (1988).

Theory,of Electronic Band Structures in Semiconductors T o understand the optical properties of semiconductors, such as absorption or gain due to electronic transitions in the presence of an incident optical wave, we have to know the electronic band structure, including the energy band and the corresponding wave function. Knowing the initial and final states of the electrons, we may calculate the optical absorption using Fermi's golden rule (Section 3.7). In this chapter, we discuss the calculation of the band structures. For optical devices, most semiconductors have direct band gaps, and many physical phenomena near the band edges are of great interest. W e focus on the conduction and valence band structures near the band edges, where the k . p method is extremely useful.

4.1 THE BLOCH Tf-i LOREM AND THE k FOR SIMPLE BANDS

. p METHOD

For a periodic potential, the electronic band structure and the wave function can be derived from the Hamiltonian, which satisfies the symmetry of the senliconductor crystals. The general theory follows the BIoch theorem [I], which is discussed in this section. Numerical methods to find the band structures and the wave functions include the tight binding, the pseudopotential, the orthogonalized plane wave, the augmented plane wave, Green's function, and the cellular methods. Many texts on solid-state physics have detailed discussions on these methods [I, 21. Our interest here is near the band edges of direct band-gap semiconductors, where the wave vector k deviates by a small amount from a vector k, where a local minimum or maximum occurs. T h e k . p method was introduced by Bardeen [3] and Seitz ' [dl. The method has also been applied by many yesearchers to semiconductors [5-101. Here we discuss Krtne's model [7, 81, which takes into account the spin-orbit interaction, 2nd Lurtinger-Kohn's models [9] for degenerate bands. These models are very popular in st~lclyingbulk and 'quantum-well semiconductors a n d have beer) used during the past three decades. They are much easier to apply than most other nunlerical methods. For a general discussion? see Refs. 11 - 1.4. 1 :3 .F

,

I .1

6'.

'1.1i,... -12 BT L.>( :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
where, assuming k

=

k2 (see Problem 4.4)

and the Kane's parameter Y and the spin-orbit split-off energy A are defined as

4.2.3 Solutions for the Eigenvalues and the Eigenfunctions of the Hamiltonian Matrh Define the reference energy such that E, = - A/ 3, and E, the band gap energy. T h e Hamiltonian in (4.2.8) becomes

-

=

I:, , which is

-

T h e deterrninantal equation d r t l g - ~ ' 7 = 1 0 gives four sigenvalues for E'. We can see t h a t the last baac! irr (4.2.11) is decoupied from , t h e first three bands: ( 1)

E'

-0

( i . c ., band-eclge ericrgy is zero)

.

(4.2.1.h)

and

The second equation gives three roots. Since k 2 is vely small, we expect that the roots of Eq. (4.2.12b) will be very close to E' = E g , E' = 0 and E' = - A , the three band edges. (i) Let E'

=

E,

(ii)

Let E'

-

0

(iii)

Let E'

=

+ & ( k 2 where )

+~

-A

E

<< A and E,. We find from (4.2.12b)

( k ~~ ~) i. l a t i o(4.2.12b) n gives

+ s ( k 2 ) .We find

Since E' = E,,(k) - h 2 k 2 / 2 m , , we obtain four eigenvalues from (4.2.12a) and (4.2.13)-(4.2.15). They are, starting from the highest energy level,

These results arc not cc>xnplete sincc ci--ii: eifcctl-., af higher bands have noc bee^ included; they will be considec.i=clwtier?. wr: discl.1~~ the Luttinger-IlCohn 1-zlode.l. Note t h ~ the t zbove r-esul! gii,e:: an ii~cil::sect ,effective Inass for the hcauy-hole band.

The eigenf~lnctionscan be obtair~cd f r o n ~(4.2.13) for the first 4 x 4 matrix,

X + iY [ ) -r ) h,~~,~ =

hh band

-

4 ,=

X i

n

-..

iY

)

+

(4.2.17a)

, l h , s o (4.2.17b)

and the second 4 x 4 matrix, h h band

(4.2.18~~)

T h e eigenvectors are obtained by substituting each eigenvalue into t h e eigenequation

and then normalizing such that ( n : + b,: T h e results in the limit k' --+ 0 give

+ c:)'l2

=

1

4.2.4 Summary of the Eigcne1:crgies and Corresponding Band-Edge Basis Functions

We summarize the resr!lts b ~ i o w2 n d ii: 'Fig. 1.3. The cornrrlonly used parabolic band models for the enzrgy r,!isper:+ii~i~s arc aIsc redefined ir, the parentheses.

4 .

IO'LNE'S MODEL FCX2 Z.'II-iDSI'RUCTURE

Ehh

Figure 4.4. The band-edge energies E,, 0, 0, and - A for the conduction, heavy-hole, light-hole, and spin split-off bands with their corresponding bandedge Bloch functions. Note that the dispersion relation E - k for the heavy-hole band Ehhshould curve down as shown and follow the result of the

-3

-s ~ -

,

El

>

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?,
Spin-orbit split-off band

The Kane's parameter P can also be related to the effective mass of the electron nz; using

Sometimes the h 2 k 2 / 2 m , term is ignored, since rn: = 0.067m -=Km , for GaAs; therefore, the term nz:/?no in (4.2.26) is ignored. W e note that these wave functions in (4.2.2,l)-(4.2.24) are eigenvectors of the Harniltonian

with eigenenergies E = E,, O,0, - A as k + 0 for the conduction, heavy-hole, light-hole, and spin-orbit split-off bahds, respectively. 4.2.5

Genera! Coordinate ~ i r e c t i ~ n

If k is not along the z direction, k

=

k sin

6)

-'

cos 9 . F i%: sir1 3 s i n ? $

the f ~ l l ~ b v i transfox.rrl:ttions ~lg cr?n be iiseil

Li)

+k

cos 0.2

(4.2.28)

find the basis functions in the

general coordinate system: 6' e-'4/2 cos-

2 - e-irb/zsin-

H

2

lt]=(

cos 9 cos C$ sin. sin 8 cos C$

(4.2.29)

6, eid~/2COS 2

cos 9 sin C$ cos C$ sin 8 sin 4

-sF~]J] cos 8

(4.2.30)

and the spherical symmetry function S(rl) = S(r), since r ' = r , the length scale is preserved in a unitary transformation. The above transformation will be useful in Chapter 9 when we discuss optical matrix elements for quantum wells. -_I

4.3. LUTTINGER-KOHN'S MODEL: THE k FOR DEGENERATE BANDS 19-13]

. p METHOD

Suppose we are mainly interested in the six valence bands (the heavy-hole, light-hole, and spin-orbit split-off bands, all doubly degenerate) and ignore the coupling to the two dagenerate conduction bands with both spins. It is convenient to use Lowdi.: perturbation method and treat the six valence bands in class A and put the rest of the bands in class B (Fig. 4.3b). Note that Luttinger-Kohn's model can also be generalized to include both conduction bands in class A, especially for.-narrow.band-gap semiconductors. 4.3.1

The Hamiltonian and the Basis Functions

Write the total Harnjltonian In (4.2.6) for u,(r) (dropping the band index n for convenience):

A'k

H where

=

2

A

H , -t-+ ,. .,VV I 2rrr, 4m;;c-

X p . I T t- HI

(4.3.lb)

where

-

Note again that the last term in (4.2.6) is much smaller than the fourth term because Ak << p = I ( u k ( pilk ) 1 h / a , since the electron velocity in the atomic orbit is much larger than the velocity of the wave packet with the wave vectors in the vicinity of ko( = 0 ) [Ill. We expand the function

where j' is in class A and y is in class B. O r specifically, we have in class A

from (4.2.22)-(4.2.24). At k

=

0, the band-edge functions (4.3.3) satisfy

H(k

=

O ) L L , ~= ( ~E) i ( 0 ) ~ l j O ( ~ )

(4.3.4)

where

since E,, = - A / 3 . shown in Fig. 4.4.

Those band-edge encrgics i:nd b:lsij filnctions are also

4.3.2

Solution of the Manliltoninn Using Lo~vdin'sPerturbation Method

With Lowdin's method we need to soIve only t h e eigenequation

instead of

where

HIJ Y

=

A (rr,.,l-k mo

Jllrr,,)

-x cu

Rk, -pE

mo

(j

EA,y

EA ) (4-3.8~)

-

where we note that II,, 0, for J ,j' E A , and I;, p; for j E A and A. Since y # j, adding ihe unperturbed part to the perturbed part in y does not affect the results, i-e., Hjy= Hi,,. We thus obtain

Let

u:! J J = Dj j * . We obtain . the matrix of the form

Djjf

where Dzp is defined as

- -

j' the singIewhich is similar to (4.1..13), the ::ingls-k>:inri ::;tse (where j band index 1 1 ) . Here we have gi~~t.r-sIlzc:3 (4.!..i.l!) to Inz!~~(:le the degenerate bands.

'IHEOKY C>FELE.CTR(SNl~:7PJ:-\?-!DS-I'RUi--i'tiRES !C-4 SSE;'VIJCC)r\'DUTTOF,S

140

4.3.3 Explicit Expression for the Lutting-er-Kohn Ilarniltonian Matrix Dj,il

To write out the matrix elcments Djj. in (4.3.10) explicitly, we define

C,

=

" C -,

P.;? i1;;y

+ p,:,! P;,"

E,, - EY

172;

and define the band str-ucture paramzters y ,, y,, a n d y , as

:

-

-

= D, denoted as HLK,in the

..We obtain the Luttinger-Kohn Hamiltonian basis functions given b y (4.3.3) -

-

HLI'; = -

-

P+Q

-S

-S+ R+

0

-s+/v~ -

CR'

P - Q

R O

R

0

P-Q

S

R'

S+

P -t Q

-&Q+

m f S

b r n ~/i+0Y-

0

'1.

-OR -- ..5 1i. L.!T -

-s/fi

-&a

GR J3/2~

J 3 / 2 ~ + {FQ -GR+ -S4-/fi P-tA

0

0

P + h

I

-

(4.3.14)

where the superscript 4.3.4

+ means Hermitian conjugate.

Summary

In summary, for the valence hole subbands, we have to solve only for the eigenvalue equation

or for the' eigenvalue E and the corresponding eigenvector c o l ~ ~ m n [a,,, a,, . . . , a 6 ] ,where the matrix elements H $ ~= Ej(0)ajjj.+ Ea, ~ : p k , k ~ are given in (4.3.14). The wave function $,,(r) satisfying

is then given by

and

4.4 TZIE EFFECTIVE Mi-iSS 'THEORY FOR A SINGLE BAWD AND DEGENERATE BANDS .In this section, we ~:~mm;lrize the e C-'.: ~ -. ~ ~ t mass r v e theory f o r both a single band and degenerate in ~ e m i c o ~ ~ c i ~ : c lwe o ? . outline ~. a fornlal proof [91 of the effective I T I ~ equzitions ~ S in Apperidi.~H. -,

'

4.4.1

The Eflective Mass T h m for ~ a Single Band

The most important conclusion o.f the effective mass theory (EMT) for a single band is as fc?Ilows: If the energy dispersion relation for a single band rr near k , (assuming 0) is given bjl

for the Hamiltonian H , with a periodic potential V(r)

then the solution for the Schrodinger equation with a perturbatit~lnU(r) such as an impurity potential or a quantum-we11 potential

is obtainable by solving

for the envelope function F(r) and the energy E. The wave function is approximated by

I

The most important result is that the periodic potential V(r) determines the energy bands and the effective masses, ( I / ~ z " ) , ~ and , the effective mass equation (4.4.5) contains only the extra perturbation potential U(r), since the effective masses already take into account the periodic potkntia~(Fig. 4.5). The perturbation potential can also be a quantum-well potential, as in a semiconductor heterostruct ure such as GaAs/Al,Ga, L,t AS' quan tum wells. 4.4.2 The E,Rective Mass Th.e.orj:for Degenerate Bands

In Section 4.3, we disciiss t h e k p rfietjl.ijd f u r degenerate bands such as the heavy-hole, light-hole, arncl rhe spin-orbit split-off bands of senli~oncluctors. The e-ffectivemass theory ic)r i r perturbatifin potenriai U(r) for degenerate

(a) A periodic crystal potential V(r)

(b) A periodic crystal potential V(r) with an impurity potential U(r)

(c) An impurity potential U(r)

Figure 4.5. Illustrations of (a) the periodic potentid V(r) and (b) thc sum of the periodic potential V(r) and the impurity potential U(r), and (c) only the impurity potential U(k) for t h e efkctive mass theory.

bands is stated as follows r ' degenerate bands satisfying

Ill. If the dispersion relation of a set of

is 'given by

the11 the soluticbn $(r) tion' U(r),

thc sen;i~~!~(~!~ti::l-i~.~ in the presence of

fi,:

f!

1. -

+. '(r) 1 :i,!(rj== E + ( r )

21

perturba-

(4.4.9)

?'~-li-:Cx'f(>i-'ELEC'TJt.3p:'lL" B.V.III ,'.;'I'J<~-~C"T'UI;:ES I N SEMIc.CINDII'C'-roRS

144

is given by

where <(r) satisfies

4.5 S T M N EFFECTS ON BAND S'I'RUC?.'UKES Strained-layer superlattices [16, 171 have been of-great interest since the early 1980s. It has been demonstrated that it is possible to vary important material properties-lattice constant, band gap, and perpendicular transport effective mass-using ternary strained-layer superlattices. Applications of the strained-layer superlattices or quantum wells to long wavelength photodetectors [17] and semiconductor lasers [18] have been proposed and demonstrated. For example, strained quantum-well lasers have been shown to exhibit superior perfornlance compared to that For corlventional diode lasers [19, 201 in many aspects, which is dir.;::ussed in Chapter 10. Detailed discussions on the semiconductor growth and the physics of strained-layer quantum wells can be found in Ref. 19. When a crystal is under a unifornl defornlation, it may preserve the periodic -property such that the Bloch theorem may still be applicable. The modulating part of the Bloch function remains periodic, with a period equal to that of the new elenlentary cell, since the elementary cell is also deformed. In this section, we derive the Hamiltonian for strained semiconductors and discuss their band structures. 4.5.1

The Bilkus-Bir Hamiltonian [23., 221 for a Strained Semiconductor

Suppose that Itear the band extremum k,,

=

0 of a semiconductor we have

svith r h ~Bloch function ~,b,!~,(rJ := L: ', k c L<,(r) is n pzriodic potential in the undefornled crystal. Herc l~qi.present a sirhpli picture for the strain analysis [23-251. As shown in Fig. 4.6;the tlriit vectcxs ;t?, 9, (and 2 ) (for simplicity, ;~ssumingthcy :'ire basi;; -vL;lxc!rst::~)) in. the uncleformed crystal i?'[)'

4.

S'rIC.Al

Ef=}-'Ef:i'';

01.4

B,.,.P D S7'RUc''-i'*IF,E,r;L;p,Es

(a) Unstrained lattice

ib) Strained lettic5

Undefomed Crystal

Deformed Crystal

Figure 4.6. (a) Position vector r for atom A in an unstrained lattice. (b) Position vector I-' for atom A in a strained lattice.

are related to x', y', (and z') in the uniforn~lydeformed crystal by

Obviously x', y', and z' are not unit vectors anymore. We assume a homogeneous strain and E~~ = e j i - We ca lefine sit; strain components as .

keeping only the linear terms in strain. To label a position A (or atem A ) in the underorined crystal, we have

The same aton1 in the deformecl crys1:si can be labeled either as

using tile new basis vectors x',

J:',

: E L I ~ L ?;I.'

f.!;:

35

in the original bases ir! the undeformed crystal. An example is shown in Fig. 4.6. \Vc have r = 2 + 9 (1, 1,O) in the undcformed crystal and r' = x' + y' in the deformed cr-ystal. We can also see that t h e change of the voIume in the linear strain regime becomes

The quantity E , ~ , + E , , + E ~ is , the trace of the matrix E, o r Tr(g), which is exactly the fractional change of the volume 6V/ V of the crystal under ui~iformdeformation:

Major Results for the Pikus-Mir Harniltonian for a Strained Semiconductor. Using the above relations (4.5.2)-(4.5.6) between the deformed coordinates and the undeformed coordinates, the Hamiltonian for a strained semiconductor can be derived; the details are shown in Appendix C. H e r e we summarize the major conclusions. In the unstrained semiconductor, the fu!l 6 X 6 Luttinger-Kohn HamiItonian is given by (4.3.10)

and its full matrix form is shown in (4.3.14) and (4.3.15) explicitly in terms of the expressions P, Q, H , and S. The strained Hamil tonian introduces extra terms denoted by

due to the linear strain. , V i e , : h e r ~ . f ( j ~w~e, the correspondences between ) in H,>" and b 3i n ( Hi ) j j .

,

4-5

STRA!P.i.TZF";E(rJ'SO:'J 'r3ANl.l t<'T'f'.UC1'URES

For Valence Band

Therefore, P,, Q,, R,, and S, can be added to their corresponding strain counterparts P', Q E , R E ,and S, shown explicitly in Appendix C. The energy pararrleters a,, b , and d are called deformation potentials for the valence band and are tabulated in Appendix K for a few semiconductors. For Conductiorz Barzd (isotropic case)

and the conduction band-edge disper~jonis

The inverse effective mass tensor is diagonal (a = P ) in the principal axis system. The strained Hamiltonian has been used extensively in the study of the strain effects [26-381 on the band structures of semiconductors. 4.5.2 Band Structures Without the Spin-Orbit Split-OH Band Coupling Next we illustrate how the strain nlodifies the valence-band structures, including the band-edge energies and the efTective masses, which are among the most important parametel-s characterizing any semiconductor materials. For most 111-V serniconc!uct-,rs, t h e split-off bands are several hundred millielectron volts below the l~eavy-hole and light-hole bands. Since the eilerzy range of interest is only several ten:; uf n~illielzctronvolts, it is usual to :!ssume that the sl~iit-of!' band:; ca:: he igriorecl. 'The band structures of the heavy-hole and light--11c.11~ bti~ldsiirt: : i p p r - o x i m ; clesci-ibed by t h e 4 X 4

Harniltonian [37, 381:

The Hamiltonian in Eq. (C.24) in Appendix C is written for an arbitrary strain. For simplicity, we restrict ourselves to the special case of a biaxial strain, namely, &xx - Eyy

f

6 2 2

Ex), - E y z -- F , ,

=

0

Thus

which essentially covers two of the 'most important strained systems: (1) a strained-layer semiconductor pseudomorphical.ly grown on a (001)-oriented substrate and (2) a bulk semiconductor under an external uniaxial stress along the z direction. For the case of the lattice-mismatched strain, we obtain Ex,

-

-

a, - n

Eyy

- --

-

a

where a, and a are the lattice constants of the substrate and the layer material (Fig. 4.7), and C , , and C , , are the elastic stiffness constants. Equations (4.5.15a) and (4.5.1.5b) can be derived by.usjng the fact that in the plane of the heterojunction, the layered material is strained such that the

(a) Unstrained

(b) Strained (biaxial tension)

X

Figure 4.7. A layer materi:ll wiih a liltrice constnni ~i to be grown on a sirbstsate w i t h ;L i;:ttice col~stantL : , , . ( a ) unstrainzd; (b) s t r a i n e d .

:

lattice constant along the plane of the layer is equal to a[,. Therefore, - ( n o - n)/rt. Since the stre.ss tensor is related to strain by the c,y-v- E,.? elastic stiffness tensor with elements Cij,

we find 5, tion:

= T),, -

5,= 0. There should also be no stress in the z direc-

Therefore, E~~ = - ( 2 C 1 2 / C 1 1 ) ~ x rFor . the case of a n external uniaxial stress T along the z axis (7,: = T and T,, = T,, = O), we have

. '

The results and conclusions presented here can be generalized to other crystal orientations or stress directions. As an example, we discuss one of the most important systems: strained In,-,Ga,As on InP. High-quality and highly strained samples of this system have already been grown and widely studied for optoelectronics applications. All of the material parameters used are listed in Table 4.1, where rn; is the electron effective mass and nz, is the free-electron mass (also see Appendix Kl. The parameters for In,-,Ga,As are taken as the linear interpolation of those of ZnAs and GaAs, except that for the energy gap, E,(In, . G a , A s ) = 0.324 0 . 7 ~ 0.4x2. We have

+

-

+

:::here rr, = 5.8688A for the I n f s~iSstrate. 0.47: rr(i'1.468) = o , , arid the strain is zero. In- this case At .r == 0.463 in,,.-,.,Ga,,,,As is lattice ir~stcllzdtu InP. Whcn .r. > 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, <: 0 ) in front of the square root, because Q, can be c i t l ~ c r~legative!compr.essive strain) or positive uf a positive quantity is conventionally (tensile strain), while the square r~:~ot: taken as a positive. As k apprt:t;;cl~eszero, the band-edge energies of the helivy hoie and light holc in (4.5.19) s h ~ u l dbc rccavered. N'ote that for the

!

unstrained case, the following expressions,

give t h e heavy-hole and light-hole dispersion relations. The dispersion relations for the heavy-hole band EHH(k)and the light-hole band ELH(k)vs. the crystal growth direction k, and the parallel direction k , can be obtained analytically from (4.5.27~1)and (4.5.27b), respectively. AJong the parallel plane, e.g., the k , direction ( k , = k , = 01, we have for Q , < 0 (biaxial compression) and k , is finite

--

from (4.5.27a) and

for the light hole from (4.5.27b). Tn the case of biaxial tension (Q, > O), we havs

(4.5.29b) Again, at k , = 0, EHH(0)+ P, = - Q, < 0 and ~ ~ ~+ (PC 0= + ) Q, > 0 , which agree with (4.5.19). Along the k z direction: we obtain for both compression and tension,

The results of the valence-band structure for E k I Hand EL,+versus k, and k, for both compression and tensio~lare shown i n - Fig. 4.8 for Ga , I n l -,As grown on InP substrate. \We cari see clearly that the heavy-hole band has a lighter efiective mass than the light-hole band in the li, direction near k, = 0 for the compressioll case (( 0.468). For a finite and fixed strain, the small-k expansion of the above dispersion relation can be written as

where the transverse wave vector has a m;! itude k , (4.5.31) we immediately obtain the band-edge energies

ztnd the ef-fective masses parallel

(I( or

=

dk: + k : . From

t ) or perpendicular (1or z ) to the ,ry

plane

'I'hesir: arz the w ~ f 1 - l ~ ~ o wyes\:its r - l [31;]i,ll?.e_r; , th;: ~.r:upfille to the snin--cjrbit split-oft' band is negicctzd. .

15-1

' ~ ~ - I E : o R OF Y E.LI:C'TRC)NIC' F A N D STRUCTURES IN SEMlCONDUCTORS

(21)

COIvLPRESSION 4 x 1 > a,

(c) TENSION

(b) NO STRAIN

a(x)
/--

Eg (x)

Figure 4.8. T h e energy-band structure in the nlornentum space for a bulk Ga,ln,-,As malerial u n d e r (a) biaxial compression. (b) lattice-matched condition, and (c) biaxial tension for different G a mole fractions .r. T h e heavy-hole band is above t h e light-hole b a n d a n d its effective mass in the transverse plane (the k , o r k , direction) is lighter than t h a t of t h e light-hole band in the compressive strain case in (a). T h e light-hole band shifts above t h e heavy-hole b a n d in the case of tension in (c). (After R e f . 37.)

4.5.3 Band Structures of Strained Semiconductors With Spin-Orbit Split-Off Bands Coupling [351

If the coupling with the spin-orbit split-off bands is taken into account, the 6 x 6 Hamiltonian (C.24) in Appendix C has to be used. For the band edge at k = 0, the Hamiltonian H is simplified to

I

1 " ' I '

-

/

unstrained

-

/*

-

C-LH (SO)

"- E9( I n A s )

I

I

l

b

.

L

l

,

,

n

I

Figure 4.9. T h e energy b a n d grip of a bulk In,-,Ga,As vs. the Ga mole fraction x: -.-, unstrained ln,-,Ga,vAs; -, transition energies from the conduction band ( C ) t o the heavy-hole (F-IfiI) and light-hole (LH) bands for a bulk I n , -,Ga,As pseudomorpl~icullygrown on InP; ---, the conduction t o light-hole transition energy calculated without the spin-orbit (SO) split-off band coupling. (After Ref. 38.)

Clearly, the heavy-hole bands are decoupled from 2 rest of bands, while the f}) are coupled with the split-off bands (It, i)) light-hole bands . :1( '.. through the strain-dependent off-diagonal terms. This coupling would be totally trnaccounted for in the 4 x 4 approximation. For In, -,Ga,As on InP, the transition energies from the heavy-hole and the light-hole bands to the conduction band with and without the SO coupling are shown [38]in Fig. 4.9. The comparison demonstrates how important it is to include the spin-orbit split-off bands, because the error in the light-hole energies could be as large as several tens of millielectron volts. The error is comparable to the heavy-hole and light-hole energy splits and is certainly too large to be ignored. As a consequence of the coupling, the eigenvectors corresponding to the energy E(O)(=: E ,_,(O) or Es,(0)), determined by

+

are not a p1Lt.e ligtlt-hole cjr split-iE :;L:!tc, Itat art adi~:ix';!~tr< o f . the l i g h t - h r ~ ! ~

,

;.

and split-off states. The band-edge energies can be readily solved from (4.5.33) or (4.5.35a),

If the split-off bands are included in the 6 x 6 Hamiltonian, the E-k relation determined by (4.5.26) becomes a sixth-order polynon~ialof E, which apparently can be decomposed into two identical cubic polynomials because of the symmetry property of the Hamiltonian. However, an attempt to expand and factor directly the determinantal equation is tedious. T h e details are given in Ref. 38. The most important results are obtained from the series expansion of E u p to the second order of k near the band edges:

where f

+

and f'- are dimensionless, strain-dependent factors

-

where .r = Q,.,/h and I<' A,: + !;'. The hand-edge energies ( 4 . 5 . 3 7 ~a) r e given in (4.5.3:ia)--li.S.-3!;c!.

111

(4.5.37a)-

From (4.5.37a)-(4.5.37c) we obtain tlle effective nlasses perpendicular (I= z ) and parallel (11 I ) t o the x-JJ plane:

-

Note that f = 1, f - = 0 for the limiting case of zero strain ( Q , + 0, x + O), and the effective masses derived from the 6 X 6 Hamiltonian become iden.tical to those derived from the 4 x 4 Hamiltonian. +

-...

4.6 ELECTRONIC STATES IN AN ARBITFL4RY ONE-DIMENSIONAL POTENTIAL

In this section we show how the Schrodinger equation for a one-dimensional p,'tential profile with an arbitrary shape can be sol d using a propagationmatrix. approach, which is similar to that' used i,. electronlagnetic wave reflection or guidance in a multilayered medium [39, 401. An arbitrary profile, Lr(z), can always be approximated by a piecewise step profile, as shown in Fig. 4.10 as long as the original potential profile does not have singularities.

THEORY OF EL; JCTT
!58

4.6.1 Derivation of the Propagation Matrix Equation and Its Solution for t h e Eigenvalues

'

For a Schrodinger equation of the form

we find that in the region 1, z,-,

where V(z) t h e form

=

$[ (

6 and -.. = A,

r)

nz(z)

-

_(

z I z,

rrz, in region 1. The solution can be written in

e ' * , ( ~ - ~+OB, e-i"(z-a)

for z,-, I z 5 z I

(4.6.3)

where the wave number in region I is

biatching t h e .boundary conditions in which $ ( z ) and (l/l?t(z))d $ ( z ) / d z are continuous at z = z l , we find

Define

,

+

1. a n d r, , - z i = / r , + I s the. thicL:~:l;ss of thc rzgion I We can express A , , and H i in terms of A , and B, in a matrix form:

,

+

,

where we have defined a forward-propagation matrix

The reIation can propagate from layer to Iayer:

For bound-state solutions, we have E

< Vo and VN+,.Therefore,

The solutio~lsin region 0 and region N form

+ 1 must be decaying solutions of the

where A, = 0 and BAr+, = 0. Note that the last position z N + , is introduced for the use of the propagation matrix and it is arbitrary (can be sct equal to z,v )*

We write the product of the matrices

and therefore

For nontrivirii sulutions. w c nzl.ist ha.vs tI:t. eigcnequation satisfied,

since B, cannot be zera (otherwise, all field amplitudes are zero). Solving the is a eigenequation (4.6.13), we obtain the eigenvalues E . I n practice, f,,(E) complex function and the eigenvalues E are real. Therefore, a convenient method to find the eigenvalues is to search for the minima of 1 f,,(E) (or log1f2,(,Y)l) in the range of energy given by the lowest and highest energies of the potential profile V(z). b

4.6.2

Self-consistent Solution for a Modulation-Doped Quantum We11

Consider a quantum-well structure with a built-in potential

ancl a doping profile N D ( z )and

N,(Z).The Schrodinger equations are

where the total potential profiles for the electrons and holes are, respectively,

and the Hartree potential is given by

111 7 ) a s ill !4,6.14t)) such t h r ~ all~ energies are measured Note [.hat we deiine v/'( upward. Therefore, the s;lnl~; c:cp;-e:;sicns for th? ticld-induced potential and the Hartree potential L1j-C i~.;cJi n (3.6.15'1) o::td (3.6.16b) without a n y sign change. Here t.' is the c,u:c:-n:tlly applieii fizitl ( - O t1el.e) ar,d & ( z ) is the

4.1

EI,E.CTKC)NIC STA.TES IF:-OP;E-Dlil/l.3j,:SIC.,N.\I.. PdJ'illN'fIkL

161

electrostatic potential, which satisfi~sGauss's law or Poisson's equation: 7 '

-

(4.6.13)

(EE)=P(Z)

For a one-dimensional problem, we have the electric field

E = 2 E ( z ) and

E(z)

=

d4

--

dz

(4.6.19)

Therefore,

The charge distribution is given by

wh. -e N,*(z) and N;(z) are the ionized donor and z :ptor concentrations, respectively. The electron and hole concexltrations, n ( z ) anci p(z), are related to the wave functions of the nth conduction subband and the ?nth valence subband by

where the sums over n and nz are only over the (lowest few) occupied s u bbands. Here the surface electron concentration in the nth conductiod subband is

where

and

have been used. T h e surface hole concentration in the m t h valence subband is

- .,

where F, = F,, = E, in a modulation-doped sample without any external injection of carriers. T h e paraboIic energy dispersion reIations

have been used in the summation over the two-dimensional wave vector k in the X-y plane. The Fermi Ievel is obtained from the charge neutrality condition:

4.6.3

N-type Modulation-Doped Quantum Well

For an n-type modulation-doped quantum well (Fig. 4.11) we have A$ = O a n d p ( z ) -- 0. We o n l y have to solve for- t h e electron wave function f ( z ) and the electrostatic potential q!~(z self-consisten tIy. T h e electric freId E ( z ) is obtained by integration

(

I E

z

+

- I., ,/ 2

[ 41 - -

4.5

ELECTRONIC STATES 1J: ON~*Z-,~JkIEN310<\T1.2.2~ ;'u'J-'E!.T~'~AL

(c) E,(z) = V(z) = VH(z) + V{i(z) Figure 4.11.

(a) T h e built-in potential, (b) the charge density, and (c) the screened electron potential energy profile of an n-type modulation doped quantum well.

For a symmetrically doped quantum well without any external bias, we have E( - L / 2 ) = 0 by a symmetry consideration. Since VH(z) = - lel+(z), we have

V,(z)

=

l e l i ~ ( z 'd r) ' + VH(0) -L / 2

(4.6.28b)

Here V,,(U) can be chosen to be zero since it is only a reference potential energy. Exnmple We consider an n-type modulation-doped quantum well with doping widths L,/2 = 10 A at both ends (see Fig. 4-11), and ND = 4 x 10'" cm-' in a GaAs/Al, , G a , , A s quantum-well structure. The surface carrier -concentration in a peiiod is N,,L, = 8 X 10" cm -. The conduction bancledge discontinuity A E, is 251 meV and the built-in potential profile V&(z) is shown as the dashecl line in Fig. 4.12:~.'The final conduction band potential profile <,(z)= V,yi(z) -I- Vk{(Z)alter solving the Schrijdinger equation ancl the Poisson's ecluation self-consistently is shown as the solid curves together with the corresponding eigenenergies of the lowest two states, E; and E,, -3

164

THEORY 3 F ELEC7'KONlC,!L4ND S'I"K[IC'TU!iES I N SEMICONDUCTORS

- 5 0 t . s - - ' . . n . ' a . . . ' s m * n 1

-100

.-

-50

0 Position

(A)

Position

(A)

50

100

Figure 4.12. (a) Self-consistent potential profile (solid curve) Vc(z)and the built-in potential profile (dashed) V i i ( z )for a G ~ A S / A ~ ~ , ~ Gquantum ~ ~ , , A well S with modulation dopings within and 90 A 5 z _< 100 A. The two solid a t both ends of the profile - 100 A I z I -90 10 horizontal lines represent the energy levels E, and E2 of the self-consistent potential. (b) T h e corresponding wave functions f , ( z ) and fz(z)of the self-consistent potential in (a).

plotted as two horizontal solid lines. The wave functions f, ( z ) and f,( z ) for the self-consistent potential V , ( z ) are plotted in Fig. 4.12b. We can see that the effects of the charge distribution with positively ionized donors at the two ends and negative electrons in the wells create a net electric field pointing toward the center of the quantum well. Therefore, the band bending is curved upward since the slope of the potential energy profile gives the. electric field and its direction. More examples of the modulation-doped . potentials with an externa1ly applied electric field F and their applications to intersubband photodetectors and resoriafit tunneling diodes can be found in Refs. 41-43. B

(c)

EJz) = vii(z) + VH(z)

Ev(z)

4.6.4

Figure 4.13.

Same as Fig. 4.11 except lor p-type.

P-type Modula tion-Doped Quanturn well

Fo; :he hole distribution in a P-type modulation-dope,.. quantum well ( N L = 0, n ( z ) = 0), Fig. 4.13, it is also possible to take into consideration both the heavy-hole and the light-hole dispersion relations:.,

4.6.5

Populations by Both Efectrons and Holes

1x1 a laser structure: both electrorls and h(3lzs are injected by a n external electric or optical pumping. Two qur-tsi-Fermi levels, 4; and F,, can be used to describe r i ( z ) anti p ( z ) , respsctive!y. One .~isualiyuses the injection

current density to firld N,,then

is used to determine F,. T h e cliarge neutrality condition

is used to determine P,. Then Fu is determined from

4.7

KRONIG-PENNEY MODEL FOR A SUPERLATTICE

In this section, we apply the propagation matrix approach discussed in Section 4.6 to study a one-dimensional periodic potential problem [44]. T h e results are the fionig-Penney model for a superlattice structure [45-451. T h e superlattice structure is shown in Fig. 4.14. Each period (or cell) consists of one barrier region with a width b and a well region with a width w. T h e period is L b + w . T h e potential within a period 0 < z < L is given by

-

and V ( z + n L ) = V ( z )for any integer rz.

4 Period 1-*Period

2>3

Figure 4.14. A pericdic potential (Krni~ig-Penl.icyrnc;del) consists of' one barrier region with a thickness b ancl a q i ~ ; , ~ n t ~ ~ n ~~.dgic>rl - w c l l wi!h a wiclrh :.I.. The ,period is I, = rz: + b .

4.7.1

Derivation of the ]Propagation M 2 b . i ~

The wave function in the period rr is given by the plane wave solution of the wave equation: i k ( z - I I L ~+

B,, e--~ x ' z - J Z L ) for nl, - w Iz InL

e i k b [ z - b - ( ~~ 1 ) L ] + L)

r1

,-

ikl,[z-O--(n- I)[>]

for(zz - l ) L

Z I

5 (n - l ) L

+b

(4.7.2)

More specifically, we write

Using the boundary conditions in which tbC/(z)and (l/nt)(dt,b/a~) are continuous at r = 0, we find the matrix ecluation for the coefficients of the wave functions in region 0 and the barrier b:

where

Similarly, the boundary conditions at z 1-egion 1 and the barrier b

- b give the matrix equation for

162

TliEOTiY OF EI_E,CTRC;;GIC' E,A 1\11) STi{l;C"]'URES iK SEMICC)NDUCT(;~RS

where

Define

We have the transition matrix for one period consisting of one we11 and o n e barrier

--

where

a n d the matrix elements are given by

-

Note that the determinantof the

T matrix is unity: detlTl

=

I

If we continue t h e rclstion (4.7.8a) t o the n t h period, we find

Solutions fur thc ELilrig~nva1.tl.e~ and ESgenvertors

4.7.2

The eigenvalues and eigenvectol-s of the 2 x 2 matrix determinantal equation:

det

tll - t

t12

t21

t22

-

=

-

T are solutions of the

0

We obtain a second-order polynomial equation for the eigenvalue t:

-.-

Since detlTl

=

1, we obtain two roots:

,, +

?+I + or If J ( t t , , ) / 2 1 > 1, t + and t - are real and either lim,, ,It lim, -,It:/ -+ a. In that case, the condition in which the wavk function (i.e., IA,l and 1 B,I) must remain finite as n -+ fm is violated. Hence, the eigenva'tie E must satisfy the condition that

-

ltl1

I"

1 =Icos

kw cosk,b -

1

-2

sin kw sin k b h

c 1 (47.15)

The eigenvalues can be written in the forms t + = e igl,

since It +I written as

=

and

t-.--

--

i q i-

(4.7.16)

1. and t + t - = 1. That is, the deternlinantal equation can be

I

THEOR Y O F t1,ECl'liOT.i~L'HI'iP:C S7'RUt :TL) RES 1I.i SE~lICOi\!DLPC'TOI)RS

ill

\Ve conclude that the eigenequatio~:is 1

cos qIl,

=

cos kvv cos k,h

sin kw sin lc,b

- -

2

(4.7.15)

and the two eigellvectors

corresponding to eigenvalues, eiCf'and e - ' ' ~ ~respectively, , satisfy

-..

and

The eigenfunctions can b e obtained from the following:

I. The ratio A ; /B,f following (4.7.19a) and the normalization condition ~~-I~,!J(z)I' d z = 1,

+(P

-

sin qL

-

-

l / P ) s i n k,b eil'"'

sin kw-cos k h b

-

!(P

+

l / P ) c o s kw sin k b b

(4.7.20)

where the eigenequation (4.7.18) has been used. from (4.7.19b) with q -t - q in (4.7.20) and the 2. The ratio A,/B, normalization condition. k , is piirely imaginary:

For bound state solutior~s~ E' <. C:,

----

' 2}?1,:

k,,

= icu,,

@/,

=

f -~

i b ' -~ b, )

~1.7 MI:@N.rG--jt'Erql"l":E\rMODE:> FOR

.P,

SIJPERI_ATT;('k;

1r.r

The determinanta! equation is given by

cos qL where

1

=

sin k w sin11 a , b (4.7.22)

cos kpv cosh a,b -!- -

2

is defined in the fol1owin.g equation when P is purely imaginary:

In summaiy, the Kronig-Penney model for a superlattice can be obtained from solving the determinantat equation cos qL

=f

(E)

(4.7.24a)

where the eigenecluatiw is defined as cos krv cosh cu, b

f(E)

=

1

+ --2 1

cos kw cos k,b - 2

sin kw sinh a, b sill kw sin k , b

A method -I find the solution E is to plot f ( E ) vs. E 2 0. 2 2 region of E such that the condition ( f ( E ) (I1 is satisfied will be acceptable. The cluantum number q is obtained from

where f( E ) is real. For a given E in the acceptable. region (called the miniband), the solution q is used to find the eigenfunction from the ratio A: /B,T(or 4., / B ; ) and the normalization condition of the wave function. GaAs/Al, Gn,.7 As Sirperlnttice We consider a GaAs/ Al,Ga,-.,As superlatticd with x = 0.3, a well width w = 100 A, and a barrier width b = 20 A. The barrier height V, = 0.2506 eV is obtained from the conduction band discontinuity V,, = A E,. = 0.67A E,(+), A E,(x) = 1 . 2 4 7 ~ (eV). LVc see in Fig. 3.15 that. therc arc t-kvu minibands corresponding to the bound statcs ( E < Z<,) of the supcrlattice with I f ( E ) JI 1. The vertical asis is f'( E i and the horizontal axis is the electron energy E in rnillielectrvn volts. Since cosha,b '1 at krr =-= Nn-.where hi is an integer,-we obtain j(E ) == coshc~,h 2 1 at k !V~-j"r.. 'Tilcrcfore, !1: - N . j r / w alwzys occurs either in the forbidden riliir:ib;.l.i;c.l! yap or at th= ec'ig~: of the m i ~ i b a n d . Exnrnple: A

-

.

l . . . . l . . , . l . . . . ~ . . , . I . . . . l . . . . I . . . . I . , . . l

0

50

100

150 200 250 300 Energy (meV)

350

400

Figure 4.15. A plot of the eigenequation f ( E ) ( = cos (7L) from (4.7.24b) VS. tile energy E . Only the miniband ranges such t h a t - 1 5 f ( E ) I 1. a r e acceptable solutions. T h e plot is for the conduction minibands of a GaAs/Alo,,Ga,.,As superlattice with a well width ~v = 100 A and a

barrier width b = 20

A. ?'he

barrier height is 250.6 rneVT'

In Fig. 4.160 we plot the energy spectrum for the electron minibands for a superlattice with a well width w = 100 A and the barrier width b varying from 1 to 80 A. We see that the width of the miniband energy decreases as the barrier width is increased because the coupling among wells becomes weaker f'or a thicker barrier. T h e heavy-hole and light-I:.;:le minib~mdsfor the same GaAs/A.lo,,Gao,,As superlattice are shown in Figs. 4.16b and c, respectively. We use A E , = 0.33 AE,, and rnk, = r n 0 / ( y , - 2y,), m;, = m 0 / ( y , + 2y,), where y , = 6.55, and y, = 2.10 for GaAs, y , = 3.45, and y, = 0.63 for AlAs. For A1,rGa,-,r As, we use y , ( x ) = 3 . 4 5 ~+ 6.55(1 - x ) and y , ( x ) = 0 . 6 5 ~+ 2.10(1 - x ) . More discussions on semiconductor superlattices using the Kroniy-Pknny model can be found in Refs. 44-50. I 1

4.7.3 ,'

..

Extension to an Arbitrary Periodic Protile

For a periodic profile with the potential f~lnctionV ( r )within one period of arbitraiy shape (Fig. 4.17), we proceed as follows. W e divide a period L into N regions with a width Az = L/N,and assign C,, and D,, for the forward and the backward propagating waves. Here

-

where the matrices F(,.,

,:

;ire obtr~ifie~t f r o m (4.6.7b). Since the prcdile is

1 " * ' 1 " ' . ' 1 " " 1 ' " .

Conduction minibnnds

-

0 0

10

20 30 40 50 60 Barrier width (A)

70

80

Heavy-hole minibands

HH3

0

10

20 30 40 50 60 Barrier width (A)

70

'3

Light-hole minibands

I

LH3

LH2

d

-

-

0

10

20 30 40 50 60 Barrier width (A)

70

SO

-

of a CaAs,/.A:,,,.1<j~1,,,7;\s su~rer1;itticewilh a well width 1.i 100 A Figure 4.16. The n~iflihai~cls arc plotted \IS. [he barrier wid111 h. WI: s t l i , ~ \ ~( a ) the cc~ilclu<~ion n~inib;lrit.!s, (13) the heavy-hu!i: r?liniband:;, and (c) the light-l~oli.rnir,ih:inds.

I

l

l

1

>z

1

0123

One period

z=L

z=o Figure 4.17.

A periodic potential profile.

chosen such that the 0th region is identical to the N t h region, we have

where

and A , = C,, B , = D N , A, = C,, and B, = D o have been used (comparing Fig. 4.17 with Fig. 4.14). The rest of the steps are the same as those for the Kronig-Penney model, . that is, we solve the eigenvalue equation

and the eigenvalues have the form

and

For a periodic potential with

=

yV, :hat

i:;. the period L consists of N

subdivisions with Az

=

L / N , we expect

which can be proved from a more general argument from the time-reversal of a real potential profile V ( i ) . -It can be readily explained that if condition detlT1 f 1, the infinite periodic potential will lead to limn ,,It:) - + m, or -,It - + m, which are not allowed solutions. lim,, ,

4.5

BAND STRUCTURES OF SEMICONDUCTOR QUANTUM WELLS

Ip Section 4.4, we discussed the effective mass theory for a single band and degenerate bands. In this section, we study the calculation of the band structures of semiconductor quantum wells. Many papers [51-671 on the recent development of the effective mass theory or the k - p theory for quantum-we11 structures have been published. Theoretical methods such as the tight-binding [68] and the bond-orbital models [69, 701 have also been introduced. Here we focus on the k - p method of the Luttinger-Kohn Hamiltonian together with the strain terms of Pikus and Bir, as discussed in Sections 4.4 and 4.5. 4.8.1

Cr.nductionBand

The effective mass theory for the conduction band is obtained frorn the dispersion relation

where the effective-mass of the electron in the conduction band is rrz* = m,* in the barrier region and nz* = nz*, in tfie quantum well. In the presence of the quantum-well potential,

where the energies are all rncasur-ec'r frorn the cond~lctionband edge. T h s

176

I'HEOKY 0.:ELECTT:..(?NIC : l3Ai.il.l S1-RIJCTL'P.I2S Il'i SEMICONDUCTORS

effective mass equation (4.4.5) for a single band is

where ( l / n t ) ( d / d z ) appears inside d / d z current density

to ensure that the probabiIity

is continuous at the heterojunction. In general, the wave function $ ( r ) can be written in the form

and I

T h e eigenvalue and the eigenfunction are obtained from the above equation, foIlowing the procedures discussed in Chapter 3. Here we ignore the k t dependence of $ ( z ) . Equation (4.8.5) is usually solved at k ! = 0 for the n t h subbancl znergy E,,(O) with a wave function $ ( z ) = f n ( ; . Then we have E n ( k , ) = E,,(O) f ~ ~ k : / 2 r n , .

+

4.8.2

Valence Band

Band-Edge Energy. F o r a given cpantum-well potential,

let us find the band-edge energy at k , = 0 first. The Luttinger-Kohn Hamiltnnian (4.3.14) or (4.5.13) is diagonal for k,= k,, = 0:

Define

Since the parameters y , and y 2 in the well are different from those in the barrier regions, we solve

where ( m )= (hhm) or (Ihm). Therefore, the band-edge energies for the hhm, or lhm subbands can all be found from the equation identical to that for the pa~mabolicband model. We only have tc! use the appropriate effective masses (4.8.83) and (4.8.8b) in the corresponding regions.

50 100 Well width

0

4.1.

.

.

%;Y.ei! width

200

1 SO

2G0

(A)

f (39

50

0

150

(A)

( a ) IJu;~ut!.i~11~-\?:~:11 pr;)tii.::s i.rii. !I-:. ~','tilcai.:c::r:!i ;:nr.l ~ : i l t t l l ~ t b:11<1~ ' of ;1 GaA:r!~~t:~-i!:rns::bi>:~r:ri ~.;:.::-:.:ic::. L:(-,.. . , and ( c ) valt.n~.t:subbanrl ... energies I?',,;.~,. . . . ., ; ~ i ? i l L , ~ ~F--I. { : :..,: . . . ::<. i he x v r l ! t . y i ~ i i h L d l t i

I

,

Exnrnple As an example, LVC consider a GaAs/Al,,,Ga,,,,As cluantum we!l as shown in Fig. 4.18a. T h e energies for the electron subbands (Fig. 4.18b), and for the heavy-hole and light-holc subbands (Fig. 4 . 1 8 ~ )are plotted vs. the well width L,. W e can see that Fol- thc magnitudes all energy levels decrease C! with an increasing well width. Valence Subbands Dispersion Relations. The effective mass equation for four degenerate valence bands (two heavy-hole and two light-hole bands) follows Eq. (4.4.11) for a quantum-well potential V,,(z) given by (4.5.6):

where EL,^ is from the first 4 x 4 portion of (4.3.14) or (4.5.13). and the enve,lope functions F,, F,, F3, and .F; can be written in the vector form

The wavz function in component form is expressed as

where

Y =

We write

3 1 3, 2,

- 21 , and - +. Denote

Theoretically, a model for a clriantum weiI with ,infiliite barriers has been rrsed and Inally of the results can t ~ cexpressed in analytical fornls [60, 63, 64, 711.

4.$.3 Direct Experimental Meassfre~nentsof t h e Subband Dispersions Experimentally, low-temperature magnetoluminescent ineasurement [72], resonant magnetotunneling spectroscopy [73] and photoluminescent measurements of hot electrons recombining at neutral acceptors [74, 751 have been used to map out t h e hole subband dispersion curves. In Fig. 4.19, we

QUANTUM w ELLS VALENCE BAND DISPERSIONS

GOAS/AIGOAS

- 100 I

I

( [ I d

I

-

-

-

-

-

75A QW

- (c) I

I

I

I 4 . . Espil.ir!l~l~i.al d;lta (clots) aricl :I?<~>I.~II!.*::! ca!i:ill:ttic!r?~ r.--,,. wnlvevector along the wavevector nlnng the i l 101 (.lir-ect;ciil?for :I!? hule suh0ancl energies of [ I 001 directic,n: ( ; ; ~ A S / A ~ , ~.-.,As G ~ , qiranttim ivellr; for i n ) 3:; A c;ii:?ni.c!~-nwcli> ;
-

-

show the experimental results (d,tta points) of Ref. 75 compared with the theoretical calculations (solid curves [ l O O ] direction and dashed curves [I101 direction). The well width L,, , the aluminum mole fraction x in the barriers, and the p-type doping concentration J in the wells are (a) L,, = 33 A, s = 0.325, Nd4 = 1 0 ' ~ (b) I d , , = 51 A, x = 0.315, N, = 10" cm-), (c) L , = 75 A, x = 0.32, NA = 2 X 10" ~ m - and ~ , (d) L , = 9 8 A , x = 0.38, N, = 3 x 1017 cmP3. Each well is Be-doped in the center 10 A except for the 98 A wells (doped in the center 35 A) at the above indicated concentration NA. It is noted that these nonparabolic subband dispersion curves exhibit the valence band mixing behavior in quantum wells. We can see that the coupling between the heavy-hole H H 1 and the light LH1 subbands leads to warping (or nonparabolicity) of the band structure. These valence band mixing effects cause the wave function to exhibit both the heavy-hole and the light-hole characteristics. Here the labeling of H H and LH is according to their wave functions at k , = k , = 0, since I;, f +) are assigned to that of the 11e;lvy hole, and it, f f) are assigned to that of the -light hole. \

4.8.4

Block Diagonalization of the Luttinger-Kohn Hamiltonian

In this section, we consider a transformation that bIock diagonalizes [37, 54, 551 the 4 x 4 Hamiltonian (4.5.131, which includes the strain effects. Define the phases 8, and Os of R and S by

x 4 Hamiltonian in (4.5.13) can be transformed to two 2 The 4 nians f i u and E'

X

2 Hamilto-

where

and the transformiltion bztufeen the old bases n e w bases is

,;I

v)(v

=

+ 33 , i :) 1

and the

43

EAPqiJ.lS'TF
<)L,ANrI'U'G \VE LLS

LZi

'

where

and the transformation matrix -

U

*:.[

TF 0

0

a"

=

-a

1

(4.8.20)

iI !

'! i

/1

In general, k , + - i d / d i becomes an operator in (4.5.20), which is difficult to treat. For special cases such as an-external stress T11[100], [OOi], [110]? E~~ - E~~ = 0, HS is independent of kz. Therefore, the Hamiltonian can be applied to the quantum-well problem directly. For the case in which an external stress T is applied along the [I101 direction or the case in which strain is caused by the lattice-mismatch accommodated elastic strain

we have

9,

=

-4

., :

where C$

3.8.5

=

tan-'(kV/k,).

Axial Approximation for the Luttinger-Kohn Harniitonian

If we approximate in the R , term

Rk

--

-

--

( k , - i k , ) -7 + Y z - 7'3 --(k-\. - i k , ) = ]

2

!

(I

I

we find that the energy subband dispersion relation is independent of the angle 4 because

and the Hamiltonian depends only on the magnitude of the vector k , . This is called the axial approximation. Note that in this approximation, we assume that y , -- y , in the R, term only, while we still use y , and y , in the other terms. We obtain

and

4.8.6 Numerical Approach for the Solutions of the Upper 2 x 2 HamiItonian under Axial Approximation Let us look itt the upper 2 X 2 Hamiltonian in (4.8.16). T h e wa1.c' function for the hole subbands can be written generally as

This wave function satisfies the Hamiltonian equation

and R are all dii!erir!ti:!l operators i ~ n dcan be obtained from where P , 0, ( C . 2 3 and (4.8.25) with k; replaced by - L(?/dz). These wave functions g'" and g('' depend on t h e rni~gnitudi:of thr. LY;LV< vector k : and position 2 , a n d

are independent of the direction o f the wave vector (or the angle us):

The built-in quantum-well potential for the holes Vh(z) has been incorporated into the diagonal terms of the Hamiltonian. Since V/,(z) is a stepwise potential, boundary conditions between the well and the barrier interface have to be used properly. The basic idea is that the envelope function and the probability current density across the hcterojunction should be continuous. Therefore, to ensure the Hermitian property of H, we have to write all operators of the form

and

We then have the following boundary conditions:

I:[

=

continuous

?dote th;it- the l,uttinger prirarne:zrs y , , y,, anci y , in ':he corxsponding region have t~ be used.

Eq. (4.3.25) can be solved using a propagation-matrix niethocl [37],which is very efficient, or using a finite-cliffercnce n ~ c t h o d[76]. The solution to the Schrodinger equation in matrix f ~ r m(4.8.38) will be a set of subband

where the subband index rn refers to HH1, H H 2 , . . . and LH1, LH2,. . subbands. The superscript U refers to the upper Hamiltonian. 4.8.7 Solutions for the Lower 2

X

2 Harniltonian under Axial Appropimation

Similar procedures hold for the Iower 2 wave function is

X

2 Hamiltonian of (4.8.16). The

which satisfies the Hamil tonian equation

The solution will be a set of dispersion curves labeled as m in E k ( k , ). For a symmetrical potential, we find that E;(k,) of the lower Hamiltonian is degenerate with that of the upper Hamiltonian ~ L ( k , ' for l each subband m. The wave functions g(3'(k,, z ) and g(')(k,, z ) can also be reIated to g")(k,, z ) and g ( ' ) ( k , , z ) , respectively, if we change z --+ -r.

,,,

Example The valence subband structures for a GaAs/Al ,,,Ga As quantum well with L , = 100 A and L , = 50 A are calculated using the propagation matrix method in Ref. [37] and are plotted in Figs. 4.20a and b, respectively. The horizontal wave vector k i is nor~nalizedby 2?i-/a, where ( I = 5.6533 A is the lattice conslant of GaAs. These subband dispersions can also be compared with the experimental data $01. similzir GaAs quantum well structures with y-type dopings in Figs. 4.19d and b, respectively. The axial approximation gives very good r.esu.lts for ;.I small k , , and the results are exactly the same a s thosc using the (.)[.igin;~l4 X/ 4. Hamiltonian a t k , = 0. 0

S ~ ~ . I I ( : ~ ~ I ~ D QUANTUM ~ C T Q I ~ Wl?I..L'i 185

Figure 4.20. T h e valence subbands for (a) a 100-A and (b) a 50-A GaAs/AI,,Ga,, As quantum well for k , in the plane of the quantum well under the axial approximation. The vertical axis is the halt. s u h b a r ~ denergy ad! the horizontal axis is the in-plane wave vector k, normalized by ( 2 ~ - / u )where , ri = 5.6533 A is the lattice constant of GaAs. These results in (a) and (b) can bt: compared with the experimental data for similar GaAs quantum-well structures in Figs. 4.19d and b, respectively.

This approximation is attractive, since the q!! dependence in the k , - k, plane can be .-).ken into account analytically in the basis functic ;, and the valence subba~idenergies are independent of 4. These wave func~ionscan be applied to study optical absorption and gain in quantum-well structures. ..-

a

4.9 BAND STRUCTURES OF STRAINED SEhIICONDUCTOR QIJANTUM WELLS /

For a strained quantum well, the conduction band edge is given by

E , ( k = 0)

-

=

n , ~ r ( Z )= n , ( ~ , ,

+ s,, + e,,)

(4.9.1)

EYY ( u a -- G ) / " and P -. ... =. - , ~ ( c ~ ,,K. ~ /Here c ~a ;, is) ~the of the subsrratc, and a is ttlc lattice constant of the quantum cr;>nc;ta~it well. The energy is rne;~stl~-i_.d frcin tile conduction Sand edge of an unstrained quantum well, as :;~IOLL'II in Fig. 4.3i. Vt'e nssume th.at the barriers are lattice matched to the xnt.~:;ira.~e; therefcre, their strains are zero. For an unstrainecl clu;intum wel:, VJC have th.: vat-enti31 ectlrgy,Icor the eI,ectxon,given

.,.,-

.-

(a)

( c ) Tensile

(b) Zero strain

Compressive strain

Figure 4.21. Band-edge profile for (a) a compressive strained, tensile strained q u a n t u m well.

(13)

swain

a n unstrained, a n d (c) a

For example, E,(ln,--,Ga,As) = 0.324 + 0.71 + 0.4x2 for an unstrained In, -,Ga,As compound, and E,(lnP) = 1.344 eV at room temperature:

AE, n(x)

=

1.344 - E,(In,-,Ga,As)

= .ra(GaAs)

+ (I

-

(4.9.3)

x)n(InAs)

From previous experimental data, we take

At x = 0.468, In,-,Ga,As exists when x # 0.465. 4.9.1

AE,

=

0.4 AE,jx)

AE,.

=

0.6 AE,(x)

is lattice matched to the InP substrate. Strain

Subband Energies in a Strained Quantum Well

The conduction band-edge energy for a strained quantum well is given by

Si~nllarly,the valence band-edge energy for the tinstrained cluantum well is defined as

The valence band-edge energies f'or the strained quantum well are obtained in (4.5.36) ignoring the coupling with the spin-orbit from EHH(0)and ELkf(0) split-off band.

where

,.'

The subband edge energies can all be obtained from a simple quantum-well model using the band-edge effective masses:

where the subscripts w a x d h in the effective mass parameters designate the well and the barrier region, respectively. If the spin-orbit split-off band is included, we find that the band-edge energies for heavy hole E,,,,(z) and mi, are not affected, but

and

where f+ is defined in (4.5.38). 4.9.2 Valence Subband Energy Dispersions in a Strained Quantum Well.

Using H from (4: .13), we can solve the valence subband energi-- . and eigenfunctions using (4.5.14):

wherea"v,(z) is given by the unstrained (built-in) potential E ; ( z ) from (4.9.6), and the band-edge shifts due to -PC and Q , have all been included in for (21 5 Ll,,/2, and bcth P',and Q p are zero for l z l > 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:
0

., 0.02

0.04

0.06

x Figure 4.22.

(Con[i)zued)

I

PROBLEMS 4.1

Derive Equations (4.1.6) and (4.1.7) for the wave function u,,(r) in the k . p theory using (4.1.4) for $,,(r).

4.2

Derive (4.1.8) .using the perturbation theory in Section 3.5.

4.3

(a) Show that the eigenvalues of the cleterminantal equation (4.1.16) i11.a two-band nloilei are

one is the conduction band heavy hcjle in the valence band (:I = C ) and the other is tb(: ( T I ' = u). We have k , == 0 for a dircct band-gap material, E,, E,, and Ei, = EL,.Show tllar the equation

($? Consider the two-band modci, whcre

-

leads to

-

-

where E,., 0, E,, = E, the energy gap. (c) Show that if we use the perturbation theory directly near each band for (4.1.15),

E,,

=

E,(O)

+ HI;,, 4~ 7 it '

EtI(O)-- Enl(0)

rr

t,?k?_ fl 2 . -b E,, t -. ., lk . p , , ~ ~ ' for the conduction band 1 E,m, ,

I 4

f12k2

-2 ~ 1 -',

fi 2 .1

Jk - p c L r E,,rzt;

for the valence band

and

Check \vhctl:,;:y i:hc: e,i.::-gj::*.; ~!:-:i,;;g ti:? pcri::-lrb:c!iorl thcol-y are the ...-, . same a:., fhijstr: ctbi:~.lis(:r i 1;:: :1.1,;.: .; i j \ i i : : j i : . . . ~ : , * ; . i i : ~ espnnsion of ths rc!jsi<)fisi;; p:lrl ,i' .?:;). :::;sL!!'!-:,~?-LP - '(;it!< ' I\-:.? ( k . P ,,,,) term is sma!.!. "

-

4.4

a

?'he matrix in (4.2.5) ce:1 be equivalently obtained from t h e symmetry properties of a cubic crystal using the wave functions IS) = S(r.), IX) = x f ( r ) , I Y) = y f o - 1 and 1%) = z f ( r ) , and the properties of the operators p = ( A / i ) V , V = x^(d/ax) + ? ( d / d y ) + z^(d/dz). For example,

+ -k ~

-H,, = (is J, IHo

-

- p

o

+

..

o -V V X pliSJ,)

4m;c2

where only the first tcrm contributes ( E s ) , since the other two terms vanish.

and

vv X p = O-<(VVx p ) , + u y ( V V x p),. + u z ( V V x

0'

Noting that

0,.ancl

p),

(3)

u,,change the spins from (4.2.4), we find

< L lcJ-rl.L) = (.Lluyl.L~,= 0

(4)

The last term i l l H,, is zero since it changes a sign if we exchange x with y. However, the symmetry of the crystal requires that the Hamiltonian matrix element should be the same with respect to the exchange of x with y. Thus, that term should vanish. (a) Following the above procedure, show that

v-l11el-e P is ileiinccl in (4.2.9). T h e last tcn71 \:itnisilrs ,by u s i r ! ~;he symmetry prope rti.::; (3f t h e :;i;tvil: I'Lnctions.'

(b) Show that

where the second term in ( 6 ) vanishes, and the last term can be expanded into four terms ((XI... IX), (YI - - - IY), ( X I - . . I - iY), and ( - iYI I X ) , of which the first two vanish. ( e ) Derive the full m i t r k % in (4.2.8) for Kane's model. .-•

4.5

I

I

N is obtained from the normalization condition

-

(b) Show that in the limit k 0, the above eigenvectors lead to the wave f~lnctionsin Eqs. (4.2.21)-(4.2.24). 4.G

Derive Eq. (4.2.26) fiir Kane's paramettr 1'.

-1.7

Tabulattr :.ill t l ~ t . t~ani!..-;.i"lg i:iii?:'gii.s and tl-ltlii- corresponding bvave , functiorls in Kane's !-rio.gc:! ;..>t. 1 i>.: .;..,l;dac:tir-)i--1?heavy-hole, light-hol~, and spin-orbit split-uiF l!i!~!d:3. r

"

!

I I

(a) Show that the eigenvector for (4.2.19) can be written as

where

I I

'THEORY .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

-

.

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24.

25. 2(i.

27,

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30. 31.

32. 33.

34.

'55. Sf:).

.;7.

36.

39. ,413.

-I.:.

.

(.\I.,

42.

;<,

T. Krn. :<.s. Lee.

i710di1lati0;1-.d~.)pc1J, hhl'i-602.4 ( i 9 9 1 ).

;!.:;(-; 3 .

r (.>.

,,

.-.

;

9

, i:L:.;i~z,~

\.: . '

1 . 5

~ ~ ~ : ~ . . 2 ~ - ; T ~ ~ i ,_,plicni ~ - ~ i ~ . i ahst):-piion ~~~~~ j 11 :I . . ! . 1 . .p/~y.<. 69, <-.-

43. F. Stern and S. Das Sarmn, "Electr:)n

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59. C. Mailhiot and D. L. Smith, "k . p theory of semiconductor superlattice electronic structure, 11: ,4pplicatio11 to G a , -,As-Al, -,,In ,.As[100] superlattices," Yhys. Rc[?.B 33, 8360-5372 (1986). 6 0 . L. C . Andreani. A. Pascl~iareiio,and F. Bassani, - ' H o l e subbands in strained GaAs-Ga, -,Al, As q u a n t ~ r nvvells: Exact solk:rion of t h e effective-mass equation," Wrys. Rc.~l. B 3$, 5587--5594 19S7). b i . L. R . Ram-Mohan, I.;. F1. Y.uj, and R. L. Agg:liwal, “'Transfer.-m2tri.x algorithm for the calculation of thc band structure of' ';crnicood~:ctct- superlattices," Pllys. Re/!. B 35, 6!52-6159 (198s).

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62. G . Y. \Vu, T. C.'. McGill, C:. Mailfiiot, and D. L. Smith, " k - p theory of seniiconductor supcrlatlice c!cotronic slruciui-e in' an appIied mr;gnetic fielij," Pltys. Rer.. B 39, GOGC)-Ci07C! ( 19>;?). 63. B. K. Ridley, "The in-plane efrective mass in strained-layer quantunl wells," J. Appl. Plzys. 68, 4667-4673 (1.900). 64. I . Sueinune, "Band-edge hole mass in strained quantum-well structures," Phys. Re!.-. B 43, 14099--14106 (1991). 65. M. Silver, W. Batty, A. Ghiti, and E. P. O'Reilly, "Sti-ain-induced valence-subband splitting in Ill-V semiconductors," Phys. Rec. 3 46, 6781-6788 (1992). 66. T. Manku and A. Nathan, "Valence energy-band structure for strained group-IV semiconductors," J. App1. Phys. 73, 1205- 1213 (1993). 67. D. Munzar, "Heavy hole-light splitting in GaAs/AIAs superlaltices," Plzys. Stlitris Solicli B 175, 305-401 (1993). 68. J. N . ~chu1manand Y. C. Chang, "Band mixin2 in semiconductor superlattices," Phys. Reu. 3 31, 2056-2068 (1985). 69. Y. C . Chang, "Boncl--orbital niodcls for superlattices," Phys. Reu. B 37, 8215-8222 (1988). -.70. M. P. EIuung, Y. C. Chang, and W. I. \Vang, "Orientation dependence of valence-subband structnres in GaAs-Ga, -,A], As quantuni-well structures," J. Appl. Phys. 64, 4609-461 3 (1988). 71. S. S. Nedorezov, "Space quantization in semiconductor films," Sou. Phys.-Solid Stnte 12, 1814-1819 (1971). 72. E. D. Jones, S. K. Lyo, 1. J. Fritz? J. F. Klem, J. E. Schirber, C. P. Tigges, and T. J. Drummond, "Deterrnin;~.tion of energy-band dispersion curves in strained-layer structures," App1. Phys. Lerr. 54: 2227--2229 (1989). 73. R. K. Hayden, L. Eave M. Henini, T. Takamasu, N. Miura, and U. Ekenbel., "Investigating the cubic anisotropy of the confined hole subbands of an AlAs/ GaAs/AlAs quantum well using resonant nlagnetotunneIing spectroscopy," Appl. Phys. Lett. 61. 84-86 (1992). 74. J . A. Kash, M. ~ a c h a s ,l;nd M. A. Tischer, "Anisotropic valence bands in quantum wells: Quantitative comparison of theory and experiment," Phys. Rec. Lett. 69, 2260-2263 (1992). 75. J. A. Kash, M. Zackau, hi. A. Tischlet, and U. Ekeilberg, "Optical measurements of warped valetlce bands in quantum wells," Surf'. Sci. 305, 251-255 (1994). 76. D. Ahn, S. L,. Chuang, and Y. C. Chang, "Valence-band mixing effects on the gain and the retractive index ch;inge of quantum-well lascrs," J. Appl. Phj~s.64, 4056-4064 (1988).

Electromagnetics In this chapter, we discuss a few important results using electromagnetic theory. In Section 5.1 we present the general solutions to Maxwell's equations for a given current density J and a charge density p which satisfy t h e continuity equation. We discuss the gauge transformation including the b r e n t z gauge and the Coulomb gauge; the latter will be used in the HamiItonian to account for the interaction between the electrons and the photon field in semiconductors in Chapter 9. The time-harmonic fields and the duality principle, which will be useful in studying the propagation of light in waveguides and laser cavities, are presented in Section 5.2. The duality principle allows us to obtain the guided wave solutions for t h e transverse magnetic (TM) polarization of light once we obtain the solutions for the transverse electric (TE) polarization of light, as we show in Chapter 7. The plane wave reflection from a layered medium is investigated in Section 5.3 for both TE and T M polarizations. These results are useful for understanding the ray-optics approach to the optical waveguide theory. T1. radiation of the electromagnetic tield is pi-esented in Section 5.4 with the aini that the far-field pattern from a diode laser and laser arrays will be derived once we know the laser mode on the facet of the laser cavity. In Appendix E, we discuss the EC1-arners-Kronig relation, which relates the real part of a permittivity function to its inlagiriary part. This is useful in optical materials because often the absorption coefficients of the semiconductors a r e measured, then the real parts of the refractive indices are calculated based o n the Kramers-Kronig relation. T h e Poyntjng's theorem for power conservation and the reciprocity theorem are included in Appendix F.

5.1 GENERAL SOLUTIONS TO ;MAXWELL'S EQUATIONS AND GAUGE TRANSFORMATlONS [11 In this section, we study the general solution5 to Maxwell's equations:

where the sou;-ce terms 1, :mcl .J satisfy the continuity equation

Since

for any vector A, we can write B in the form

However, this equation does not uniquely specify A. If we add V t to A, where 6 is an arbitrary function, we find that

still satisfies V x A' = H . From the fundamental theorem of vector analysis, to uniquely define a vector A, we have to specify both its cur1 and its divergence. Substituting (5.1.7) into (5.1 .I), we obtain

Since O x ( - V + )

-

0 for any ftlnction #, we may write

Assuming an isotropic honlogeneous medium

and substituting the expressions for B and E from (5.1.7) and (5.1.10) into (5-1.21, we find

has been used. Gauss's I a v ~for the electric field (5.1.3) gives

Thus, generally speaking, we should find the solutions for the scalar potential 4 and the vector potential A from (5.1.13) and (5.1.15) in terms of the sources p and J. We still have to choose a gauge to uniquely specify the potentials 4 and A. Since for any 4 and A satisfying (5.1.7) and (5.1.10), the following gauge transformations

atso satisfij (5.1.7) and (5.1.10), where ( is an arbitrary function. Thus, we have to specify V . A. There are two common ways to specify V - A: the Lorentz gauge and the Coulonlb gauge. Lorentz Gauge. In the Lorentz gauge, we choose

We then have

That is, the vector and scalar potentials satisfy the wave eq~rationswith sources J and p, respectively. The solutions can be written generally as

where the time delay due arguments o f the

t o pi.opag:iti,;n. Ir. --

S O I I I . C ~[ ~ T I I I S .

~ ' ,/c., 1 app9al.s explicitly in the

C o n l ~ ~ mGange b [ 2 ] . In the Coulcn~bgatlee, we choose

Therefore,

which is the Poisson's equation. The solution is

The solution for A has to be found from (5.1.3.3) and (5.1.23). For an optical field, p = 0, we also find 4 = 0 and E = - d A / d t . -..

5.2

TIME-HARMONIC FIELDS AND DUALITY PRINCIPLE [I]

Often, the excitation sources J and p depend on time sinusoidally. We may write the source terms as

and similar expressions i,.tr the fields E, H, B, and D, where Re [ ] means ti-.,. real part of the function in the bracket. For example, an electric field in the time domain of the form

can hz expressed in terms of a phasor E(r, w ) 01 simply E(r):

Vv'e then obtain axwe well's e(1u;rtions in the frequency domain:

where all quantities E, f4, 8, D, p . and J are i n phasor representations and their depcndencies on the angular frzqucncy o are implicitly assumed. The relation of D(r, t ) to E(r, t ) is

In he frequency domain, we have

In a source free region, we find

If we make the following chitllges .I

E-H

H+-E

p

4

~t ' + p

(5.2.13)

then Maxwell's equations stay unchanged. Thus, for a given solution (E, H) in a medium (,u, E ) , we have a corresponding s o l u t i o ~( ~H , - E ) in a rnedi~im (E,p). This is the duality principle. Since the b o u n d n ~ yconditions in which the tangential components of the electric and magnetic fields are continuous across the dielectric interfaces and stay the same in both media, we have a dual situation such that the simultancous replacements in (5.2.13) give the soIutions to Maxwell's ecluations .as well. If including both electric and mngne tic sources, we write

Once we make the following identification

we find that Wlaxwell's equations are invariant. As an application, if we know the solutions (E, H) to the original Maxwell equations in a meditlm ( p , E ) due to the electric sources p, and J, wc can obtain the "dual" set of solutions due to the magnetic sources p, and M in a medium ( E , p ) with "dual" boundary conditions. For example, tangential magnetic fields are continuous vs. tangential electric fields ct~ntinuousin the original problem with electric sources. Applications of (5.2.13) a r e discussed in Chaptcr 7, where we explore the optical waveguide theory for*both the TE and T M polarizations. T h e idea is that once tve obtain the field solutions for the TE polarization in a dielectric waveguide from Maxwell equations, we can use the duality principle to write down the T M solutions silllply by inspection. . -.-

5.3 PLANE WAVE REFLECTION FROM A LAYERED MEDIUM [I, 31 In this section, we first discuss plane wave reflection from a single planar interface between two dielectric media. We then present a propagation-matrix approach to plane wave reflection from a multiple layered medium. 5.3.1

TE Polarization

Consider a plane wave inci,:!ent o n with the electric field giver; by

3

planar boundary, as shown in Fig. 5.13,

ELE :::'TE
!2('6

and the magnetic field given by

where the components of the wave vector satisfy the dispersion relation:

Here k o

=

w\lFi=

W / C

is the wave number in free space, and

fiI~l

nI = is the refractive index of the material. Usually for dielectric materials. The reflected fields are given by

p , = p,,

The transmitted fields are

where

Matchins the boundary conditions in which the and n , =,{-. tangential' electric field ( E , ) is continuous at .r = 0,

for all z , we obtain k,,

- A-;,

-+ I-

= r

tTqnation (5.3.10) is Snell's I;LW OK

=

A ,-. -

[:-ti pi~rtsi-'-in;itchir~g concli tion

,k, sin 19, == ,k, sir1 8,. = k ,

sin 8,

(Fig. 5.1.b):

(5.3.12)

or 8,

=

8,. and

n, sin 8, = r l -, sin 8,

where k , = w JG = koni, i = 1,2. The other boundary condition in which the tangential magnetic field ! H z ) is continuous leads to

Solving (5.3.11) and (5.3.13) for r and t , we obtain the reflection coefficient r=

1 - PI1 1 + PI,

~lk2.r P I , = -~ 2 k l . r

and the transn~issioncoefficient t=

1 + P,,

for the electric field. We note that when k , > k , , the total internal reflection occurs at an angle of incidence larger than the critical angle B,, given by the condition when 8, = 90":

k , sin 8,. or 8,

=

sin-'(n,/n,).

=

When Oi > B,, k,,

k,

=

(5.3.16)

k , sin Bi > k , , we have

. c

; k i - , = k ; - k -f a

,,

Thus, k is purely imaginary, k 2,t- = along the - x diiection:

The reflection coefficient beconies

where

=

k: - k:;

<0

(5.3.17)

+ i a2,a Z> 0, and the field is decaying

The fraction of the power reflected from its boundary is given by the reflectivity R :

T h e transmissivity is given by

Power conservation requires R + T = 1, which can also be checked by substituting (5.3.14) and (5.3.15) into (5.3.20) and (5.3.21). It is clear that if Oi > 8,, total internal reflection occurs; the tra~lsmissivityis zero since k,, is purely imaginary, and the reflectivity is unity, ) r 2 = e-i'4121 = 1. However, the reflected field experiences an optical phase shift of an amount -2412. This angle is called the Goos-Hanchen phase shift. Special Case for Nornlal Incidence. At normal incidence, 9; = 0°, k , , = k , = koiz, and k , , = k , = k , n , . We find the reflection and transmission coefficients for the field are, respectively, IZI

r =

-

nl +

5.3.2

117

2n 1

t = Ill

n2

+

rl,

TM Polarization

The result of T M polarization can he obtained by the duality principle using the exchange of the pllysical quantities'in Eq. (5.2.13). T h e results are

H,

where k -, -

=

=

$t-,,

H,e - i k 2 , , x + i k 2 , r

k , ; and /- - 1-bl

i - P12 = ---

1

7-

PI2

El

-1 2 --

~1'7.r

2? k I,,.

(5.3 2 5 )

-.

For total internal reflection, k,,

=

ia !, and r,,

=

Tbl

eLi2+12 : where

The magnetic field experiences a Goos-Hiinchen phase shift of an amount -2+-:p when the angle of incidence is larger than the critical angle 6,. 5.3.3 Propagation Matrix Approach for Plane Wave Reflection From a Multilayered Medium If the medium is inhomogeneous along the x direction E = ~ ( x and ) p = p(x), we can discretize the permittivity and permeability and use the propagation matrix method, as shown in Section 4.6 for the Schrodinger equation. We use El'! E~

=

l-4~~)

=

E(x[)

for - d r - l z x 2 - d ,

as shown in Fig. 5.2. For a TE polarized incident wave

we have the reflected wave

In the t th layer, x ( - , z x where

> x,,, the electric field is given

- -krx ( A ,e -

-

by EL-= FE:, .C

i k c . r ( . ~ + d f )-

Bf

e i k l ~ ( . ~ + ~eik..2= .))

(5.3.31)

UP f 7

e.

and k , , = h 2 p , ~ , - k g , , k,; = k u , for all The boundary conditions in which .E,, and H -, are continuous at s = .-rl, lead to A ( + Br

,

A,,, e - ' " t ' . - " , " - ~ ' ' + ~j-/ , 'f$- ',,,'

,eik(*..I)r'-d/-k"(+,i

Region 0

z

Figure 5.2. A plane wave with TE polarization E rnul titayered medium.

=

YE,

-

y ^ ~ ~ e - ~ ~i k uo- ". r is~ +

incident on a

Again, sirniIar to the propagation method in a one-dimensional potential in the Schrodinger equation, we d-.fine

and the thickness of the region

+1

We have

-

where the backward-propaglltion m ~ i t r B,,, i ~ ,.iT[) is definec.2 as

-

Altcjmatively, we can definc a forvlard-propagatio matrix F(,+,,, similar to that for the q~~azltutn potential problem (4-.6.7):

-

and the product

-

Bf(!+ F(!+

,,[

is a unity Note that + = I / + matrix. The amplitudes of tlle incident and reflected waves are related to those amplitudes -.. in the transmitted region N + 1 by .

,

wilere B,, = 0, since there is :o incident wave from the bottom region in Fig. 5.2. We obtain the transrniAsion coefficient of the multilayered medium

and the reflection coefficient of the rnultilayered medium

r ,

1,he electric field in the trr~nsrnitteclregion is given by

Since we use (iik,;:ts the bot:,c.,!;: i:r:.,ii~lcl;ic;, the l a s t region N + 1 can be chosen su,ch that A , , . == rl,+, :i,. . = O f o r co~i~:enience.I t means that the field will be n i e r - ~ s ~ ~wi r st11 d t i ! ,.: ! - ,! !I.

,

-

.

.

,

,

.-x.--.---

'

.. ...,-- s. ...

1 .

I

I

I

,.... ,,

_

.- .

-7.

.-

..

.. .

-,.

--.----

. I

- -

-

...-.. ...- .... ..-.......-. :

Figure 5.3. A plane wave reflection from a dielectric slab with a thickness d.

As an example, we consider a dieIectric slab _with a thickness d as shown in Fig. 5.3. The backward-propagation matrices lol and B,, are -

B,

[--

I

1 ( 1 + P,,)e--'~ii.rd (1 - Pal)eikllcd 2 (1, - ~ , , ) e - ' * 1 , ~ " ( I + P,,) eikl,rd

= -

We calculate the matrix product

- ?jUIBL2, then

( 5 . 3 -41)

the transmission coefficient

and the reflection coefficient

where

is the reflection coeficient of a piar:ar .;LIT-facebetween medium L and medium E + 1. Note that ,,,. , j ---. - r; .. ,,. siace Pti,+ = 'l./P,+ For T M polarization, we use the daa!it): pricciple again: exchange E H,

,

,,

,,,.

-)

-+

- E , P ~ + Eand ~

,cf

->

,u,,

E.-cnmple A common formula used in determinirrg the quantum eficiency of a photocletector is the absorbence 11 defined as A = 1 - K - T , where R is the total reflectivity of the power froin the slab of an absorption depth d , and 7' is the transmissivity of the power. Here R iT + A = 1 by power conservation. Consider normal incidence k , , = k , = k,,. + i ( a / 2 ) , where ct. is the absorption coeficient of the optical intensity and a factor of accounts for

the absorption coefficient for the optical electric field: Assuming that the transmission region 2 is the same as region 0 (Fig. 5.31, we have

Equation (5.3.20) and (5.3.21) also Iead to (neglecting the small imaginary part of Y o , ) 1=R,,

+ 7;, = ~

r + ~~

~

l~ ~l

t

~ (5.3.47) ~ l

The power reflectivity of the slab is found by using (5.3.44):

if we ignore the phase coherence i k , , . d in the exponent, assuming that the absorption is the dominant effect. ':'he transmitted power is found by using (5.3.43):

Therefore, the absorbance of the slab absorption region with a thickness d is A=1-R-T=

1 -R

) - e-"")

( I - R,, e-"")?

(I

+ R,,

e - 7 U " ) (5.3.50)

Somctirnes, the multiple reflecf:ion is negligible if n d >> 1 and the above al.:ir,;bance is a p p r ~ x i ~ a t ehy c?

~

5.4

5.4.1

RADWrION AND FAX-FIELD 1PA'I"TERN 11.1 General Expressions for Radiation Fields

We write (5.1.19) and (5.1.20) in the Lorentz gauge ( V - A the frequency domain:

= -p&d4/dt)

W e find A(r)

-4%-/ P

=

e i k l r - r'/

IF

J(r'> 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
.:--I

k, = k sine

X

.

Table 6.1 Representotions of Vectors, Dyadjcs, and Their Corresponding Matrix F O P ~ S

Vectors and Dyadics

a

=

u,.?

Matrices

+ a,9 + a , i

a - b = a,b, (a b)(c d)

.

a = a' b

["I

2

+ a,b, + a,bz = a . Cbc) . d

(a' b)(cfd) = at(bc')d

+ ibxcy;Gy^+ h,c,,.2

bc'

where .. .-

bc = ' b , c , E

=

I:1

b, [c.,

cy

cZ]

and the n ~ a t r i vrepresentation (see Table 6.1) for the cross product k X , we have

I. TGHT ~ % O P A G A . ~ ~I NO VARlOUS I~ MEDIA

2 3 ~

The left-hand side of (6.3.4) is

The matrix equation (6.3.4) can be written as

T o have nontrivial solutions for the electric field, we require that the determinant of the matrix in the above equation be zero. That is,

There are two possible solutions. Solution 1.

k l + k l - w p 2. E = O If (6.3.7) is true, we find from (6.3.5), which contl-]ins three algebraic equations, that

E, can be arbitrary except zero (for nontrivial solution)

(6.3.8a)

and X

= E -, = O

That is, if the electric field is polarized only in the y direction, the wave vector must satisfy the dispersion relation given by Equation (6.3.7). The solutions of the fields are, therefore,

where

k- r

- k , x + k z z - Ji(xsi1l0 + z'cos0)

(6.3.9e)

\Ve can see that for an electric field polarized in the y direction, it will propagate with a wave number k = and its polarization remains y-polarized. Therefore, this polarization (6.3.43) is a characteristic polarization of the uniaxial medium. - .

WG

Solution 2.

which is obtained by setting the square bracket in (6.3.6) to zero. If (6.3.10) is true, we find inlrnediately from (6.3.5) that

E,

-

=

0

(since k.:

+ k:

- oZP# &0 if (6.3.10) is true)

(6.3.11)

and

(k:

-

w ~ , x E )E .k~, k , E z

=

0

(6.3.12a)

Using (6.3.10) and (6.3.12a), we obtain

or simply

k,&E,

+ k,&,E, = O

which is kaD=O The complete solutions of the fields obeying (6.3.11) and (6.3.12~)are given by

LIGI-TT P R o P ~ \ G x T I O ~I N VAP.IOUS MEDIA

232

where

k r

=

k,x

+ k,z

=

k ( x sin 0

+ z cos 0)

(6.3.13e)

The polarization of the electric field given by Eq. (6.3.13a) is another characteristic polarization of the uniaxial medium since it is a characteristic mode of the medium from the fact that the fields (6.3.13a)-(6.3.13d) satisfy all of Maxwell's equations. This wave propagates with a wave vector determined by Eq. (6.3,10). We note again that

k- H

=

0

since (B/lH)

(6.3.14~)

Thus the wave vector k is not perpendicular to the electric field E. The complex Poynting vector, E x H*, is therefore not in the direction of k. The solutions of the fields (6.3.13a)-(6.3.13d) are called extraordinary waves. The dispersion relation of the extraordinary waves is given by (6.3.10). On the other hand, the solutions of the fields (6.3.9a)-(6.3.9d), for which the wave vector k is perpendicular to E, H, D, and B, are called ordinary waves. The complex Poynting vector points in the direction of k. The ordinary waves behave the same as those in isotropic media with a permittivity E . A summary of results of the ordinary and extrziordinary waves are shown in Fig. 6.3. 6.3.3

k Surfaces

We may plot the dispersion relations such as (6.3.7) and (6.3.10) in the k space. For the ordinary waves in case (11, Eq. (6.3.7) is a circle with a radius w 6 in the k,-kz plane. For the extraordinary waves in case 2, Eq. (6.3.10) is an ellipse in the k,-k, plane. We summarize the results as follows. 1. 01-clinaryware surface

k + ki = wZIL&

( E m u s t b e p o l a r i z e d i n t h e y direction) (6.3.15)

In other words, E must be polarized ir, the direction perpendicular to the plane containing bath the optical ( s \ :t.uis and the wave vector k for w nG , j c and the refracordinary waves. The ivave number X = ~ I ,=~ tive index n, =

J/ri)pOt-:I.

-

(a) Ordinary waves

kz I

ik s i n 8 7 z kcos 9

kx

Figure 6.3. A sumrnary OF the k vrclnr and the field vectors for (a) ordinary waves and

(b) extraordinary waves.

2. Extraordinary waue sz~rface

k,Z w 2p a ,

k;

+ ----- 1 02p&

( E must lie on the X-z plane) (6.3.16)

06

Here = w n e / c . The extraordinary wave surface has ~ w princio pal axes given by o n , , / u and w r r , / c . The electric field E must be polarized in the plane containing both the optical axis and the wave vector k. The results are shown in Figs. 6.3 and 6.4. Remember we have made use of the symmetry of the media with respect to the z axis (the optical axis). The plane contahing the optical axis and the wave vector k is chosen to be the x-r plane. If we allow the wave vector k to rutate around the z axis, we vvou!d expect the following:3.. The ordinary waves will have the electric iield perpendicular to the wave vector k artd t112 optic31 ::xi:,- and tht: dispersion relation i s a spherical s~li-face, in thc k space:

LIGHT P:IOp,2(.:ATlON I N VARIOUS MEDIA

k, (optical axis)

t

k-surfaces for E, > E

Figure 6.4.

A plot of the k surfaces for t h e ordinary wave and the extraordinary wave of a

uniaxial medium.

2. The extraordinary waves will have the electric field polarized in the plane containing the optical axis and the wave vector k. The dispersion relation is given by-an ellipsoid in the k space:

which is a result of rotating the ellipse of (6.3.16) around the optical axis.

Special Cases 1. If the wave propagates perpendicularly to the optical axis, for example, k = fk, k, = k, = 0, we have from (6.3.9a) and (6.3.13a) either

E

=

9~~ e i k . r

=9

~ e i,k , x

k,

E

=

i ~ e i ,~

= . ~i

~ ei*,." ,

kt,=

= wfi;

(ordinary waves)

(6.3.19)

0r

WK(extraordinary waves) (6.3.20)

Note that when k, = 0, one finds E, = 0 from (6.3.12c), and (6.3.13a) should reduce to (6.3.20): 2. If the wave propagates p~tra'llelto t h e optical axis, for example, k = z^k, ii, = k , = 0, we have from i6.3.9a) and (6.3.13n) again

E

=

EE, e ik, .:.

(

=

- c &kd:~ ~

( For t': polarized in the x - z plane)

for E polarized in the y direction)

( 6 . 3 -21) ( 6.3 - 2 2 )

Both electric fields propagate with the some wave number k , = o f i and both are pcrpanclicular to the direction of pi-cspagation! The ordinary wave and the extraordinary wave bc.come degenerate (both are ordinary waves now). 3. In general, the wave propagates in a direction with an angle 8 to the optical axis. We have the dispersion relation

k L k z sin20 + k Z Cos2 8

=

2

(6.3-23)

p~

for the ordinary waves. The phase velocity

is independent of the angle d for ordinary waves (remember that the -. polal-ization of the electric field is perpendicular to k and the optic,al axis). For the extraordinary waves, for which E is on the x-z plane, we have ,

The phase velocity is

which depends on the angle of propagation. 6.3.3 Index Ellipsoid

,

-

For a given permittivity matrix g7we define an inlpermeability matrix K as its inverse

.j.l:j~er.eK 1 i ;ire trlc eie;!lcr?ts el'

till;:

r!i-tr:-i:.:

Pi. 2nd n - 1 = x ! .v,

-

y, x ,

-

z.

I11

I.IGHT J'ROPA3A?'I(')N IN VARIOUS MEDIA .

236

the principal axis system such that

is diagonalized

the index ellipsoid equation (6.3.28) becomes

which is plotted in Fig. 6.5. If n , = n , -- n o and n , medium as described by (6.3.1), we have

=

n , in a uniaxial

It should be noted that the index ellipsoid is directly defined from the inverse of the permittivity function, and it contains information about the refractive indices of the characteristic polarizations. O n the other hand, the k surfaces depend on the polarization and the direction of wave propagation. For a uniaxial medium, there are two k surfaces, one for the ordinary wave and the other for the extraordinary wave, with only one index ellipsoid to define the medium. T h e concept of index ellipsoid will be used again when we discuss electroptical effects in Chapter 12. 6.3.4

Applications

Quarter-Wave Plate. Based on (6.3.19) and (6.3.20), we Bssume a plane wave incident from the free space to a slab of uniaxial medium with

Index elli~soid z

Figure 6.5. An index ellipsoid i n iI-12 prii:zipal axis system f o l wllicli !he pirrnittiv;ity matrix i s di3gon:llized.

Consider an incident electric field from

fret3

space given by

-----

which is linearly polarized and k , = wifp(,e,, is the propagation constant in free space with a subscript "zero" (Fig. 6.6). We also assume that the permeability of the uniaxial medium p is'the same as that of the free space E , L ~The . electric field makes an angle of 45 degrees with the y and r axes. After passing through the uniaxial medium, the electric field will be (ignoring the reflections at the boundaries)

since the y componeni of the electric field will be an ordinary wave that 6 (the subscript "oh" stands for propagates with a wave number k, = w ordinary wave), and the z component of the electric field will be an extraordinary wave with a wave nuniher k , = w 6 . If we choose the thickness d such that the phase difference between the two compollents of the electric field to be 90 degrees or multiplied by an odd integer (assuming E ; > 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

'
find the normalization coefficient C , to be

for both TE even and odd modes. The above coefficient can be derived using the expressions for E , (7.1.3) and (7.1.7), and the eigenequations (7.1.10a) and (7.1.10t3) for the even and odd modes, respectively.

7.1.8

Optical Confinement Factor T'

An important quantity, called the optical confinement factor, is defined as the fraction of power guided in the waveguide: 1

r'-

jinSid,Re(Ex H:"

2 total

. 2 id x

Re(ExH*)*2dn

Using the expression in (7.1.3) for TE even modes, we find

It is obvious that r < 1 always. As the waveguide dirnension.'beconles smaller, d -+ O 01- k , d -;L 0, we find that for 1LE, mode

C;?TICAL 7;V IVEi.;U.ID : THEORY

354

Since ( k , d / 2 ) ' + ( nd / 2 ) 2 = x', we have a d / 2 k , d / 2 . Thus k , d / 2 -, K. Therefore,

+

( p / p I ) ( k Id / 2 ) 2

-CC

For p = p 1 and k , = 2 7 ~ / A , , the following formula is useful in estimating the optical confinement factor of the fundamex~talTE, mode in the weak guidance limit:

7.1.9

Sumrna~yof TE and TM Modes

( p i = p , inside, and p i = p outside the guide)

(7.1.33)

1. TE even nzodes

E,

=

C

1.

I

COS

k,x

1+

I

sin k,d kxd

The eigenequation is

The optical confinement factor is

cos2( k , : l / ? )

---.-..---

(1 t

sin k,vil,lk ,d)

(7.1.34d)

2. TE odd nzodtrs

E

=

C , ei";Z

Y

d 1x1 I 2

sin k , x

+

.

(7.1 -35a)

d

a ( x + t i / 2)

sin k,d

-

x

< -2

[$)($)sid

(K.rd/2)

I

"'

sinz(k,d/2) ( 1 - sin k,d/k,d)

TM Polarization. For TM polarization, we may obtain the results by the duality principle: replacing the field solutions E and H of the TE modes by H and -E, respectively, p by E and E by p as discussed in Section 5.2. We obtain

( E = ~ E .l

H,

=

CIe'*:'

inside, ei = e out-side the guide)

cos k , ~

(7.1.36)

The eige~lequationis

The optical confinement factor is

I- = [I

+ (:)(lrd)

2

cos'k, d / 2

(1 + sin k , d / k , d )

I

-'

(7.1.37d)

2. TM odd modes

d 1x15 -2 I

sin k , x

-.

(7.1 -3Sa)

n

X j--

ea(x+d/Z)

2

1-

sin k,cl k d

+

-):[

(;)~in~(k,~d/2)

I

ll2

The eigenequation is

Figure 7.7.

0ptic:tl confinemrnt fit(r.!:)rs fc)r [he ?'lZ!,; !solid curves) anr! the TM,,, (dashztl i i Function of ?;r:/.,'.\ ,:. 'The rcf~.;~.clive indicts a r e !r I 3.590 2nd ,l 3.3S5.

curvss) modes :is

-

-

The optical confinement factor is

For dielectric materials, p = p 1 = pg,we can replace a11 factors containing p/pl by 1 and E / E ~ by rzZ/n: in the expressions for the eigenequations and the optical confinement factors for both TE and TM polarizations. In Fig. 7.7 we plot the optical confinement factor r for the T E (solid cu~ves)and TM (dashed curves) modes. All optical confinement fkctors start from zero at cutoff and approach unity at the high-frequency limit. 7.2

A S W M E T N C DIELECTRIIC S

W WAVEGUIDES . -.

7.2.1

TE Polarization, E

= fE,

If the dielectric constant in the substrate E~ is different from that of E in the top medium, or p 2 # p , the structure becomes asyn~metric(Fig. 7.8). In this case, the field solutions cannot be either even o r odd modes. The general solution can be put in the form

.'

Aghin, the transverse mode profile is decaying away from the guide with different decay constants a and a 2 in region 0 and region 2, respectively. The standing-wave solution inside the guide is written as C,cos(k,,x + qb), which is equivalent to A cos k , , x + B sin k ,,x. We choose this form because it is easier to match the boundary conditions later and the phase angle 4 is also related to the Coos-Hiinchen phase shift (discussed in Section 5.3) when

total internal reflection occurs.

N, = (1 / i w k ~ ) ( d E , / d xis) given

by

where

The next step is to match the boundary conditiom.

1. E, is continuous at x

=

0 and x

= -d :

2. HZ is continuous at x

=

0 and x

-

-_

-d:

Dividing (7.2.k) byf (7.2.4a) and (7.2%) by (7.2.4b), we obtain

We add 17777- in (7.3.7t~) since there car1 bt: multiple solutions to (7.2.6b) especially when a' is thick. We do not add any 171'71. in (7.2.7a) simply because it will be conv
Eliminating q5, we obtain the eigenccluation. for the TE,,, mode: A I .r(1

=

y~ ( Y +- tall---l tan-' -

A L ~ . ~

A"

la2

-t- ~ 2 . ~ : -(UZ =

0 , 2 , 2 , .. . ) ( 7 . 2 . 8 )

E - L ~ ~ , . V

Alternatively, tan k , , d

(~I.a/Pk l x+ )I P I

tan k,,d

-..

~

~

/

P

Z

~

~

~

X

)

(7.2.9) I -(~~a/~k,,)i~la,/~Zkr,)

= -

=

( a + aZ)kIx k;, - aa2

After solving for a , a , , k , , , and k, from (7.2.3) and (7.2.9), the complete electric field can be obtained from the normalization condit ion

We obtain

E,

=

c,e'l::

I

cos 4eU""

X Z O

C O S ( ~ ,+ X4 ) COS(- k , , d 4 ) e"2('+d)

-45 X 5 0

+ C, = [ 4 w p / A , ( d + I / a + 1/a,)l1/' if

5

(7.2.12)

-d

y = p , = p 2 ; 0th. where wise, it can be cast in a more complicated analytical expression.

Cutoff Condition. For the wave to be guided, E , has to be larger than both E , and r . Let us assume that E~ > ~ ( =pp L = p Z ) . When reducing the fre= korz2, the decay constant in region 2, a , , will quency until k, = vanish before a does. Thus, at cutoff

o~K,

for the TE,,, mode at a ,

=

0,. and k,

=

k0r2,. We also have

at cutoff. The cutoff frequency is clctermined f r o n ~

4

Figure 7.9.

the dispersion relations for an asymmetric dielectric waveguide.

where ko = w\/= = W / C . T h e dispersion curves for the TE,, modes are plotted in Fig. 7.9. They start from the cutoff condition k , = k,n, and approach the upper limit k , -+ k , n , when the frequency is increased. Note that if n, > n , >> n,-[hen

We obtain

which is equivalent to the cutoff condition of the TE,,,,,,, mode in a symmetric waveguide with a 2 d thickness. This can be easily understood from a comparison of the two electric field profiles in Fig. 7.10. Since the decaying

X C

E2 A

Ax El

fasr decay

E

El

j /

T'd

I

,z 2d

I E1

E2

(a) An asymmetric u.;iv.'guidz with t.l >: r , " --". (- and a thickness d . (b) A symmetric waveguide with a thickness 2ri. Thz cutoH condition k)r the TE,, mode i+(a) is equivalent to that of the TE,,,,, , mode in (bl.

Figure 7.10.

,,

constant a in region E cl~caysvery fas,l in Fig. 7.:LOa, its field profile loolcs vely much like half ol' Ihe field pl-ofilc of Fig. 7.1.0b. 7.2.2 TM Polarization, W

= 9%

We again obtain the solutions for the TiM polarizatioil using the duality principle. We obtain from (7.2.12) -

cos 4 e ff Y

=

.

x>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'<

-t-&,

Region 1 Region 2 Region 3 (7.4.10)

jeU"

Region 4

C,em5.rcos(k,y + +,,)

Region 5

C4cus(k, x \

where k: + k f + k i = 0 2 p , e l , kf - - a$ -:- 4 f = w ~ ~ ~etc., E ,which , are obtained by substituting the field expressions (7.4.10) into the wave equation (7.4.5) using the material parameters , L L ~ and ei for region i. The choice of the standing-wave solutions cos(k,x -t (b,? or. decaying solutions is ~ n a d e .simply by inspecting Fig. 7.12 ancl keepin;: in mind that we are looking for the guided illode solution. The sarxe s depc.n::lence is usecI for regions I and 2 because bctth repic~nsshare the s;in,te i.~o~iridr~;y at JJ == ktJ, and the phasematching condition or Sncl't's /;:hi! requires tile sagre dependence on the spatial variztble u-, Simil;ir r.i.;lc.; i ~ p p i y it? the ;tis!d ex~rressions in other r't:gions.

Boundary Conditions. T o derive the guidance conditions and the relations

between the field coefficients C , -

- C,,

we match the boondary conditions.

1. At x = 0 and x = d, H z and E,. are continuous. The other boundary condition for 6, continuous is ignored since E , is small. At x = 0,

-k,C,sin

4,

=

a,C,

(7.4.11a)

Eliminating C,, C3, and C,, we find

Eliminating #, again, we find P ( ~ ~ 2-pk tI) a~Z I

_I

k,d

=

tan-'

+ tan-'

, 5 1 ( ~ ' F 3 ~ 3- I C ; ) ~ , ~

+ p77 (7.4.14a).

Tlle above formula can be furtl~ersimplified for k , << k , , k 3 , and k,:

2. At y = 0 and y = w , match H,and E.. The other conditions that H z and E , are continuous give identica! results as H, and EAt J) = O

C ' I ~ ~(/I :,,j" :-",,

(7.4.15a)

Thus,

Therefore,

kyw =

IT

+- tan-' .

--

t'.Iff2

E~

ky

+ tan-I

&la4

~4

ky

(7.4.18)

I

In summary, the two important guidance conditions are

k,d

= p7i

+ tan-'

Fla3

-

+ tan-I

F3kx

k,w

=

q~

+ tan-'

Elff?

-

~~k~

pla5

--

! j

(7.4.19)

F5k.r

+ tan-.'

ElU4

-

%ky

which can also be seen from the ray optics approach. An intuitive way to understand the approximate boundary conditions is that for (i) at x = 0 and x = 11, the optical power density of interest is its x component bouncing back and forth between the two interfaces at x = 0 and d. We therefore need E,,Hib and HTE,. However, H, = 0. We need only E , and H_ and drop E, since it is small. For boundary conditions (ii) at y = 0 a n d w , we use only Hi and E x , or N, and E- for the y component of the optical power density. Both pairs give the same results since H, is not of concern. 7.4.2

EH,, Modes (or E;,,

,,,,+,,modes)

For the E q , , mode, the domlnant field components are E , and H,. We can assume that H, =. O and H , havs exactly the same form as in (7.4.10). Expressing all other components in terms of Hy only, we then match the boundary conditions. iVt: use

I I

which are derivable fi-om Eqs. (7.4.1.)-(7.4.4). T h e rollowing guidance conditions are found ai'ter matching the boundary conditions, kxcl

E la3

= prr i -tan-'

-

+ tan-

1 &lQ5

~3kx

k,w

=

+

q ~ - tan-'

c 2 ( w 2 p I ~-I k I ) a 2 " l ( ~ ~ P 2-~ k:)k." 2

]+

-

~5kx tan-'[

& L ~ ( ~ * I I~ Ek;)a4 ~ .1(u2P4"4 -

k:)ky

I

where (7.4.22) can be further simplified for k , (< k,, k , , and k,: k,w

=

qrr

+ tan-'

PI(-='? -.

Pzk,.

+ tan-'

Plff4

-

PJk?J

Alternatively, we may assdme that E, = 0 and Ex is of the form (7.4.10), and then express all the other components in terms of the Ex component using Maxwel,lYsequations. These results will give almost the same eigenequations and mode patterns.

-

Figure 7.13. (a) Normalized propagation parameter P vs. the normalized frequency B = ( d / kt - k : ) ? / ? of a rectangular waveguide with an asprcl ratio wr/d = 2. The refractive index in ihc. guide is , I , = 1.5 a n d o ~ ~ t s ~ d e '?!,, i s = I . (Reprinted from Ref. 33, 01939, published by VSP, T h e Netherlands.) K a r t t h e width w is along the r direction and the thickness cl is along t11c ,t direction. (b) Plots of t h e tr:ms\;ersz field cornponznls E:," 2nd FITb' For the lowest n i n e nlodes for a rrctangi~lal.w;1vtfg~11clt'in (;:I. i\Jo;d the cI:~sli;v relation betwcen the transverse electric field of the k/E,, 111oclz a n d the t1.3nsverst'n-ingi~t'tii:field of the EH,,, mode. (Rcprintecl (c)T h e schematic intensity p~lttzrns from Ref. 23, 61959, publi.;hl-:cl by VSP. 'T'lle Nc.:l~cl-lni~ds.) for the mocles in part (b). Since the intensity patterns of the ME,, mode are aalmosi indisting~iishnblefrom t h a t of the l..'?!,,,,n ~ o d c\~,c' . only st1c;w orir set or p a t t e r n s h e r r .

(b) (c>

HEoil (EY,)

X

Intensity patterns

EHoo (EIXI)

EHol (El;)

HE01

(q2) HEOl or EHOl

HE10 (El,)

EHl0 (E;,) I

EI-Ioz (E h)

Figure 7.1.3. (Corlti,lrrc~c/)

We consiclcr a rectanguiar waveguide where t h e aspect ratio w / d = 2, the refractive index of the waveguide rrl is 1.5, and the b a c k g r o u ~ d medium is free space = 1. We plot in Fig. 7.131, after Ref. 23, the vs. a normalized frequency B = V / a , normalized propagation constant Example

p

and

where the V number is defined the same as that of a slab waveguide with a thickness d ,

The corresponding transverse electric and magnetic field patterns inside the dielectric waveguide are also plotted in Fig. 7.13b. Note the dual relations between the transverse electric field E : ~ of the HE,, mode and the transverse magnetic field H : ~ of the EH,, mode. We can see that the field patterns provide the information on the intensity variations of the E x , E,, H,, and H , components of the waveguide modes (Fig. 7.13~).T h e intensity pattern of the HE,, mode is almost indistinguishable from that of the EH,, mode. Therefore, only one set of intensity patterns is shown. These mode patterns should be interesting to compare with those of the metallic waveguides [24]. The results in Fig. 7.13a have been compared with those of Goell [22] with a good agreement. T h e original labels of the modes in Ref. 23 are slightly different. In Figs. 7.13a and b, we use our convention for the ) modes, and the HE,,( = ETP+,xu+ 1) ) modes and t h t EH,,(= E;,+ coordinate system is shown in Fig. 7.12, where the vertical axis is the x axis and the horizontal axis is the JJ axis. This coordinate system is chose11 such that when the width w + m, lhe resulls here approach those of the slab waveguides discussed in Sections 7.1-7.3 using the same coordinate system. Our convention is the same as those for Marcatili [20] and Goell [22]using modes, I, except that our aspect ratio is the E?p+ ,,,,+I, and E & + ~ x ~ + ~ / c l> 1, where w is the width along the y direction instead of the x direction. ,

7.5

THE EFFECTWE INDEX IVIETROD

The e-ffective index methocl is a useful ischiiicl~ie to find the propagation c o n s t ~ ~ noft a dielectric vi;tveguidc. As an exampie, assurne that we arc

(a) A rectangula die tectric waveguide ) , x 1

(b) Step one: solve for each

i)

"8

"

11)-

"3

y the effective index n ,fdy). iii)

"7

"4

"1

"2

"9

"5

"6

(c) Step two: solve the slab waveguide problem using neff(y)

Figure 7.14. (a) A rectangular dielectric waveguide to be solved using t h e effective index method. (b) Solve t h e slab waveguide problem at each fixed y and obtain a n effective index profile n,,,(y). ( c ) Solve t h e slab waveguide problem with the etfective index profiie n e f f ( v ) .

interested in the solutions for a rectanguIar.dielectric waveguide as in Section 7.4. Consider first the El; , modes for which E , is the dominant component (see Fig. 7.14a).

,+ ,,

Step 1. We first solve 2 slab w;!veg~iide problem with refractive index n , inside and t ~ ; rr, , or~tsicieih:: c\laveguiile (Fig. 7.14b). The eigenequation for the E,, component y$;ill b:: that C:IC tht: TE n:r.>des o;f a siab guide

,

OPTICAL. VY.c\'dCG'UII?E THEc31I?'

272

discussed in Section 7..1., since E,, is parallel to the boundaries:

k,,tI

-

tan-'

CLjcy;

+ tan-'

~ 3 k l l

Pla5

+P

r

~~5kl.V

where is the decaying constant (outside the waveguide) and p i is. the permeability in the ith region. From the solutions of the slab waveguide problem, we obtain the propagation constant, therefore, the effective index n,fi,l. For y > w o r y < 0, similar equations hold if n , > n,, n,, and n , > n,, n 9 , Otherwise, approximations have to b e made by assuming n, when the fields in regions 6, 7, 8, and 9 are that l z e f f , , = n 2 and n,,., negligible. Step 2. We tnerl solve the slab waveguide problem as shown in Fig. 7.14c, and obtain

,

k 1Y w

=

-

tan-'

Eeff. 1ff2

+ tan-I

~eff,zkly

Eeff, 1 ( Y 4

+ (7m

(7.5.2)

~ e f f , 4 k l y

where E ~ = ~r l 2, F f, , i ~ O~ Note that the E,. component is perpendicular to the slab boundaries; therefore, we should use the guidance condition for the TI'vl modes of the slab waveguide problem. In step 1, for each y, we find a modal distribution as a function of x. This field distribution will also vary as y changes. Thus we call this function F ( x , y ) . In step 2, the solution for the modal distribution will be a function of only y , called G ( y ) . Thus the total electric fieId E , can be written as

It is interesting to compare the two eigenequations (7.5.1) and (7.5.2) obtained here with Eqs. (7.4.19) and (7.4.20) derived in Section 7.4. T h e difference is that the "effective permittivities" appear in (7.5.2) in the effective index method, while the material permittivities are used in (7.4.20) in the previous approximate Marcatili method. When comparing the propayation constants with the numerical approach from the full-wave analysis, the results of both Marcatili's method in Section 7.4 and the effective index methocl agree very well with those of t h s numerical method, generally speaking. 'The effective index rilcthod us~ia!lyhas a better a g r c e m m t with the numerical method, especially nz:ir cutoff.

!

I I

I 1 1

The effective index method ha!:, also been applied 125, 261 to the analysis of cliode Iasers with lateral var-iai-ior-1sin t'ilick~~ess o f t h e active region or in the complex permittiv-ity. The resultant analytical for-mulas in many cases are helpful in understanding the semico~-rductorlaser. operation characteristics.

7.6 WAVE GUIDANCE IN A LOSSY OX GAIN MEDIUM In an unbounded medium, the propagation constant k is given by

when the permittivity is complex,

that is, the medium is lossy propagation constant becomes

(E"

> 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(
where the horizontal wave nnmbers are

and their col-responding vertical components can be obtained from the dispersion relation

The horizontal wave number k:,,, for a particular nz differs from its nearby value by 2 ~ / i I .We can generalize the above discussions to the case of a penetrable dielectric grating. A phase-matching diagram is shown in Fig. 8.4. "These plane waves for the ref.ccted and tra~srnit-tedfields are calleh the space h;lrnlonics. It can be seen that for a pal-ticular m such that k,,,, > , k,,,,, becomes p~lrrlyimaginary a;lJ wz choose the imaginary part of k,,,, to he positive such that it d e s y s :way f i o n ~the surface. In the transmission region with ti ref]-active index j ; I .= vrp-i ~ / (l p , , , ~ ~, the ) transmitted wave vectors k:,, wi!l ha:-e :he same liui.izontal wave numbers k,,,, a s

Figure 8.4. Phase-matching diagram for a plane wave (a) reflection and (b) both reflection and transmission from a grating structure with a period A.

'

-.

those of the reflected waves (8.1.6):

k,,r

=

k,,,

=

25k z o +- v1A

For a general theory to find the reflection and transmission coe,fficients of the space harmonics, see Refs. 4 and 5. Phase Matching Condition for Grating Couplers. A dielectric grating can therefore be used as a waveguide input or output coupler, as shown in Fig. 8.5. Suppose we have a dielectric slab waveguide with a propagation constant fl for the f~indamentalTE, mode. If the incident angle Oi and the grating period A are chosen such that

for a particular integer m, then the incident wave will be coupled to the u~ildeci?'E,, mode (Fig. 8.5b). The coupling strength clepends on the incident 3 beam profile and its distance D from the edge of t h e grating, as shown in Fig. 8 . 5 ~If D is t9o long, the incident power coupled illto the waveguide with t11c grating section can leak back into the incident region. If D is too short, half of' the beam proti!e will be reflecteci f'rrxn the pianal interface in the section r > 0, and the coupling into the g~iidsd~noijein the region z > 0

Lnput coupler

I1

Output coupler

Figure 8.5. (a) A grating coupler. (b) Phase-matching diagram for P (c) Grating input a n d output couplers.

=

keno sin 8,

will be small. W e can use the grating as an output coupler based on the reciprocity concept. The guided wave in the center waveguide section in Fig. 8 . 5 ~can radiate back into. region 0 in the right grating section while propagating along the waveguide. The exit angle can be determined using (8.1.9).

8.2

COUPLING OF MODES IN ' T H E TIME DOMAIN [6]

Consider first a resonator consisting of an induclor and a capacitor, as shown in Fig. 8.6a. The differential equations for the voltage ~ l ( t and ) the current i ( t ) are

Coupling capacito~

Resonator 1 Resonator 2

Figure 8.6.

(a) An LC resonator. (h) Two coupled resonators.

or in terms of only [ A t ) (or i ( t ) )

Their solutions are

where the resonance frequency is o, = l . / d L ~ .If we define the two new variables n + ( t ) and a _ ( t )

we see immediately that

Therefore, we have two independent equations. The squares of the magnitudes are

which is the total time-aver;igec! energy stoi**:d in r h e circuit. Since the tT"vo

variables a + ( t ) and a - ( t ) are decoupled, we may choose c d t ) describe the resonator. 5.2.1

= a,(t)

to

Energy Conservationl Condition

Let us consider two resonators, 1 and 2, coupled through a capacitor as shown in Fig. S.6b. Without the coupling capacitor, each variable a , ( t ) or ~z,(t) satisfies the differential equation of the form (8.2.5) with a resonant frequency o, or w,, respectively. With the coupling capacitor, ,they satisfy

This assumes that the coupling is weak; thus--the coupling coefficients 1K,,I ((: w , and IK,,( -co,,and the energy stored in the coupling capacitor is negligible. A similar coupling system, a coupled pendulum system, is shown in Fig. 5.7. Therefore, from energy conservation in which the total energy in two resonators should stay as a constant (neglecting the energy storage in the weak coupling capacitor),

Figure 5.7. A coupled pe~.~c!uli~n; S Y S ~ C I I I : ( a ) in-pi~i!:;c mc)iie with a 1.esonant freqllency (b) out-of-phase rnocle w i t h a reso~li\.n:R'ecll~c.ni:y !*: ,.

to-;

we find that

n,a,"j(K,l

-

K:7)

+ n'l'a, j(

K 1-, - K!'.LI )

=

0

(8.2.9)

Since a , and a , are arbitrary (defined by the e.nergy and phase in each resonator), we conclude that the two coupling coefficients must be related by

8.2.2

Eigenstate Solutions

To find the eigenstate solution of the coupled resonator, we assume the solution to be

where o is the- new natural resonant frequency of the-entire coupled system. Substituting (8.2.11) into (8.2.7), we obtain

Since the determinant of the above linear equations should vanish for nontrivial solutions for A , and A,, we find (id

- o l ) ( o-

0 2 )-

K12K71 = 0

(S.2.13)

The second-order polynomial equation has two solutions:

I

where K Z 1= Klp has bee11 used. Let

The energy spiittings between (0, and w - for o , # w Z and .w, = w 2 are plotted i n Figs. S.8a and b. The ratio nf the two amplitudes A , and A2 s.;ttisfies A,

.A 7

where

LJ~

= w

+.

or

PJ

K,,

-0:

-

W1

--

C?

--

cd

7

(8.2.16)

a

K?,

.-.Thus, t h e soluti,ons c!,Ct 1 2nd

cz,(,t!

can bc represented

Cr:)i;PLER:, AKD CC)UIJL,ED-MODETHEORY

'wA':EGi1i'i.>j,

0

o+=+ + n

a-=Q - n Before coupling

After coupling

o, = o + IKI (b) The frequency splitting in (a) asynchronous coupling a n d (b) synchronous coupling.

W-=

Figure 8.8.

Before coupling

W-

IKI

After coupling

as a summation of the two eigensolutions. Each solution is called a system mode with a corresponding resonant frequency. We obtain the ratios of-the two amplitudes for both system modes: For o Forw

=

A:

K,,

A;

A + 0

-=

o+

A , ---

= o-

A,

K12

A - 0

-A -

general solution as

and

(8.2.17a)

K21

- -A

-

I : : [ .I: [

which determine the kinenrectors

+0 -

a (8.2.17b)

K2,

We can write the

.I

which also implies that the total solution consists of the beating of t h e two natural modes w + and w - . Suppose initially that a,(t = 0) = a,(O), and a,(t = 0) = a,(O). W e can find c , and cz in terms of a,(O) and a,(O) from (8.2.18), n,(O) = (c, + , c 2 ) KLZ,aZ(0)= c,(A + 0 ) + cz(A an2 obtain

a),

A I-

' R

sin

sin i11:

.

If initially n.,(O) = 0, wc have

Let us consider two cases, assuming that (

1. Asynchronous corlpling

( w , f w,).

0

=

1 for simplicity.

Equation (5.2.20) gives

< 1, we find the peak energy and /a,(t)lZ= 1 - la,(r)('. Since I K z , transferred to resonator 2 is lK,, /L!lZ,which means that the energy transfer are from resonator 1 to 2 is never 100%. The energies la,(t)12 and l~-1,(t)1~ plotted in Fig. 8.9a. We see that the energy transfers periodically between two resonators with a period T = rr/n = 2 7 1 - / ( w + - w - ) . The "beat" angu- w = 2!2 determines the energy exchange rate. This lar frequency w, = w ,frequency splitting, as shown in Fig. 8.8a, is larger than the original energy difference before coupling, w , - w , = 2 A . I n other words, the two frequencies appear to "repel" each other in the coupled resonator system. 2. Syndrronous co~ipliizg( w l = w ~ ) .We find A = 0, !2 = I K , , / ( K ) , and

'Therefore, at t = ( n + + ) v / K , where n is an integer, the energy is completely transferred from resonator 1 to 2. At t = n7i-/K, the energy is transferred back to resonator 1. The energies ~ n ~ ( tand ) i ~ la,(t)12 are shown in Fig. 8.9b. The time constant T = T / K is called one beat period, and the ', beat angular frequency is w , = w + - w - - 2 K . as shown in Fig. 8.8b. The frequency splits from a degenerate system ( w , = w 2 ) into a nondegenerate system with two system resonant frequencies i c d # w -). The energy splitting , the magnitude of the coupling coefficient. is 2 ) K J twice

,.

One way to determine the phase of K , -, is to consider the two system modes. Assume K,, < 0: L

'\.'

'.4 'IEG'JiDI- ~':)ii1'Lv-.RS 4 T ;D COIIPLED-M0I)E THEORY

(a) Asynchronous coupling la, (0)12=1

r'-

(b) Synchronous coupling

w,= w2

Figure 8.9. The energies in resonators 1 and 2 as functions o f time, assuming that at r = 0 only s ( w .f wZ). T h e energy transfer from resonator resonator 1 is excited. (a) A s y ~ l c h r o ~ l o ucoupling 1 to 2 is incomplete. (b) Synchronous coupling ( w , = w z ) T h e energy transfer from resonator 1 to 2 can be complete at t = ,rr/(?K), 3 ~ / ( 2 K ) and , so on.

We find experimentally that for the in-phase mode as shown in Fig. 8.7a the resonant frequency is smallel- than that of the out-of-phase mode. Therefore, the assumption K I 2 < O is true. An analogous system to the coupled resonators is a coupled quantum-well system. T h e in-phase mode for two coupled quantum wells has a lower eigenenergy than that of the out-of-phase mode, as shown in Ref. 7.

8.3

COUPLED O?1-ICL4.I,'LJrAYXGUII)ES 16, 8-32]

The couplod-n~odt: theory i\ alsu ~ ~ . w f r iinl i~nderstanding the coupling nlcchanisms in parallel wavegu!~les.11s .;how11 in Fig. S.10. T h e guidance is along the 2 direction. ~ l ' i ~ t e g u i c l c s azcl O c , : ~be two rectangular- or slab dielectric guides. For ;i yuicled rr~ode in the + z direction in a single

Figure 8-10. Two coupled optical waveguides u and b.

waveguide a , the electric and magnetic fields in the frequency domain (Chapter 5 ) can be expressed as

, and M(")(x, y) are the modal distributions in the xy plane, where ~ ( " ' ( x y) which satisfy the normalization condition:

Note that the time conversion exp(-iwt) in Maxwell's equations has been adopted. We also see that

The total guided power for the fields in (5.3.1) and (5.3.2) is

If the mode is guided in the

-2

direction, then a ( z )

=

n0exp(-i/3,z):

and tI-le Pojinting vzctol- E >.: 11''' tvilf t ) ~i l l [he - - 2 ciirection. The relations hetween the field vectors ( E " . , H t) of t . 1 1 ~forwarci propagation mode and those of the backwarcl pri.)p;iga:ion mt:de are shown in Appe.ndix 1-1. If we still clefi~lethe power to be the -4- .? cornpc~r!i:rltof the guided power, we the11

296

obtain

for the guided power in the - 2 direction for (3.3.5). Consider two parallel waveguides, a and b, for which the total field solutions can be written as linear combinations of the individual waveguide modes:

T h e amplitudes n ( z ) and b ( z ) satisfy

where K,, and Khn are the coupling coefficients. T h e expressions for K,, and K,,, will be derived in Appendix H. T h e total guided power is

lvhere the cross overlap integrals

1

E ( q ' ( ~y,)

i(

H ( p ) a ( ~y ,) . f d x d y

(8.3.10)

clj

and s,,, s, = + 1 for propagation in the +z direction, and - 1 for propagation in the - r direction. If we assume that the coupling of the hvo waveguide modes is very weak so that the cross overlap integi-als C,, and C,, are negligible, the total power is then

Substituting (8.3.11) into (8.3.3 2) ancl using ti?$ c-oupled-mode equations (3.3.5), we find

K,,

=

K,,

=

Kzlz

if sash > 0

(codirectional coupling)

-K&

if sash < 0

(contradirectional coupling) (8.3.13)

The coupled-mode equations can be expressed in a matrix form

where

8.3.1

Eigenstate Solutions

The solution can be obtained by substituting

into (5.3.14). We find

where I[ is an identity matrix, or equivalently,

For nontrivial solutions to (8.3.16), the determinant n-iu.st vanish:

'There are two eipenv;iluzs fur ,G:

205

where

For codirectional coupling, K,, = K:',, $ is real for a lossless system..As a n example, we assume two arbitrary dispersion relations

The P , ( w ) and P h ( w ) VS. w curves are shown as dashed lines in Fig. 8 . l l a . Both curves have positive slopes and intercept each other at ( w , , P o ) . The eigensolutions p + and p - are the solid curves. At w = w,, P , = Pb, = JK,,K= K ) , and the splitting ( P + - P - ) is 2 K . Since

we find if K,, > 0 (the expressions for K O , and Kb, will be derived in Appendix HI:

For p

=

PI

A-

-B-

.-

KQb A-*

- A -.

*<

0

(8.3.23)

Khrr

Furthermore, if the two guides have the same propagation constants, Pa we find (for real K,, and codirecrional coupling) ---- 'I

B+

for ,B

=

,B+

= P,,

(8.3.24)

We thus have two possible system modes, the ,~3+(in-phase) mode and the (out-of-phase) mode, which are ?lotted in Fig. 8.12. Fur contradirectional coupling (Fig. 8.I lb), I/, = JAI - lK,,,,~"he propagation constants p, and ,8 .. are cor!~plesin the region:

I

3-0

00

(a) Codirectional coupling

1

I

b0

00

(b) Contradirectional coupling The dispersion curves j3 + and .* rcctional coupling.

Figure 8.11.

P - for (a) codirectional coupling and (b) eontradi-

are shown as the dashed semicircles. The imaginary parts In1 /3+ and Im We also see a stop band with its bandwidth ho determined by J(/3,(o) -P,(w)/21 = IK,,I. 8.3.2

~ e n k r a lSolution of the Coupled Waveguides

After finding the eigerlvectors v , and v,, where

p+ (in-phase) mode

p-

(out-of-phase) mode

Figure 8.12. A simple sketch o f the t w o system modes: the (out-of-phase) mode.

P + (in-phase) mode and the P -

tlie solution for n ( z ) ailcl b ( z ) , given the initial coridition a(0) and b(O), car1 be obtained -in the same way as (5.2.19). It can also be derived using - the eigenmatrix V, where its columns are the eigenvectors of the matrix

a:

The general solution is

The derivation is straightforward. The general solution is a linear combination of the eigensolutions (see (8.2.18) als?):

Therefore, a t z

=

0, we firrc!

Substituting (8.3.30) i1ltc2 i8.-: 291, we obi air^ thc. solution - (5.3.28). The sol~itiom c a n also be exprc.,.;til in terms of :I matrix S ( z ) a n d the initial

-

...... .< : COU?LF.U OPTII.:,,4.!- TJAVE,.j:IID.ZS

conditions:

where

A cos $12 - i--sin +bz

1

@

i -sin gI,z K;a

*

cos +z -k i-sin

$2

I

If at z = 0 , the optical power is incident only in waveguide I, (n(0) b(O) = 0), we find

Therefore, at +z = 7im/2,37;/2, . . . , (2n + 1 ) 7 ~ / 2( n transfer from guide a to guide 6 is maximum. Since

-

=

1,

an integer), the power

for p,, # p,,, the power transfer is never complete (Fig. 8.13a). The minimum distance, L , = T / $ , at which the output power reappears in the same guide as the input is called a best length. If P , = P,, we have $ = J K L r band J , the solutions are

-

where h ' li',,i, = K b L lICj, = FLI= ,Rb ~ I ~ Vb2en C useri. Cornplkte powez- transf2r C)UCUSS for S Y ~ U ~ ~ T O I I O Uc~)uplin,o :; !Fie. - Q.13b). For K ?I = (3ri 4- 1)7~/'2, complete power transfer occu!-:.;: this is called a crc~ss state. For K t j s r , there is 90 power- transfer- ! ' T C ) ~ - I gl~id: ii t i ? .:/lid!: !.I, this is called a p;lr;illsl state.

- c-:; L

:a

'~A';'\\~EC;U 1I)rt ..'O;JIJLEKS !\NC C I)l.rPLED-MODE THLOIIY

(a) Asynchrenous coupling

pa# Pb

(b) Synchronous coupling

Pa= Pb

Figure 8.13. G u i d e d powers 1n(z)l2 a n d llr(z)12 vs. the coupling distance z : (a) asynchronous coupling f i , st P,, (b) synchronous coupIing f l , = Pb.

8.4 IMPROVED COIJPLED-MODE THEORY AND ITS APPLICATIONS [18-321

A detailed formulation of the improved-coupled mode theory to account for either asymmetry # K , , ) or the finite overlap integrals C,, and C , , is shown in Appendix H. The major results are that we have the same form in the coupled-mode equations:

----t?( z ) Liz

where the cueff-icients

:I.,.

y!,.

=

I/<,,',~z( :,) -k

1~

/ , bz( j

and iil,,, ;ire all deiinecl i n (H.30). T h e

transverse fields have been assumed to be

and all the transverse field comporlerlts are real for a lossless system, we find yn, y,, k a b , and k,, to be real. Power conservation leads to 0

d

=

=

-P(z)

dz d -Re[nn" dz

=

d 1 -- R ~ / E , x Hr i d x d y dz 2

+ ab"Cb, + bn*Cab+ bb*]

Here the overlap integrals Cab, C b U rand C = (C,[, + Cb,)/2 have all been defined in (H.11). Since both a and b are arbitrary, we conclude that the coefficients in front of ab* and ba" must vanish. Therefore,

which is identical to the exact relation (H.31). What is important is that when the overlap coefficient C is not small, the total guided power should be given by

where the last two terms due to the cross powers between E(")and H(" or E'"' and HI") are not negligible anymore. 8.4.1: Applications of the Improved Coupled-Mode Theory [23, 24, 291

To illustrate the applications of the improved coupled-mode theory, we corkider two parallel waveguides that are not necessarily identical. We define the ssynchronisnl factor S in terms of the parameters y,, y,, k a h , and ki>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<,(,, k l , , , , a n d arc plotted i;crsus A/;. ( d ) l ' h e outpiit po\ver2s i n g ~ ~ i r lr rt.: P,, (solid CL~I.\J<)iinr:I in pui:.le B , I>,, (idashecl cur-ve) LILYplvttetl vcrsus A I ZThe . par-arnrtcl-s arc tl,, = ii!: =. 2 ~ 1 1 1 wavegi!ic!c , rclgr-to-edge sep:ir:ttiirn 1 =. 1.9 p m . wavelengtl~,i = i.Oh )),[TI. t ! , , '7.1+ Lrl/'.?.: I , , = 1.2 -- t \ - '7. Tile olitsidz refsac,iivs indes is n,, 2.19. The cc:i:p!cr Ic-ngth i:, il.lS! ; il!m. (,5i'te!- Ref. -111, ,?; 1987 Il3,EE.j

-

-

-

waveguide a , i . 1 , 2.2 + A 1 1 / 2 , the eiTective 1x:fractive index in waveguide 6,n,, = 2.2 - Ar1/2, where the refractive index difference

An

=

n,, - no

is proportional to the externally applied voltage V across the two waveguides. The refractive index outside the two waveguides is assumed to b e constant, no = 2.19. The waveguide dimensions are d, = d , = 2 p m ; the edge-to-edge separation t = 3..9 p m . The wavelength A is 1.06 p m . In Fig. 8.14a, the asynchronjsrn 6 is plotted versus the refractive index difference. We see clearly that IS1 is linearly proportional to IAnl- The overlap integrals Cob (dashed line) and C , , (dotted line) with their average C (solid line) are shown in Fig. 8.14b, where they vary between 0.168 at A n = 0 to around are shown in 0.178. The coupling coefficients k,,, k,,,, and Fig. 8.14c, and agree well with the qualitative results of Ref. 23. The output powers P, (solid curve) and Pb (dashed curve) are shown in Fig. 8.14d. Note that the minimum of Pa does not occur right at A n = 0 (where Pb = 1-0) due to the crosstalk. The asymmetry of P, and the symmetric properties of P, versus Ail or the applied voltage agree very well with the experimental data of Marcatili et al. [24], shown in Fig. 8.15. Note that the polarity of the voltage V in Fig. 8.15 is opposite to that used in Fig. 8.14d. More discussions and experimental results on three coupIed waveguides and the crosstalk can be found in Kefs. 33-38. Extensions of the coupled-mode theory for anisi -ropic waveguides or,multiple quantum-well waveguides are reported in Refs. 39-41..

- 20 -60 - 4 0 - 2 0

0

2Q

4C

60

80

VOLTAGE Y . of the modulr~tul-s-for PC, and P1, in Figure 8.15. Solid and dashed curves r l : ~the p o v i e ~.>u:puts tile i n l e t (1s Fig. 8.14~1,rcspt.ctivkly. rrh-:~arc n?eas~~~-ei'l ;IS t'linctions of t h e voltages b' applied to the eiectsodes The dois LI1.e [he res~:!tjpredicted by t h iiriprr~~ccl ~ coupled-rnocle the~u-yin Ref. 23. (After 'Ref. 24.)

.


c

i F PL.1 C.'kV171!lNS OF' OPI'I:2Al, %'A~.I'EGT -1I):,,COIJFL!
8.5 MPLI CATIONS OF OPTICAL \r,17AWGUIDE(COUPLERS In this section, we discuss sonie applications of thi: coupled waveguide theory.

8.5.1 Tunable Filter C42-441 Consider two parallel slab waveguides a and b with different re.fractive indices n o > n , and thicknesses d, < di,as shown in Fig. 8.16. The dispersion relations of the TE, waveguides of guides a and b will have their low-frequency asymptotes approaching P = k,n, and their high-frequency asymptotes approaching either P = lion, and p = k,n ,, respectively. For d, < d, the two dispersion curves intersect each other at (o,, Po)- Therefore, for an incident optical wave from waveguide b at z = 0, b(0) = 1 , a(0) = 0, we find

Kt, a ( e ) == i-sin

*

$!,I

exp

(b)

-

(;I) Two coLtp[2ddiclcct ria: s1~15 v.i:~vtguicler; ( 1 :\nd b w i t h the i-cf~-it~.+'-+ indizt:~. '4 h e l l s i o n s of [lt,c: dl,. { b ) Tkc Li!:j[3t:r:;i~i1 cur it:^ P,,: w ' ) ant1 ,8;,((.)) f ~ the i :;trilctures in ( a ) inlersecl at n p o i n l (,to,,. Pu).

g

,1,,

8-16

;1lh

LII

PI,, ktrh= Knl,, ancl

-

lib,, &,, have been assumed here. I n general, we can always use the more accurate parameters y o , y , , k , , , and k,,, if necessary. The power transfer from waveguide b to waveguide a is determined by

Therefore, we have complete power transfer at Kt' = w/2, 3 ~ / 2 ,5 ~ / 2 , etc., only at w = w,, P,, = P , , A = 0 and 4 = K,, = K,,,(= K ) . For a fixed coupler length L' = ~ r / 2 K , for example, an incident light with a wide spectrum on waveguide b, the output optical power in waveguide n will contain signals of frequences close to w,. T h e directional coupler acts as an optical filter. Furthermore, if the materials are electrooptical so that we may apply a voltage to change the refractive indices n , and n,, the intersection frequency w , will be tunable. Optical wavelength filters using waveguide couplers have been reported for Ti:LiNbO, and InGaAsP/ InP materials. T h e In ,_,Ga, As,, PI / InP material system is especially interesting because of its applications at 1.3 or 1.55 p m wavelengths and its potential for optoelectronic integrated circuits. The experiment reported in Ref. 44 has two-coupled waveguides: one has a narrower guide width d , = 0.42 p m , but a larger refractive index n, than n , (obtained following Ref. 45) with y, = 0.127; the other has a guide width d, = 0. .: p m and y , = 0.075. The galIium mole fraction x u (or x , ) depends on the arsenic mole fraction y, (or y,) for lattice matching. The input power is assumed to be I in waveguide b. The output power from waveguide a , P,, peaks at 1.12 p m . We have applied the improved coupled-mode theory and compared our theoretical results (solid and dashed lines> with the experimental results (open circles) in Fig. 8.17. T h e reported in Ref. 44 used for the theoretical calculations are within the measurement accuracy. ?'he possible reasons of discrepancy may be that (1) there is still some difference between the tl~eoreticalmodel in Ref. 45 and the experimental values for the refractive index, and (2) the losses in the waveguides are not taken into account. However, the comparison shown in Fig. 5.17 shows very good agreement in general.

-,

8.5.2

Optical Waveguide Switch

Cur-isiilcr ;t clirectional coupler with an inciilznt light into waveguide 11, as shown in Fig. 8.18. Assume tr(O! = I. The o~ltplltpower from waveguide b is

OUTPUT PO'/\/ERS

WAVELENGTH (MICRON) Figure 8.17. T h e output powers of two coupled In,Ga, -,As, P,-,/ I n P waveguides used as an optical wavelength filter. T h e input power is assumed to be 1 in waveguide b. T h e theoretical results for output powers at waveguide a , P, (solid curve), at waveguide b, Pb (dashed curve), are compared with the experimental data of Ref. 44 (open circles) for P,. (After Ref, 29, Q 1987 YE.)

No power is transferred to waveguide b at the exit if

$I?

.

= n7i-

n

=

1,,2,3, . . .

We can plot A f and K t in the plane to represent this state, ,called the parallel state @ since all input power into guide n at z = f exits from the same waveguide. I11 other words, the equations

1

V/h~aV/A a

i

8-12. An opticn! wavr:guide switch.

Figure 8.19. A switching diagram fur a n optical waveguide switch. T h e @ slates are represented by tllr solid cur.vzs and the @ states are discrete points o n the k-t' axis. The horizontal dashed line shows a switching from a 8 state toward the = state.

0

represent a set of circles on the K t vs. A t' plane, as shown in the switching diagram, Fig. 8.19. O n the other hand, complete power transfer can occur 0 and Kt' = (2n2 l)7r/2, where = 0, 1 , 2 , . . . . W e call this only if A" a cross state 8 when the total power is transferred from one waveguide to the other. These cross states are represented by only discrete points labeled as @ on the vertical ( K O axis. Using ti:,. electrooptical effects, which are discussed further in Chapter 12, the refractive indices in the waveguides can be changed by an applied voltage bias. T h e refractive index difference between two waveguides, or the mismatch factor A = ( p , - P,)/2, can be tuned by the voltage. Therefore, we can switch the directional coupler from a cross state at Kt' = ~ / (and 2 A = 0) to a parallel state, shown as the horizontal dashed line in Fig. 8.19.

+

8.5.3 The

ilp

Coupler

It is difficult to achieve switching in the design of a single-section waveguide coupler since the coupling coefficient is a function of the spacing and the material parameters; these parameters are harder to adjust in the fabrication processes. It is generally easier to control coupling by an applied voltage to change Ap = Pb - P,,. A two-section waveguicle coupler has been proposed [46] to achieve switching w i t h more flexibility, as s h o w in Fig. 5.20. T h e amplitudes a ( z ) and b ( z ) for a coc~pledwaveguide configuration as shown in the first half szction of Fig. 5.21) are related to thc initi:~lconditions by

hj z )

=

S( I ; p,, . p,,)

Figure 8.20. A A P coupler with two sections where the propagation constants exchange positions because of the change in the applied voltages.

where we keep the order p, and /3, in the arguments of the

.K u b

I --

-

3 matrix:

sin t,bz

$1

cos $2

P, and P,

A

+ i-

dJ

sin I(,z

1

The i.atput mode amplitudes at the exit, z = t ,are obtained by the product of two matrices because of the cascade connection:

3 i .4

If K , ,

-

W/.VF,C;LilDL:

!-:C.I.JPLL:RS kP.iD Ct.)'C!PL.EG-PilODETI-IEC)I'?Y

-

I<,, = K , ( ~ ( 0 ) 1, and h{O)

b ( t ) = i-

2K $

@f

sin-

2

[

- 0,we find A

- i-sin-

cos-

IIJ

iit) e i(,b r 2

The full power transfer (cross state) occurs when n@

=

0, or

For example, if A = 0, we find q!~ = . -. K , and c o t 2 ( ~ b / 2 )= 1. Therefore, K t = ~ / 2 ,3 ~ / 2 ,57i-/2, etc. No power transfer (parallel statej occurs if \b@12= 0. This can happen in either of the following cases:

Case 1

Case 2 :

7 Ge cos-2

$& = 0 2

sin--

+

a2 . -sin-$2

?

fit 2

=

0

For case I, @ e = 2nz-rr7or

which are circles in the K t vs. A t plane for rrr = a nonzero integer. For case 2, it is satisfied only when A = 0, and c o s ( K t / 2 ) = 0, i-e., K t = (2m 1)r,m = an integer. A plot of the switching diagram is shown in Fig. 8.21. The switching diagram shows the degree of .freedom to switch from a parallel (3 - to a. cross @ statt. or vice versa. 'The theory for coupled waveguides has ;\IS(:, been extended to gain media [47,481, coupled fiber Fabry-Perc?t resonators [49, SO], laser amplifiers [5 1, 521, and coupled laser 2irr;l:ys [53---551.11; Section 10.5, we d isci~sscoi~pled serniconclc~ctorlaser arrays, :\ad tlie couplecl-n:odt. theory is used to investigate the superrnode bel.laviors.

+

27~

7.c

Figure 8.21.

8.6

37r

47r

Switching diagram for a AD coupler.

DISTRIBUTED FEEDBACK STRUCTURES [6, 8,56-641

Consider a corrugated waveguide structure as shown in Fig. 8.22. The permittivity function ~ ( r can ) be written as an unperturbed part &(O)(r)and a perturbed part A&):

For the unperturbed part, assurne that a particular guided mode with a propagation constant P, or --P, is of interest. The amplitude for the fonvard

*

..

r ~ g n l - e8.22. ( a ) A c ~ ~ ~ u g : iw;ibeg~:icllte~l tvitl: a pdr-n;itlivity function t:(r) can he w r i t i c ~ las the .
-

-

WAVEGUIDE COUPLERS AND COUPLED-MODE THEORY

316

propagating mode a(z) is aoe;/l,Z and backward propagating mode b(z) is b oe-;/3,=. Since g(r) is periodic in 2, we may write, for Fig. 8.22, using the Fourier series expansion

';g(x , z)

L p_

dp(x)

(8.6.2)

e;p(h/ A)z

-00

where A is the period. For a lossless structure, ';g(x , z) is real; therefore, (8 .6.3) 8_6.1 Derivations of the Coupled-Mode Equations for Distributed Feedback Structures

From Maxwell 's equations, we find, for E =

y2Ey -

J.L o£(O ,(

a2 aT

x, z) - 2 Ey

=

yE y , a2 2Ey aT

J.Lo';g( x, z) -

(8.6.4)

For a time harmonic field with an e-;wr dependence, we find (8.6.5) We expand the solution E,. in terms of the unperturbed waveguide modes ( 8.6.6) m - - co

where m refers to the mode numbers, TEo, TEl ' TE 2 , and so on, and E;,m)(x) is the transverse modal distribution with a corresponding propaga-

tion constant

13 m : (8.6,7 )

We substitute the general expression C8.6.6) in the wave equation and ignore the iJ 2A mCz) / ilz 2 terms, assuming that AmCz) are slowly varying functions. We find

" ' p_

- 0 '.

317

8.6 DISTRIBUTED FEEDBACK STRUCI1JRES

If we multiply both sides of (8.6.8) by (f3, / 2wJ.L)E~')*(x), which is related to the x component of the magnetic field of the guided sth mode by H;' )(x) = (-f3,/ wJ.L)E~')(x), and make use of the nonnalization condition,

we find

( 8.6.10) where we have assumed that e(O)(x, z) is real ; hence , f3, is also real. If we focus on the coupling of the forward propagating f3, mode to the backward propagating (m = -s) mode so that f3 - s + p(2r. / A) is close to f3" for a particular p = t where f3 _, = - f3" assuming the couplings to the other modes m -s are negligible, we then have

*

dA , Cz) _--,-,-c.. dz

=

where the coupling coefficient

.

IK

(z)

A ab_

Ka b

.

e- '2(/l,- (~/A)'

-s

(8.6.11)

is

(8.6 .12) A phase-matching diagram is shown in Fig. 8.23, where we consider the first-order grating t = ± 1 such that f3 , = - f3 , + 2r. / A and A _,(z) is coupled to A,(z)"tiy (8.6.11). Equation (8.6.12) defines the coupling coefficient for TE modes in a corrugated waveguide. Extensive research work has beeri done on both TE and TM mode coupling coefficients in distributed feedback structures and applications to DFB lasers [58-64]. If we multiply (8.6.8) by the wave function H;- ')(x) for the -s mode with a propagation constant f3- s = -f3" we find the second coupfed-mode equation dA

-,

(z)

dz where the coupling coefficient

-·K A ( ) i2(/l,- t~/I\)' -Ibasze Kba

. (8 .6.13)

is

(8 .6.14) We also see that the condition for contradirectional coupling in a loss less

Figure 8.23. Phase-matching diagram for a forward propagating mode with a propagation constant 0 , coupling to iI hnckwilrd propagating mode with a propagation constant P..., ( = -0,) in t h e negative z direction, p , - p , + 2 - r r / A is the phase-rnalching condition fur the z c o t ~ ~ p o n e nof t s the PI-opagation wave vectors. The difference 4fl = (0, - T/A) is the cletuning parameter.

-

coupler

K,,

=

-K:,,

(8.6.15)

is satisfied since d - , = d:. As shown in Fig. 5.23, the forward propagation amplitude A , ( z ) is coupled to the backward propagation amplitude A-,(z) by (8.6.13). 3 i : :refore, the phase-matching condition requires P '-,= P, 2 ~ r / A . Define

We obtain

5.6.2

Eigenstate Solutions of the Coupled-Mode Equations

We follow a similar procedure as before for coupling in two parallel waveguides by assuming the eigensolution of the form

The eigenecluaiions become (using K = K,,,)

:

The eigenvalues are

where

In the original expression for the electric field (8.6.61, we keep only two counterpropagating modes, nz = s, - s (for the forward and backward propagating TE, modes):

The magnitude of the electric field for the forward propagating mode is

The magnitude of the backward propagating mode is

Ttle total electric field (8.6.23) can be written as

I

The propagation constants of the system modes are

Far away from the point p,(o~,,) = !ir/;l_when Ifi,,(o) ( (?r/l\)l > I Kl, j3A

is real, and the PI-opagationconstan; apprxiches t h e following Iin?.its:

A t w = w , defined by P,(w,) = e n / A , the real and imaginaly parts of the propagation constants 3/, satisfy

and S

=

( K [ .For w in the range I P S ( u )- E r / A l < IKI, we have

In Fig. 8.24, we plot the dispersion relations for e = 1. W e see that the stop band occurs at the vicinity of the frequency w,. For the bvo eigenvalues 4 is, their corresponding eigenvectors are determined by

Tfie general solution can b e written as

8.6.3 Reflection and Transmission of a Distributed Feedback Structure

For a wave incident from the l e f t side of the distributed feedback structure at z = 0, as show11 in Fig. 8.25, we may flnd the general solution using the fact that at z = L , B ( L ) = 0, sincz there is no incident wave from the right-hand side. Therefore,

Figure 8-24. (a) The dispersion curves for the contradirectional coupling of modes in a perio?i2 waveguide structure; ( b ) the enlarged portion of the intersection region.

Incident &A@)

L

*+A(L)

Transmitted

Figure 8.25. A periodic cosrugaied w:ivcgilicl~structurr with a n inciclrnt wave :I! z = 0. AiO? is the ficlcl a n l p l i t ~ ~ doef the incident wave atirl D(O) is the field :~rnplitucfe of the reflected wave. .:l(L) is t h e field anlplit~ideof the tl-:x~srnittedivave L : : I ~ 13! L ) = 0 since v ~ asslime r n o wave is incidtnt from the ripht-hnnd side.

For a given A(O),we have

-K(c,

+ c,)

-

(8.6.34)

A(0)

Solving for c , and c, in terms of A(0) and substituting into (8.6.32), we obtain A(z)

=

B(z)

=

-Ap sinh S ( z - I.) +- is cosh S ( z Ap sinh SL + i S cosh S L K*sinh S ( z

-

t)

A ( 0 ) (8.6.35)

- L)

+ iS cosh SL 4 0 )

A p sinh SL

The reflection coefficient at z

-

=

0 is

-K'* sinh S L

Ap sinh S L

+ iScosh S L

which approaches 1 if 1 KLl is large enough. Also, when becomes purely imaginary and 1 r(0)l2 vanishes when

Figure 8-26. A plot of the reflectivity IN-!.For

1x1-ge v:llue

I/-(())/' vs. i f l e iii:io

o[ IN-\. thr: t.randwldt11 fur

IA

> I K 1, S

ng/iI
is about

A plot of the reflectivity 1r(0)1*vs. A,f3/IKI for thrae different values of 1 KIAI (i.e., three lengths of DE'B rellectors) is s'hosvn in Fig. 8.26. \,'\re see that if (filincreases, the reflectivity for small Apt in~reascs.Wc should note that when IS/ -- 0, lA,bl - IKI, and lr(0)i2 + IKLI'/[II(LI'+ 11. The transmisL is sion coefficient at z

-

t ( L ) = -A( 14

A(0)

=

is Aj3 sinh SL t- 15'cosh SL,

The theory developed in this section will be further applied to the study of distributed feedback lasers in Section 10.6. PROBLEMS

8.1 A TE tight ( A = 0.8 ,urn) is normally irlcident on a prism from air as shown jn Fig. 5.27. T h e waveguide width d is chosen such that the light can be resonantly coupled into the waveguide, thereby, the phasematchkg condition is satisfied. (a) Find the propagation constant k,. Cb) Find the Goos-Hanchen phase shifts for the guided mode for both the air-waveguide and substrate-waveguide interfaces. Cc) From the ray optics picture, determine the width ci from the propagation constant k Z obtained in (a).

"a

d t -

&ng

n,

Air Waveguide Substrate

Figure 8.27.

$ 2 Consider an InP/ In,,,,Ga ,,,As/ InF dielectric waveguide with a diri~ension(I= 0.3 p n l . 'The wavelength in free space A,, is 1.65 p m . The refractive i ~ ~ d i c ea.re s T I ,(ln ?Ga As) = 3-56 and !I 3.16. T h e propagation constarlt for the TE,, mode is found to be ,3 12.7 ( ,u~ I Y ( i d If' a grating coupler is cicsig~ledusing the same materials, Fig. 8-28, so that tht; first-orc!ci- space h a r i i ~ i : ) ~n~ocle ? i ~ (1.7; = I ) is cr.)uplcd t o the TE,, mode at d; = 45", find thc grating pe~.lrjc!i3.

,,., ,,,,,

'.

- -

(b) Repeat part (a) for rrz = 2. Draw a phase matching diagram to show all of the propagating (nonevanescent) space harmonics (called FIoq~ietmodes) in the InP region for this case and show their reflection angles measured from the x axis. ( c ) Consider a grating output coup1e.r shown in Fig. S.25. If the output coupler is designed so that there is one and only one Floquet mode propagating in the x direction (this configuration also has applications for surface emitting diode lasers), what should the period A,, be?

K n

Output

InP Figure 8.25,

8.3

Consider a three-parallel-waveguide coupler shown in Fig. 5.29. Using thq coupled-mode theoly, the electric field of the three-guide coupler

where E ? ' ) ( X ) describes the x dependence of the TE,, mode of waveguide j ( j = 1,2,3) in the absence of the other two guides. The coupled-mode equation can be written as

where the coupling between guides 1 and 3 has been neglected since they are far apart. (a) Find the three eigenvalues and the corresponding eigenvectors of the three-waveguide coupler. (b) Sketch approximately the mode profiles of the three eigenmodes as a function of x . (c) Given the initial values, n,(O), n,(O), and u,(O), find a,(z), u,(z), and a,(z) in terms of their initial values. Put your results in a matriv form. 8.4

The coupled-mode equations for N identical equally spaced waveguides, assuming only adjacent waveguide coupling, can be written as

where

(a) Show that the general solutions for .the normalized eigenvectors and their elgenvalucs for an arbitrary iV arc

where

p,

=

/3

-+

2 K cos

N

1

1

T h e order of /3/ is chosen such that Hint:

I

2 cos 0 1

det

I 2cos 8

P , > P, > P, >

0 3.

1

2cos 8

1

1

-

. > 3/,

sin[(^

+ 1)8]

sin 8

where N is the dimension of the nlatrix. (b) Find the eigenvalues and their corresponding eigenvectors for N = 4, and plot the transverse field profile E(x) = X:= ,aiE ~ ( xfor ) all four eigenvalues. 8.5

(a) Sketci-;qualitatively the dispersion relation PC,of the TE, mode vs. w for a single slab waveguide in Fig. 8.30a with a fixed refractive index r z , at two different thicknesses ( d , < d,). Explain your sketch of P vs. o curves using the graphical solution for a d / 2 vs. k,d/2. ., (b) Consider the two coupIed slab waveguides (Fig. S.30b). with 12, > n,,, derive a condition between d, and d , such that the dispersion curves /3,(o) intercept at a frequency w,.

5.6

Consider a distributed feedback structure as shown in Fig. 8.37a, where

E - & I

if--h<x<0

and

A

O
and A E ( x , z ) is periodic in z , Fig. 8.31b. ( a ) Find the Fourier series expansion of A E ( x , z ) , i.e., d,(x), where

(b) Evaluate the coupling coefficients K,,, and K , , for the TE, mode. Assume that the first orders p = 1 provide the coupling. (c) Let K = K,,, = -K&. Consider an incident wave from z = 0 with

+

an amplitude A(()).Derive the general expressions for B(0) and A( e 1. (dl Plot the yeflectivity I B ( O ) / A ( O )vs. I ~ lA,!3/KI for K t = 4.

Consider a distributed feedback structure as shown in Fig. 8.32 where the hcight function is

Let the u n ~ e r t u r b e dE " ! ( x , z ) be

6'')

=

[

-d < x < O

otherwise Repeat Problcrn 8.6(a)-(d) for this sawtooth grating structure. ( E

Figure 8.32.

(a) Describe how a 4 P coupler works and its advantages compared with a single-section directional coupler. (b) Show the switching diagram ( K t vs. A e ) with as much detail as possible. (c) Consider an input with n ( 0 ) = 1, b(0) = 0 , 4 e = 7i-/2, and K t = s;-, where e is the total length of the A P coupIer. Find the output powers I n(C 11' and I bV )I'.

In! Write down the expression for the coupling .coefficient K for two identical parallel wrweguides. Corlsider only the TE,, mode. DisCUSS clcialitativeIy all possible ways and trade-offs to increase the coupling coefficiex~t. (b) Repeat (a) fur the ci?upli~lgcixflizicnt K f ~ ar clistributed feedback struct~ire.

(.c) Compare the coupling coefficients !:r:(a) and (b). What are their similarities and differences? 3-1-13 (a) Consider the slab waveguide in Fig. 8.3321 with a TE, mode profile approximated by E,(x) 6 , e - - J ~at/ the " ~facet z = 0, and E, is assumed such that the guided power is normalized. Find the far-field radia tiun pattern using an approximate formula,

-

I

Input facet

Directional

Output

fclcei t:c)

Y i ~ s i - e3.73.

330

WAVFCiU1I:)E (:'OIIP!..F!'\?I AN;] !:'01Ji'l,ED-MODE THEORY

where we have ignored all prefactorr. Find the angle O,,,- where the radiatcd power is half of its pcak value a t 8 = 0, that is, I P ( 0 = H,,,) = ;_P(O = 0). Assume that d = A,/4, and A" is the wavelength in free space. (Note: I"_ e-"-" d x = (b) Suppose we use thc approxin~ationin part (a) for the TE, mode in two coupled identical waveguides with the center-to-center separation s = A,, where A, is the laser wavelength in free space (Fig. 8.33b). Calculate the radiation pattern for the in-phase eigenmode of the coupled waveguides. Roughly plot the far-field radiation pattern of the in-phase mode by specifying the angles for the zeroes and maxima. (c) Suppose we shine a light to excite the TE, mode in guide a only on the input facet of the coupler and look at the electric field profile at the output facet at a distance E away from the input facet (Fig. 8 . 3 3 ~ )For . e = 7 ~ / 4 Kwhere , K is the coupling coefficient of the two identical guides, find the expressions for the electric field profile at the outprlt facet and its far-field radiation pattern. Use the approximates for the TEo mode profile in (a) and (b) for simplicity. Explain how one may control the radiation pattern using the directional coupler.

45.)

8.11

(a) Find the far-field pattern due to the TE, mode using the numerical values from PI-oblem 7.1 of Chapter 7 on the output facet of a diode in the 8, direction. For simplicity, assume that the TE, mode i iiniform i n the y direction froin - kV to + W, and D is very largc: (see Fig. 5.5). Plot the power pattern VS. 0, roughly. (b) Consider a directional coupler with three identical guides as in part (a). Using the eigenvalues and eigenvectors from Problem 8.3, plot (i) the three electric field profiles of the coupled waveguide system and (ii) the far-field radiation pattern P ( 0 , ) of the three eigenmodes, assuming that the waveguide center-to-center spacing is A,,/ 2, as shown in Fig. 5.34.

I

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7 5

9 3

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L\;',\VEC;~JILIE C'i>LJl'i.2E'is AP! I 1 COL'PL,EIS-T~IOT)ETI-IEORY

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39. D. Marcube, "Coupled-mode theory for anisotropic optical waveguides," Bell S,vst. Tecll. J . 54, 9S5-995 (1975). -1.0.

A. Harcly. ,W. St~.eifer,ancl

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~vaveguide systems ( 1936). 41.. I,. Tsang ancl S. L. Chi!nng. "Coiiplecl (1;:lAs multiple-quantum-well chnnrrel viuveguicles including clu3cli.atic: C ! E C ~ ~ . O O P ~~~ Cf i ~ c t J, ." I,ig/lt~'(l~'e ?'~c/LIzo/. and ZEEE .I. Q ~ L ~ L ~ EICL.II.OII. Z ~ L ~ 0: J XS33-836 (1988).

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R. C. All'erness and K. V. Schmiclt, "Tunable optical waveguide directional coupler filtcr," A p l .

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33, 163 -163 (1978);

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Optical Processes in Semiconductors The quantum theory of light has been an intriguing subject since 1900 when Planck [I] proposed the idea of quantization of the electromagnetic energy. In this theory, the light or the electromagnetic wave with an oscillating frequency w is quantized into indivisible packets of energy Ftw. In 1905, Einstein [2] proposed that the photoelectric effect could be explained using a corpuscIe concept of the eIectromagnetic radiation. With the invention of lasers, quantum theory of light also plays an important role in our understanding of many optical and electronic processes in materials such as semiconductors. Rigorous in-depth treatments of the quantum theory of light can be found in Refs. 3 and 4. In this chapter, we study the optical transitions, absorptions, and gains in semiconductors. We start with the Fermi's golden rule, using the time-dependent perturbation theory derived in Section 3.7. We discuss the Einstein A and B coefficients for spontaneous and stimulated emissions and show how they can be calculaF-d quantum mechanically. A formal density-matrix approach can also be L. :d to derive the optical absorption and gain coefficients in semiconductors and is shown in Appendix I. We then discuss the optical processes in bulk and quantum-well semiconductors. The optical matrix elements are derived based on Kane's model for the semiconductor concluction and valence-band structures. For serniconductor quantunl wells, valence-band mixing effects play an important role in the optical gain of a clunntum-well laser. We show how these heavy-hole and light-hole band mivi~lgeffects can be taken into account in the modeling of gain spectrum in quantum-well structures.

9.1

OPTICAL TRANSITIONS USING FERhII'S GOLDEN RULE

C'onsicfrr a semiconc1ucto1-illuniinated by light. The interaction between the p t ~ ~ ) t c ~:ind n s the e!ectrons in ihe sernicontl~!.~tor can'bi: described by the J-Ian~iltoni:~n [ 5 ]

where I U , is thc free electron mass, e = -lei for electrons, A is the vcctor poterltial accounting for thc presence of the elect]-omagneticfield, and V(r) is the periodic crystal potential. The above Hanliltoniarl can be understood because taking the partial derivatives of H with respect to the momenturn variable p and the positiorl vcctor r will lead to the classical equation of motion described by the Lorentz force equation [5] i n the presence of an electric and a magnetic field (see Problem 9.1).

9.1.1 The Electron-Photon Interaction Hamiltonian The Hamiltonian can be expanded into

= H,,

+ H'

-.

where H , is the unperturbed Hamiltonian and perturbation due to light:

.,

H' is considered as a

The Coulomb gauge (discussed in Section 5.1)

C-A=O has been used such that p A = A . p, noting that p = ( i t / i ) C . T h e last term e2~'/2nz,, is much smaller than the terms linear in A, since leA/ -s< Jpl for Zzk = most practical optical field intensities. This can be checked using p f r r / a , , , where a , 5.5 A is the lattice constant. Assuming the vector potential for the optical electric field of the form

-

where k , , i-, the

-

0

WLIYC V.IU~(>:-.

!:i

i:, the optic:!! a!?gular fi.eili!ency. niici e is a

unit vector in the direction of the optical electric field, we have

since the scalar potential cp vanishes ( p = 0) for the optical field. Thus the Poynting vector for the power intensity (w/cm2) is given by

o ~- ; - k,-sin'(k,,

-

- r - ot)

P

which is pointing along the direction of wave propagation k,,. The time average of the Poynting flus is simply

noting that the time average of the sin2(.) function is 1/2. Thus, the .- magnitude of the optical intensity is

S

=

l ( S ( r ,t ) ) l

o A: = -kc,, =

2~

nrw2Ai 2pc

--

~ I ~ C B ~ W ' A ~ (9.1.12) 2.

where the permeability, kr = p , , c is the speed of liplit in free space, and r l , is the refractive index of the material. The interaction Harniltonian H 1 ( r ,t ) can be written as

and tl:a superscript

+

means &heHermitian adjoint operator of N 1 ( r ) .

9.11.2 Transition Rate due to Electron-Photon Interaction T h e transition rate for the absorption of a photon (Fig. 9.la), assuming an electron is initially at state E,, is given by Fermi's golden rule and has bcen derived in Section 3.7 using the time-dependent perturbation theory:

where Eb > E, has been assumed. Similarly the transition rate for the emission of a photon (Fig. 9.lb) if an electron is initially at state E,,is

The total upward transition rate per unit volume ( s - ' in the crystal, taking into account the probability that state cr is occupied and state O is empty, is

where w e sum over the initiaI and final states and assume that the FermiDirac distribution

is the probability that the state cr is occupied. A similar expression holds for f,,where E, is replaced by E,, and (1 -- fb)is the probability that the state b

(a) Absorption

Eb

(b) Emission

F&l

t

-l

~iiti~~l

state

Figure 9.1.

( a ) Ti12 abhc,rption

electron transiliuns

;illd

E

b

I

I

ii:) tl15 e:~:is:,iorl of

n

i

t

i state a l

'=-

Final stat?

;I

pllotcjn

?\

it11

the col.responrlins

is empty. The prefactor 2 takes into account the sum over spins, ancl the n ~ a t r i velement HL,, is given by

with IHL,I = IHibl. The downward transition rate per unit volume (s-' cmP3) in the crystal can be written:

Noting the even property of the delta ft~nction, 6 ( - x ) upward transition rate per unit volume can be written as

fC

=

R,,,

=

-

2

-R

9.1.3

--•

6 ( x ) , the net

--

*

27~

C k.

b

=

-IH;,I' fl

L,

Optical Absorptii

,I

(9.1.21)

S(E, - E, - f i , ~ ) ( f -( f, b )

Coefficient

The absorption coeficierlt cu ( l / c r n ) ixl the crystal is the fraction of photons absorbed per unit distance: I . I

a =

No. of'photons absorbed per unit volume per second --

(9.1.22)

No. of photons injected per unit area per second

The injected number of photons per unit area per second is the opticaI intensity S (LV/cm2) divided by the energy of a photon h w ; therefore,

-

Using the dipole (.long w;jv(~i,:!r:~c!h) ;ippi.(>::in-iit.ii(:~i~,that A(r) = h t l i k ~ ~ 2 :l12. ~'i we find that the matt-LY,uelen!c:l!s (:irr? be written i n tcim:.; (.)f the nlomentun

matrix elernent:

The absorption coefficient (9.1.23) becomes

T h e Harniltonian can also be written in terms of the electric dipole moment:

because

and

where Eb - E,, = tiw has been used, and E = + iwA for the first term in A(r,t) with exp( - i w t ) dependence' from (9.1.6) and (9.1.7). In terms of the dipole moment, we write the absorption coefficient as

We can see that the factors containing A ; are canceled since the linear optical absorption coefficient is indepetldent of the optical intensity, which is proportional to A:. When the scattering relaxation is included, t h e delta

f~lnctionmay he repiaced by a Lorentziarl function with a linewidth 11:

where a factor n- has been included such that the area under the function is properly normalized:

/ 6 ( ~, Ea - l i w ) d(kw)

=

1

9.1.4 Real and Imaginary Parts of the Permittivity Function The absorptioi-2coefficient cu is related to the imaginary part of the permittivity function by

--- - W -

€7

nrc: E,, assuming that -=z< E , . Here a factor of 2 accounts for the fact that cu refers to the absorption coefficient of the optical intensity, not the electric field. We obtain the imaginary part of the permittivity function:

-

re'

2

nliw'

V

C XI2 . pb,12S ( E b - E,, - Rw)( f, - f b )

(9.1:34)

In semiconductors, the conduction and valence-band structures determine the energy-momentum reIatiorls E,: E(li,) and E, = E ( k , , ) , respectively. t2n important part of calcillating the absorption coefficient is to find the band structures ancl the \j72ve fanctions. 1'112 c::prical rn:tt:k element p , , is calculated fron: the wave functlens l ~ n s ~ on ( ! the parabolic band model. We far vihich tI?c band structures arc usu;tlly start from bulk scmiconcSu~t;c~~-s known. T h c n we usc the k . p perti:rb:itia:,n method anii rhc effective mass

-

-

theory near the band edges, as cliscusscd in Chapter 4, to study the optical processes near the band eclges of bulk and qUantum-well semiconductor structures. Using the Kramers-Kronig relation derived in Appendix E,

where P denotes the principal value of the integral, we obtain

If, instead, Eq. (9.1..30) with the dipole moment rnatriv is used, we have

.,(a)

= Eg f

2 I/z Cle^' k,

kh

2

3 ( E b - E,) 2

( E b - E ~ -) ( ~h w )

( f'? - f b )

(9.1.37)

and

In practice, linewidth broadening caused b y scatterings has to be considered to compare the theoretical absorption spectrum with experimental data. We replace the delta function by a Lorentzian function in (9.1.34) using (9.1.31),

The real part of the permittivity function ~ ! ( wcan ) be obtained using (9.1.36) for a zero linewidth with the factor L,/<E, - E:, - A m ) replaced by

and we obtain

Similar procedures apply t o (9.1.37) and (9.1.38).

9.2

SPONTANEOUS AND STIMULATED EMISSIONS

In Section 9.1, we presented a quantum mechanical deviation for optical absorption of semiconductors in the presence of a monochromatic electromagnetic field using Fermi's golden rule. Here we discuss the spontaneous and stimulated emissions in semiconductors and derive the relations between the spontaneous emission, stimulated emission, and absorption spectra [4, 6, 71. First we consider a discrete two-leveI system in the presence of an electromagnetic field with a broad spectrum (Fig. 9.2a). The total transition rate per unit volume ( s - 'cm-') is given by 1

R,,

=

b7

277 k

S ( E 2 - El - h w k ) Z n p h

-IH;,I2 A

where Hi, is the nlatrk element of the interaction Hamiltonian due to the electromagnetic field and Aw, is the energy of a photon with a wave vector k.

I -

Occupied

El

Figare 9.1. !a) A photon iacidrnt r)ri :I discrr?te rwo-level system where !eve1 1 is occupied zin1.l level 2 i:; empty. Ih) The k-space diagrtlr,~for the density c>f photon statcs. A clot represents onz slate with two possible pc)l:triziitions.

::4:,

OPTiCAL PF.C("ESSI:S Ti4 SEMICONDUCTORS

The expression

is the number of photons per state following the Bose-Einstein statistics for identical particles (photons), and a factor of 2 accounts for two polarization states for each k vector. T h e photon field can be described by a plane wave for simplicity as

9.2.1

Density of States for the Photons

T o define the density of states for the photon field, we use the periodic boundary conditions that the wave function should be periodic in the x , y , and z di-rections with a period L . Therefore,

The volume of a state in the k space is therefore ( ~ T / L ) (Fig, ~ 9.2b). Using the dispersion relation for the photon,

where c / n , is the speed of light in the medium with a refractive index rz,, we can change the sum over the k vector to an integral. Let us look a t the integral using the number of states with a difierential volume in the k space d 5 k / ( 2 n / ~ ) ' = k' d k d f 2 / ( 2 7 r / ~ ) ~where , d R is the differential solid angle,

V is the volume of the space,

3.2

SPOj'.JT'ANF.OUS A N C ~~TI'l?dll-!Z..:Y T E n E':;I!SSYO;xjS

347

is the photon energy, and the integration over the solid angle is 47r. We find

which is the number of states with photon energy E,, per unit volume per energy interval, crnA3(ev)-l , and E,, = E , - E , is the energy spacing between the two levels. Note that the Ylanck constant /z instead of h is used in (9.2.8). 9.2.2 Spontaneous and Stimulated Emissions: Einstein's A and B Coefficients

Define

as the transition rate per incident photon within an energy interval, (eV/s). ~ )a broad We obtain the upward transition rate per unit volume (s-I ~ m - for spectrum or incoherent light (blackbody radiation) Rll = 4

2

P ( ELI

where

is the number of photons per unit volume per energy interval with a dimension of c ~ ' - ~ ( e l ~, and )-

is' the average number of photons per state at an optical energy E,,. If we take into account t h e occupation probabilities of level 1 and level 2, f', a n d f,, respectively, the expression for R is slightly modified (Fig 9.3):

,

where r , -, ( E ) d E means t h ~ l ithe ilpwarri transition rate per unit volume has bee11 integrated for a light with a spzotriii w i d h d E near E = E?,. r , . ( E ) is thc nrimbcr c j f I -+ 2 triinsitioi~spzr x c i ~ n r : I pcr unit volume pe,r energy "1intclval ( $ - cm be~~" '-

OP'TIC'AL I'RG(:FSSES 1N SEMICONDUCTORS

stim R12

Figure 9.3. Schematic dingram for the stimulated al~sorptionrate X,,,stimulated emission rate ~;'i", and the spontaneous emission rate R;';On in t h e presence of two levels with q ~ ~ a s i - F e r m levels, i F , and F,, respectively.

.

spon

R21

R21

ho

Similarly, a stimulated emission rate per unit volume can be given:

The spontaneous emission rate per unit volume is independent of the photon density, and is given by

-.-

At thermal equilibrium, there is only one Fernli level, therefore F , = F 2 . We balance the total downward transition rate with the upward transition rate:

R,z

B,,f,(l - f , ) P ( E , , )

We obtain

=

= &I

stim

+ ~ s p o n

2I

(9.2.16)

B,,:l-,(l - f , ) P ( E 2 , ) + A2,f,(l - f , ) '(9.2.17) e

Therefore, .we find

and

Thc ratio of the stimuIated and spontaneous emission rates is

which is the namber of photons per state (9.2.131.

9.2.3 Derivati,on of thz Optical Gain and Spontaneous Emission Spectrzlm The net absorption rate per u n i t volume within a spectral width cl E is

Therefore the absorption spectrum within a spectral width d E can be written as

The ratio of the spontaneous emission spectrum and the absorption spectrum is

and A F

F , - F , is the quasi-Fei-mi level separation. If we consider the net upward transition for a monochromatic light (a single photon with E = h o ) instead of a light spectrum, we use r-$f( E ) , that is, the net upward transition rate per unit volume per energy intervaI from (9.2.21) with n,,, = 1, =

:'by (E) d E '

'nt t

=:

p' 1 2 ( f, - f 2 ) /V( E j

We ~hiingc;.1 /ci E to a dcita f u n c t i o ! ~for

..i

pair vf discrete slates Eq - ancl E l ,

,,.;'"vf;) = = N ( J ; ) B y. ( g . ? - - - E- il : ) ( j ' , f ' : ) llCl

\

I.!

(.

(9.2.26n)

,

(9.2.26b)

This has to be summed over- ail initial states 1 (using k, as the quantum nunlbe~-)and final states 2 (using k,,) for all the electron wave vectors, ."(E met

=

hm)

=

2 N ( E ) xx ~ , ~ 6 ( E E, , - h w ) ( f b - f,) (0.2.27) k,

kb

Similarly from (9.2.231,

where

for each incident photon with a n energy Ate. Therefore, we have the Poynting vector or the photon intensity-for a photon incident into a volume V with a speed c / r i , :

Thus, we obtain

and

which is proportiona1 to the optical matriu element, and the absorption coefficient is

7

r e -

C,) =

( 11.2.33b) l'l,(.6u,~;~0

The above result is identicii! to i9.1.25).

Similarly, the spontaneous emission rate per unit volume per unit energy interval (s-' cm-' e V - ' ) is given by

and the net stimulated emission rate per unit volume per energy interval is given by

When a ( A c o ) becomes negative, we have gain in the medium: s ( E ) = -a(E) -.

We can also write

The spontaneous emission spectrum r v u n ( E ) and the gain spectrum g ( E ) are plotted in Fig. 9.4. The emission spectrum is always a positive quantity. while the gain changes sign to become absorption [S] when the optical energy is larger than the quasi-Fermi level separation F, - F,. Expressions similar to Eqs. (9.2.34) and (9.2.37) have been compared with experimental data with general good agreement [9]. Note that the spontaneous emission intensity per ini it volume per energy interval should be f ~ ~ o r ~ ' " ( Awhich o ) will be dis-

I

Spontaneous and gain spectra

I

$pcln(~)

/\ L~~LT-*+% /

F.,- F,'

g u t ,.<~o:I(.

9.

,

'The spontaneous emission spectrum

and the gain spectrum g ( ~ are ) plotted.

'The r n c r g gaiu y L- Is c11angt:s 1argr:- io thani~1)sc)rption t h e clifferencc when hetween ttrs optical tile c;u:;5i-F:;ri;li

1t:vel di[frl.ei?ce F7 --

rl,'

cussed furthcr in Section 10.1. Some effects of the optical matrix clenlcnts [lo] due to doping effects or conduction-banL{nonparabolicity [9] have been discussed. T h e crossing point F2 - F , in the gain spectrum serves as a good way to estimate the carrier density in a 1ase1-diode or a light-emitting diode in addition to matching both the gain and the spontaneous emission spectra. More details 011 the gain spectrum and the population inversion condition, izo < F, - F,, are given in Sections 9.3 and 9.4. 9.3

INTERBAND ABSORPTION AND GAIN

For interband transitions between the valence band and conduction band of a sen~iconductor,we have to evaluate the optical matrix element, using either

9.3.1 Evaluation of the Interband Optical Matrix Element and the k-Selection Rule

The vector potential for the optical field is A(r) = A e i k ~ ~=. (' & A , / ~ ) ~ ' ~ O P . ' . The Bloch functions for electrons in the valence band E , and the conduction band E, are, respectively,

where cr,.(r) and cl,(r) are the periodic parts of the Bloc11 functions, and the remainders are the envelope functions (plane waves) for a free electron. T h e momentum matrix elemen t is

9.3

INTERBAND ABSORITION AND GAlN

. 353

where we noted that [u;(rXII/i) V'u/r)] and [u;(r)uu(r)] are fast varying functions over a unit ceJl and are periodic, while the envelope functions are slowly varying functions over a unit celL Therefore, the integral over d 3 r can be separated into the product of two integrals, one over the unit ceJl l1 for the fast varying part, and the other over the slowly varying part. Alternatively, for an integral

(9.3.6) where F(r) is slowly varying over a unit cell, we may use the periodic property of the Bloch periodic functions

(9.3 .7) where the G's are the reciprocal lattice vectors:

=

L LeG! e'G ·(c +R)F(r + R) d 3 r R

G

(9.3 .8)

n

the R's are the lattice vectors, and exp[iG . R] = 1. Since F(r) is slowly varying over a unit cell, we may approximate F(r + R) '" F(R) and put it outside of the integral over a unit cell. Therefore,

(9.3.9) where l1 is the differential volume d 3 r when we consider the sum of the envelope function over the whole crystal. Note that the orthogonal property fnu;u u d 3 r = 0 has been used in (9.3.4). Also note that from <"'al"'a) = <"'bl"'b) = 1, the normalization conditions for u e and U u are O/l1)fnu;ue d 3 r = (l/l1)fu~u u d 3 r = 1. From the matrix element (9.3.4), we see that the momentum conservation

(9.3.10) is obeyed. The electron at the final state has its crystal momentum like equal to its initial momentum IIku plus the photon momentum IIkop- Since

(IPTlr:'..L\.L PROCESSES IN" SEMI CONDUCTORS

3.'A

k u p - 217"/ A\), and the magnitudes k (, k t : are of the order 2 'IT / a o, where aa is theo lattice constant of the semiconductors, which is typically of the order of 5.5 A and is much smaller than An, we may ignore k o p and obtain

(9.3.11 ) which is the k-selection rule in the optical trans: nons . Notice that the interband momentum matrix element, PCL:' depends only on the periodic parts (u c and u) of the Bloch functions and is derived from the original optical momentum matrix element Pba' which , on the other hand, depends on the full Bloch functions (i.e., including the envelope functions). 9.3.2

Absorption

Using the k-selection rule, we find that the absorption coefficient (9.1.25) for a bulk semiconductor is

(9.3.12b) where the Fermi-Dirac distributions for the electrons and in the conduction band are, respectively

111

the valence band

1

j~_(k) = 1 +

1

(9 .3.13)

e(E c(k)-Fc) /k o7'

and Ft., F, are the quasi-Fermi levels. We use k to represent both k , and kL:' In the case of thermal equilibrium, F,., = Fe = E F . We assume that the semiconductor is undoped, the valence band is completely occupied, and the conduction band is empty, t, = 1 and j~, = 0, We further assume that the matrix element (e . PCI)~ is independent of k and denote the absorption at thermal equilibrium:

al)

-=---;:-l 7d

3k

f ) -I'~ . Pl' 2 I( - (., _(Ie ,() ( :ZW , ' J (J )" _7T 'I

( t;:

2k 2

L~,, " t -

"

h ...:-. , 171}'

_.- I,rl (.v

'J I

/.

(9.3,14)

wh ere

=

E t'

1 .,. +

1

m ~:

J

(9.3 .15)

m j;

Notice. that all energies are measured from the top of the valen ce band. Therefore, both E; and F; contain the band-gap energy E g . The integral c an b e carried ou t analytically:

(9.3 .16a) ( 9.3 .16b ) Therefore, the ab sorpti on coeffici ent d epends on the m omentum-matrix e le m e n t and th e reduced density of stat es:

(9 .3 .16c ) 1f we use the dipol e m oment ( 9.~ ") and define

(9.3.17) .'

we c an relate J.L C 1' to PCl' by

I nI o W C l ' J.L t "

(9 .3 .18)

e

where W cr = i E; - E) j tz = w . Th e absorption coefficien t CY.o( hw) give n b y

-

71 (t )

(

)'

-- ,-. - Ie ' ll ,. L I:: II

1

...,

).l e i!-'-

j '_

._

I'

[' 2 m,.

I T

'} "

.1

JJ I 2

(flW - Eg )

II

') -

IS

then

(9 .3. 19)

r

T h e abso rpt io n sp ec tru m is pl otted in F ig. 9.S. vV c can see th e domin ant .square- roo t behavior o f th e o p ticul c ne rgy ab ove the band gap, (hw -- E g ) 11 2, re prese nting th e Joint den ...;i ty o! stat es. Below th e band-gap energy E g , th e abso rp ti o n d oes no t o cc ur s in c-. th e ph o to ns '~ ~ C a fo rbi dd en b and g a p .

•

',"

~

~'

._ -

~ . "- "

' ~ ' ",.' - -

-.<:"-: ~' - ""

' ''' ' ' ~ p

.• _ _

.:- . ..... _ •.

.

~.

" .,

3 ~-, ;)

- • . - •....• -...

.

,

' 0 )T IC A.. .' .

_ .~ -

~

,. "

"

\-'RO C.ES~ :ES

-

.'

.

'.'

. _-

•• ,

•. •• • _ . - . _

~ _

.••

_,v~

. ., ._..

IN SE MJCONOUC r O RS

<-----'-------~tl(O

(b)

Figure 9.5. (a) Optical absorption in a direct-band-g ap semiconductor . (b ) The absorption spectrum due to the interband transitions .

9.3.3

Gain

Under current injection or optical pumping, there will be both electrons and hotes in the semicond uctors. Let us assume that a quasi-equilibrium state has been reached such that we have two quasi-Fermi levels, Fe and Fu' for the electrons and holes, respectively. Carrying out the integration as before, we get

1

- h co [/,.(k) - le(k)]

(9 .3 .20) Let

E= We find, by a change of variables, the integration is the same as in (9.3.14) with the contribution at X = 0, and E = hco - E g : (9 .3 .21) where

We note that a (hw ) becomes ne gati ve if f/ k ) - / )/\.) < O. Since

fAk ) -

jJ k )

~ '.

!:'.i .:

..

'7.: .

·~

•..

93

l >ITERB;\ND ,-\.BSORPTJON A

~f)

357 .

'- I/'JN

leads to

or (9.3.23) The population inversion condition is satisfied if (9.3.23) holds and the absorption becomes negative or, equivalently, there is gain in the medium . The above condition (9.3. 23) is called the Bernard-Duraffourg inversion condition [11]. Plots of a(tlw) and (Iv - Ie> are illustrated [12] in Fig. 9.6. We can see that gain exists in the optical spectral region E g < tz w < Fe - FL" The gain spectrum is determined by the absorption spectrum Q:'o( fz w) of the semiconductors without optical excitations multiplied by the Fermi-Dirac inversion factor (II' - Ie), which accounts for the population inversion probability. This factor, t. has a lower limit - 1 at small optical energies a nd

t. .

E

E

tz(j)

(c)

'-

Fi g u r e 9.6 . (a ) O p tical t r~l l1 ~it i () il S b e twee n co nd u c t io n han •.! a n d val ence b a nd . (h ) T ile F errm-Ti irac d is t ribu tion s (E ) a nd J) E ). Cd T h.; absorpt io n s pe c t r u m a t lu») . N ot e th at it is nega tive fur E ;; < flu) ' : F, - F, . when: g:1 il1 exis ts .

r

-r-: :-..... " ..- ....--.- .'--:... ".,...

~)FJCESS'::S

iY'),Il AL

358

IN

SEMICONDUCTOR~

has a zero crossing at the quasi-Fermi level separation, Fe - FL , then approaches +- J at large energies, as shown in Fig. 9.6. The increasing absorption coefficient with increasing energy h co and the decreasing magnitude of 1fL' - fel (dashed line) give a peak gain, which depends on the temperature and the effective masses of the electrons and holes near the band edges. 9.3.4

Summary

We summarize the final explicit expressions for the gain spectrum g(tIW) = ai h co). -r

1. Zero Iincwidth (assume no scatterings)

g( flW)

C o(

=

e . PCL')2 p,.(E =

x [fc(E = h t» 2. A finite linewidth

Eg )

-

fl,(E = h co - E g ) ]

(9.3.24)

r ~

2

g(hw) = CaCe' PCI') X

-

iu» - E g )

1o

0'0

~

r/(27T)

dEp,.(E)

2

'J

(Eff +- E - Izw) +- (f/2f

[fc( E) - fL'( E)]

(9.3.25)

where .., 'TTe~

Co

=

(9.3.26a)

2

n r Ce 0 m 0 w

(9.3.26b)

fcC E)

1 1 +- exp{[ E g

+- (m,./nz;)E - Fe] /kBT} 1

(9.3.26c)

(9.3.26d)

INTERBAND ABSORPTION A1\lD Gf-\IN IN A QUANTVl\tl-vVELL STRUCTURE 904

In this section we consider the interband absorption and gain in a quantum well, ignoring the exciton effects due to the Coulomb interaction between electrons and holes. The exciton effects a re considered in Section 13.3.

The interban d absorption or gain can be ca lcula ted from (9 .1.25) generally if s ca tt e r in g r elaxat ion is not consi dered : (9A.la)

.,

'1Te~

Co =

9.4.1

(9A.lb) ?

n,.c€omow

Interband Optical Matrix Element of a Quantum Well

Within a two-band mod el , the Bloch wave functions can be described by

(9.4.2) for a hole wave function in the heavy-hole or a light-hole subband m , and

(9.4 .3) for an electron rn the conduction subb and n, The momentum matrix given by Pba =

=

.'


il

is

Pb a

)

(u clp lll ) (5k :.k',r/~~1

(9AAa)

where

"In -- f"" d -7' P * ( Z )' o

l en

A..

II

-

6

f1I

( ~7) ,

(9 AAb )

'X

is the overlap integral of the conduction- an d valence-band envelope fu nc -tio ns . The k-sel ection rul e in the pl ane of the quantum well is s till sa tisfied . The polarization depend ent matrix element P eL" = ( ltc lpill L') is di scussed in Section 9.5. H ere we h ave to take into account the qu antizations o f th e el ectron and hole; e ne rgies Ell and E,,:

E~II -=0

r:

C h il !

( 9 ,4 .5 ~t )

--

'!

r

I, ( . :~ ••)0 I

2 m":c:

)

-r

'

)

.::; .-.,olCA 1.. PEC·C.,SSES IN

360

~EMICONDUC.TORS

Note that E"m < 0 and Eb

--

Ea

=

Eel"! 11m

+

1z 2

k; -

2m,.

= Eel"! h m (I'\or )

(9.4.6a)

where (9.4.6b) is the band-edge transition energy (k, = 0). The summations over the quantum numbers k , and k , become summations over (k',, m) and (k" n ). Since k , = k', in the matrix element (9.4.4a), we find a(lzw)

=

L

Co

2

IJ/~~112 V

n,m

9.4.2

Lie' PCLf 8(E/~;~I(k,)

- hw)(f(~l -

t;)

(9.4.7)

k,

Joint Density of States and Optical Absorption Spectrum

Change the variable from k , to E, in (9.4.7)

and use the two-dimensional reduced density of states p;D (9.4.8a)

(9.4.8b) where A is the area of the ,.cross section, AL _ = V, L _ is an effective period of the quantum wells, and V is a volume of a period. We obtain ~-

a ( t,) IW =

f '277" { ~

2:.c k dk 2

C0 L

"L

t:

I hm n 1

Ie' I

Pel I ~ (5 (

E· h cnl 7 (k ) - h w ) (j'.~11 - fen) l'{ L

0

zn,fIl

(9.4.9a) -..c

C·0 L"IJ'111~j' I~m clE c.i p ;'DI~' e II,

m

Pal'~()(Eel1 \, lim

+ E- ( -

n~ o: )(flil_!C (

tl

)

0

(9.4.9b) Therefore, the contribution is at E/~;~l -+- L'[ = !u,) anc.l the absorption edges occur at h co = E/~;:I' For a n unpumped so m icon dtrcto r , f,1iI = 1 and fn = 0,

,1

,61

we have the 'lbsorption spectrum ilt tJF:rmaJ equilibrium CYoUuu):

C''0 i.....J '\I/"'IOI' ECtl) /-;n - e . Pee 122])//(, Pi"  tl(l) -" .-:"hm

CYo(tzw)

(9.4.10)

ll, rn

where I-1 is the Heaviside step function, which has the property that H(x) = 1 for x > 0, and 0 for x < O. For a symmetric quantum welJ, we find II;;, = om"

Llsing an infinite well model and CYoUuu)/(Cole . Pc)2) is given by

nI,.

7Tfz2Lz ?

oco(fzw)

-

C Ie' . p _J2  0

m

cc I

3

r_

for EI',':;. < hw < Ed 113

2

7Th Lz

(9.4.11)

nl,.

2

7Th Lz

etc.

which is the joint density of states of a two-dimensional structure. A plot of

the above function is shown in Fig. 9.7. 'With

carrier

injection,

the

Fermi-Dirac

f,m .. f:', has to be included, and we l

above

gain

process

can

also

be

inversion

factor,

.ain the absorption coefficient

tEo, lu) hm )

The

pDpulation

_.

j'''( E' t c

understood

=

,

flu)

--

from

following analysis for carrier popuJatiolls in quantum wells.

Eetl "hm )]

Fig.

9.8

(9.4.12)

and

the

o r1':·c.'\ I.. ~ 'R ')CESSES 'E

/

. I -E - . · . •. ----·F . ·· • .• E, 1-

I

------

(b)

I

I I r- _ _ I

e2-

-

IN SEMICONDUCTORS

•

I+-Pe(E)

c

I

1---+--i---'

Eg---+--'----J.----.l..-...l...-------'_ Pe(E)fc(E)

~

neu 2D

Ph

0

--

0

0

0

_0_0_

£.

0

s ,»

- -- -

0

0

E h 1< - Fv ----

E h2 -

0

-

20

2 Ph

-t

Figure 9.8. (a) Populat ion invasion in a quantum well such that Fe - Ft. > hco > £g + Eel £"1 ' Here Fe is measured from the valence band edge where the energy level is chosen to be zero . (b) The products of the den sity of sta tes and the occupation probability for el ectrons in the conduction band p,,( £)j) E) and holes in the valence band p,,( £)/,,(E) = p,,(E)[l - f, .( E)J are plotted VS. the energy E in the vertical scale .

9.4.3

Determination of the Quasi-Fermi Levels

For a given injected electron density N, which is usually determined by the injection current and the background doping, the quasi-Fermi level Fe for the electrons can be determined using

L

N =

L f d Ep ;D( E)fcn( E)

N,l

n = o cc u r ie d subbarids

(9.4.13)

T!

where NIJ is the electron density in the nth conduction subband. The last expression in (9.4.13) sh ows that the carrier concentration is just the area .' below the function p ;D( E)// E) in the top figure of Fig. 9.8b .. Here E = E e n + Ul 2 k ;/ 2m: ) is the total electron energy in the nth subband:

i

N,I 2 Al. .-"1.

')

-=-L V

A

.

---~/d ~k

k(

J

~

!1 2k 2 I

Er

1

ll C

=

2 Ji7'i: c'

(

...., _

"T

) .~

-

21)

.0"

I

in"..:

,I

f ' 1 1I,r' L

.

Using 1

J dXi+x

-1n(1 + e-')

(9.4.15)

we obtain

N= n

(9.4.16)

The quasi-Fermi level for the holes, F"

can be determined from

P = N + tV.; - N); = I:" m

(9.4.17)

Again, the hole concentration is just the area below the function Ph(E)!h(E) in the bottom figure of Fig. 9.Sb.

9.4.4

Summary of the Gain Spectrum

We can write the gain spectrum from (9.4.10) and (9.4.12). The following is a summary of the expressions for the gain spectrum: 1.

Zero line width

11, nl

')

Finite linewidth r 'f;-

g( Ilcu) = Co I: 11;;;;,1'! dE,p;uii' . p, I:' n, in

U I" /')"

'

_/(-"_,T} X

... _ . ,.-- ---- -----,,. 'I' ,",' (1:.'). --- j",,'," (' f.'! ,I' ]

----------------.. ---------- ..---_::;-.

[_E;:-.',.( U)

,c

/! (;) 1"- -'r (l'/' 2):' _. ..

{

(G4 \ _.

'Sill

• .l -

()PT)C\~,

P}ZC CESSES IN SEMICONDUCTORS

where (9 A .20a)

?

n,.c8 o f7 z6W 111,.

Tlh 2 L

IJ~':n

=

foo -

(9 A.20b) z

dz,Jz)gm(z)

(9A.20c)

(X)

1 (9A.20d)

(9 .4.20e) The proper momentum matrix elements Ie . PeLf for bulk and quantum wells are discussed in Section 9.5. It is important to note that the momentum matrix element is isotropic for bulk cubic semiconductors, and it is polarization dependent for quantum-well semiconductors. Theory and experiments on polarization-dependent gain in quantum-well structures based on Kane's model [13] have been investigated [14-20]. The momentum matrix Ie . PCI,,1 2 should be used with care [16J since various missing factors exist in the literature. 9.4.5

Theoretical Gain Spectrum and Comparison with Experiments

The gain spectra using (9.4.18) for a zero linewidth and using (9.4.19) for a finite linewidth are plotted in Fig. 9.9 for transitions (a) between a pair of (a)

Normalized gain

+2

r------' I I

+1

-1

_ _ _ _ oJI

Without broadening With broadening

-- r ,

E

eih

~

))!

e

'~~~~~c-f

1(1

FC-F y I

-2 (b) Figure 9.9. The gain Spectra with (da»hcd CU1"\eS) and without (solid curves) scattering broudcnings for d quantum well with transition involving (<:.t) ,I single eketron and hole subbarid pair and (b) two electron .in.l t\'oh,) hole subbands. The ste pwise joint del1si[y of st.ucs

is also shown a~ dashed lines. The ~,lill is normalized by the peak coefficient of the t-irsl step ~i'-":JI hy the co e tficienr in Eq, (9.4,] 8),

+2 +1

-1

tu» y

L. __

~~_

Normalized gain

9.4

ABSORPTiON AND (;AIl' iN A QUi\NTur,r-vIELL ,;TR(n)fiE

GalnAs/lnP

IQWH

L..:7.0nm

36

T= 185X

---,-------------

" TE

300

7

HA

200

I

E

u

100 A

"-

A

0>

!-.

A

a A

-100

0,80

_9,85

0,90

energy I eV

(a)

GalnAs/lnP

n=2 ,

n=l

,/

200

E

U

"-

•

lOa

0>

!-.

a

T=100K

.. TM

300

I

lz=19nm

.. TE

400

-

MQWH

1

v

v v,..

v

v'9VVVVV V



--

-100

.........

n2o=2,27' 1 012cm-2 ----,­

0,75

0,80

0,85

energy / eV

(b)

Figure 9.10. Theoretical anJ e.\['H.-:riilwi"]UJ ;.;l.()d,J i1 -"pn-:tra fg for tkJth the 'IE and T1\tr f1olari/::Jlions of an Inu ..:1,Gar,q-c."-\s/InF mu!tjpic'-ql.L:Il_Ulll-\\'dl bser (1) with a \vetl width 70 A <"If T = 185 K, The sheet etrric!" dnitv i.'i e:,ti:;):!"cd " h,: 1.-4i) >: lUI./m:'.; (h) with a weJl width

i'/J /\ at T = 100 K and the' 11eL'r r.:1r;-::'! jejir. _.7

holt: :lIbhal1lJ:; are UCCUjl!l'U :nd t!:,,< \i\ftcr Ref. 19 (['i 19:.,Q IEEE,\

t{)!2 I-'rn:. -The second electron and

iU;;Lrl:IU': 1,' tL.: :"'.::ccnd r(::!: :l!. the high-en\:rg)' nd

O FTlC/ . 1. rR OC ·:S ~ES IN SEMI CONDUCTORS

el ectron and hol e subbancls and (b) betwe en two pairs of electron and hole subbands. E xperime nta l data for qu antum-well' lasers have b een shown to compare very well with the above theory in genera] [19-20], as can be seen from Fig. 9.10. The materials are ln O.53Ga U.nAs quantum wells latticematched to InP barri ers and substrate. We can see at a small carrier concentration, Fig. 9.10a, that only n = 1 electron and hole subbands contribute to the gain . The gain for TE polarization is larger than that of the TM polarization since mo st of the holes occupy the heavy hole subband . At higher carrier concentration in a qu antum well with a larger well width in Fig. 9.10b, the second electron and hole subbands are occupied and features lik e double steps appear in the TE gain sp ectru m .

a

9.5 MOl\tlENTUM MATRIX ELEi\lENTS OF BULK AND QlIANTUMWELL SEMICONDUCTORS In a bulk semiconductor, the optical matrix element is usually isotropic. However, in a qu antum-well or superl attice stru ctu re , th e optical matrix element will depend on the polarization of the optical electromagnetic field [14]. In Kane's model for the semiconductor band structures near the band edges, as discussed in Section 4 .2, the electron wave vector is originally assumed to be in the z direction . The wave functions at the band edges have also been obtained as I~, ± ~ ) for heavy holes, It + for light holes, and I-!, + i ) for sp in -o r bit split-off holes.

t>

.'

9.5.1

Coordinate Transformation

If the electron wave ve ct or k has a ge ne ral direction sp e cified by ( k, 8, ¢ ) in

sph eri cal co.? rdin ates,

k

=

k sin

e cos

we find . the band-edge wa ve fun ctions are as fo llow s, using th e coo rdina te . foi , .LD (4 .z'7 .,:'"'/ ) anu-I ( ~t1.'::',..., . _1 ,J') . trans ormatrons -r ,

Conduction Band

,1_I' '>

I' ;L.-. )

( 9.5.1)

J 67

Heavy-Hole Band 3 3 )'- 1 2' 2 Ii-lex' + iY')

i')

0--0'

-- 1

=

fi

I( cos () cos

(p -

i sin

(p ) x~

+ (cos () sin ¢ + i cos ¢ ) Y - sin () Z) I j

-23 ,-

3 )' 2

=

1 /2-1 (X'

')

- i Y') 1 ')

]

=

Ii I(cos e cos

¢ -+ i sin ¢ ) X

+ ( cos e sin Light-Hole Band

31) 16-]

-

-

2'2

= -I(X' -1

=

16

+

iY')

4» Y ._. sin 8Z) 11')

n

1') + - - IZ'j') 3

I(cos () cos ¢ - i sin ,~ ) X

+ (cos (j sin

-+"

¢ - i cos

V~

Isin



+ i cos (p) Y - sin () Z) I! ')

e cos q,X +

sin

e sin q,y +

cos

,.., .J

2 ' 1

/6- ICcos

f)

cos cP

+

j

sin ¢ )X

+(cos()sin (!> .- icos (p)Y' - sin8Z>ri')

./

02) Ii')

(9.5.2)

Spin -Orbit Spllt-off Band

1 1 -I(X' + iY') 1') + - IZ'I')

~2'2~)

13

13

1

- /3 I(cos e cos 4> -

i sin ¢) X

+ (cos e sin 1> + i cos q) ) Y - sin ez) 1-1') 1

13

+ 1

1 )' 2

2'

e cos
[sin

1 - -I( X'

/3

- iY')

I')

-

1

13

IZ' l ')

1

/3- I( cos e cos
- IS

e sin 4>

i sin
- i cos (p) Y - sin ez) II')

Isin 0 cos 1>X + sin

e sin (pY + cos oz; I J,')

(9.5.4)

We keep the primes in 1-1 ') and Ii') for spins in the new coordinate system for ease of calculation of the dipole matrix elements, as will be seen later. Consider a quantum-well structure with the growth axis along the z direction. In general [141, the z axis for the p-state functions does not have to coincide with the z axis for the quantum-well structures. However, when averaged over the angle in the plane of the quantum well, it is found that the same matrix elements for both polarizations give the same re sults as those obtained assuming that both z axes coincide with each other. To calculate the optical momentum matrix element

M

=

( uclp lu)

=

.t/ u. \

L

~I ox

':

I

=

II

PCI'

», ';\ +

.'

J

Y(u ,'I !' !~ I, + i / 1/, \ l, I cJ Y I ' I \ •

"i !--. 1/) (9.5 ..5 ) rJz i

with polarization d epe nclc ncc, we evaluate J . iYI for e = i or y for TE polarization, and e = :: for TM polarization. H ere u L' ix Ii S i' ) or liS J') and Ll l , can be II li li or II ih'

9..~

MOMENTUM I'vli\TPJ :(

E L.F~/ F :"l

rs

3L9

Define Kane's parameter P and a momentum-matrix parameter P, as

(9.5 .6) Then for conduction to heavy-hole transitions, we obtain M e -hll as

33)'

(

is i'! pI 2 ' 2

=

[(cos e cos

-

+ (cos

cP-

isin cP)x

e sin ¢

+ i cos cP ) Y - sin () i]

p

Ii

3 3) ( is J. ljp l 2 ,- 2 = [(cos e cos ¢ + isin ¢) x

+ (cos 0 sin q, - i cos q, LV - sin 0 i]

iz(9.5.7)

For conduction to Irght-hole transitions, M C -

(is i'lp l ,~, ~) ~ (i S 1 'I

p!~ ,- ~) ~

rr rr1 .

1I1

are

(sin 0 cos

q,£ + sin 0 sin q, y + cos Oi)P,

(sin 0 cos

q, f: + sin

0 sin tb y

+ cos 0 i) P,

3 1)' = --=- [(cos e cos 1J + i sin (P)x

(

is i'!pl - - . 2'

16

2

-;- (cos () s in
(

. .. 1.)

3 1 \, II p! -, ~) ,

2 .: /

=

ei] PI"

-- I r '-r-~'-- l ( CI ) S cl co s (l) - ! ~I]:(II)x '

(j

"

,.

Vi ( J

+

(cos 8 :~i.n:/)

j

I

+ i cos .j)).\- ---

sin

fJz ] P;

(9.5.8)

(.p ileAL FROCESSES It SEMICONDUCTORS

-:.7C

9.5.2

Momentum Matrix Element of a Bulk Semiconductor

For a bulk semiconductor, we take the average of the momentum matrix element with respect to the solid angle dn. For example, the momentum matrix element for TE polarization (e = x) due to the transitions from the conduction band with one spin (say < is 1"D to both heavy hole bands, IL~) and 1%, - %)' (one of the two transitions is zero) is I

1

-4rr fix' -1 4rr

1 3

Mc-hlf sin

e de d¢>

1"sin e de f27T d¢>(cos? 0 cos? ¢> + sin" ¢»_x p2

0

__ p2 x

~

2

0

]Y12

(9.5.9)

b

where

(9.5.10)

is the bulk momentum matrix element (squared). Notice that the above result is the same for the other spin in the conduction band. If we take = y or i, 2 and average over the solid angle, we s~ilY obtain M o since the bulk crystal is 2 isotropic. We could also take l(iS J,'iexl~,~)f + l(iS J,'lexlt- 1>1 and average over the solid angle and obtain 11,1/; as expected. In practice, an energy parameter E p for the matrix e~ement is defined: E p = (2m o / fz 2) p 2 or Afl} = (m o / 6) E p ; E is tabulated in Table K.2 in Appendix K. "

e

9.5.3

Momentum Matrix Elements of Quantum Wel1s

Next, we consider quantum-well structures. The optical matrix elements will become polarization dependent. The theory and experimental data on the polarization-dependent gain are discussed in Refs. 14-18 based on the parabolic band model. Improvement of the optical matrix. element using the valence band-mixing model in quuntun: v/,~lls is discussed in Section 9.7. TE Polnrlzation. Let (' ,=.~ .i, th:« IS, tLe optical fielo is polarized along the x (0·]" y) direction. The opt icul dipole matrix clement is averaged over the

azimuthal angle q> in the plane of quantum wells. We obtain for (is l' 'I

and the same matrix element is obtained for the other spin (is t'l in the conduction band. Similarly, for the sum of transitions of an electron with spin t' from the conduction band to both light-hole bands, we find

(9.5,12) The above result is the same for both spins (is 1 'i and (is sum rule exists:

i'l. Note that a (9.5.13)

which is independent of the angle

e.

/

TI\'l Polarization,

e=

i

1 ----') .c tr

=

fl7T J,"12' ()

. '+'

1 -T' .~ en:.; ~ 0 .~-_._--- -~----:

M

A'1 t.:~

II

., 3 .' I"' = -sin-c eA1!J2

r -- h i t ? _

(9.5.14)

(9.5.15)

.~~

372

.'~-';""""

,

-- ..

(.-.'"'~''''''''''''''''''''-'~j''''-'"",,'''''~--'~oo:,'~":'':".r.-.'

~

"t'•...;:::-;I'- 'T'I',-".,.,....., •.

OPTlC:\L PROCESSES iN SI:MICOr--;DUCTOHS

The sum of (9.5.14) and (9.5.15) for e = 2 is still 2A-I b2 • The above results, (9.5.14) and (9.5..15), are the same for both spins in the conduction band. The angular factor cos ' 9 can be related to the electron or hole wave vectors by 1.2 I\. Z

cos" ()

(9.5.16)

for -the electron wave vector k

13k! +

=

2k zn' with E en

=

tz 2 k ;n 12m; or

(9.5.17)

for the hole wave vector k = pk t + ik zm with E h m = -hk;mI2tniz. Note that the hole energy E/Jm is defined to be negative so all of the energies are measured upward. An alternative approximation is

(9.5.18)

where the reduced effective mass rri; defined by 1

1

1

+ m*e n1j;

(9.5.19)

has been used. The results of the momentum matrix elements are summarized in Table 9.1. It is noted at the subband edges, where k , = 0, that for

Table 9.1 Summary of the Momentum Matrix Elements in Parabolic 2 Band Model (Ie' Pet f = Ie . M1 )

-------------

Bulk Quantum Wel] TE Polarization (e = x or y) 2 (Ie' M, __ IJIJ / ) = +- cos ' f-J)AJi;' 2 (Ie~ . 1\1 c -II/ 1 > = (2.j -" leo' (.2 (j\M,2 \ ~ ,~ J "

tU

Conservation Rul e I'

1

I

(I.t· M e '- h 1-) + (!Y' 1\1' __ .'1!-'> +- '.,:' M, . .',I-.\ (I e. Me .. /JIf) +-
.

I

A

,")

"

,

=

...

-,.'140- , (11 = hl: or lII)

0-<,

9,',

INr':RSUEBN-1D ABSORPT1CN '

TE polarization, the matrix element) are (9.5.20) (9.5.21) and for TM polarization, the matrix elements are

(Ii· M, __ hh I 2 ) k{=O (Ii, M

C _

=

2 l h I ) k / ==O =

0

(9.5.22)

2Mb2

(9.5.23)

These matrix elements are useful in studying the excitonic absorptions in quantum wells and they are used in Chapter 13 where we discuss electroabsorption modulators using quantum-confined Stark effects.

9.6

INTERSUBBAND ABSORPTION

In an n-type doped quantum-well structure, intersubband absorption is of

interest because of its applications to far-infrared photodetectors [21-23], for example. In this section, we study the infrared absorption [24J due to intersubband transitions in the conduction.nd of a quantum-well structure with modulation doping. We assume that tile doping is not large enough so that screening or many-body effects due to the electron-electron Coulomb interaction [25J can be ignored. The far-infrared photodetector applications using intersubband transitions are discussed in Section 14.5. 9.6~1

Int ersubband Dipole Moment

Consider the intersubband transition between the ground state

(9.6.1) and the first excited state

(9.6.2) 1 = /{., I 1 / ! ., l wnerc me X +1(yY anc LP.c 1 transverse •W~lV<~ vee t ors 1\ / = ",\"\ +- \yY,~: position vector p ~'"" .l;\: + Y5; in the quantum-well plane have been used. The J

I

I

~

/'

,.

I

I~'

~

37

1

by

optical dipole moment is given

(1I,11I,.) (

= <5 k

k' "

I

e;'v,

<¢ 21 ez 14> 1 ) i

(9.6.3)

which has only a z component, where we have used the orthonormal conditions

9.6.2

Intersubband Absorption Spectrum

The energies of the initial state and the final states are, respectively,

(9.6.4 )

7 * -»,

as shown in Fig. 9.11c. Using (9.1.30) and (9.1.31), we obtain nonzero absorption coefficient a.(liw) only for = 2 (TM polarization) since the x

e

Ec

Ec

E2

E2

E}

E}

kt (a)

(b)

(c)

Figure 9.11. (a) A simple quun tu m well with a sru.ill duping conce ntration. (b) A modulationeloped quantum Well with a signlr: . . .in; amount of.;crel'ning due to J large doping concentration. (e) The subband energy diacrum in the k , space. A direct vertical transition occurs because of the k-selcction rule in the plane of quantum wells.

and y components of the intersubband dipole moment in (9.6.3) are zero:

(9.6.5)

where

(9.6.6) is the intersubband dipole moment and N, is the number of electrons per unit volume in the ith subband, which has been derived in (9.4.16):

(9.6.7a)

Or, in terms of the sheet carrier density (l / ern"),

(9.6.7b)

The absorption coefficient is, therefore,

('c.(tuv)

=

which is a Lor entzian shape with a linewidth f. In th e low-temperature limit s uc h th at (E F

"--

E)

>?»

II.IJT, we obtain

(9.h.9)

: I

•

,....

.,.

•

~

"

_ , ..

OPT' C;\L PRO CESSES IN S E ~l1 CU N b u cr() RS

376

and for occupations of the first two levels,

a(hw)

An integrated absorbence is given by

.

(9.6.11) where we have replaced the integration limits (0, co) by (- co, co) for the Lorentzian function in (9.6.5) with an integrated area equal to Tr. Example Let us estimate the peak absorption coefficient in an n-typc doped G aAs quantum well u sing an infinite barrier model with a well width L z = 100 A. The first two subband energies and wave functions are (1'7'1 ; = O.0665m o) :Ie ?

£1

=

"

-n:* (TT z.«; -i. ,

Z

1

56.5 meV

=

E 2 = 4E l = 226 meV

The intersubband dipole moment is

16

-L_

f.LZ 1 = e j

o

-cP 2(

Z ) ZeP l( z ) dz = - - 9.., els ,

TT-

o

- 18 eA = -2 .88 X 10- 28 C' m If the carrier concentration N 1S 1 X 10 18 ern - 3, we can assume that only the first subband is o ccu pie d and check whether the second subband population N z is indeed small. We calculate i

where

l.

IT 1-J

.... , . ,

... _

. "';

"

•

'1.

";

()6

INTLRSUBEAND ABSOIU'TiON

377

We can check that N z = (Ns/L)lnO + exp[CE F - E 2 )/ k BT]) = 2.4 X 10 15 em -3 « N. If N z is not negligible, we have to use N 1 + N 2 = N to determine the Fermi level E F assuming that the first two subbands are occupied. The peak absorption coefficient occurs at h co Z £z - £1 = 170 meV. The peak wavelength is A Z 1.24/0.170 = 7.3 }.Lm. Assume that the linewidth is r = 30 meV and the refractive index n , = 3.3. We find

1.015

X

10 4 cm- I

In principle, the peak absorption coefficient increases with the doping concentration N. However, if the concentration is too high, the screening, band bending, and many-body effects [25] will be important as they affect the energy levels and, therefore, the peak absorption wavelength. The occupation of the second level N z will also decrease the absorption. • Example We consider GaAs / Al O.3Ga O.7As superlattice structures, as shown in Fig. 9.12, with a well width Wand a barrier width b. The absorption o spectrum is shown in Fig. 9.13a for a structure with ~V = b = 100 A and in o Fig. 9.13b for ~V = 100 A ando b = 20 A. We can see that as the barrier width b is reduced from 100 to 20 A the interminiband width is broadened because of the strong coupling of well states. We assume a surface concentration of N, = 2 X 10 ll/cm Z , T = 77 K and a linewidth T = 15 meV. o If we reduce the well width W to 40 A, th .ntersubband transitions will occur from the bound to the continuum minibands. The resultant absorption spectrum is a broader spectrum with a longer energy tail at the high-energy side, as shown in Fig. 9.13c. Here we use the same N, = 2 X lOII/cmZ and a 0

(a) Bound-to-bound transition

(b) Bound-to-continuum miniband transitions ./// '////. "/// '//. ' / / / / / C4 C3 C2 -~

-----'--

I

-----l--

I

Figure 9.12. (:I) Inte rsubband bound-tobound tr a nsit ic.n i') ~t super lattice. (h) Inter:;t:hhanu bound-tr.-cont inuum trunsitions in

Cl

_J

~t

I I

xupe rlatticc.

OPTIC\L PROCESSES IN SEM1CONDI ioror.s

3T; , .c> 2000

(a)

E

u

'-'

....c

BOll nd -to- b

ound

transi tion

161:>0

.~

u

!=: 1200

..... ~

0

u

800

c

0

:.::::

c. 400

l-o

0

""' <

.0

0 20

80

140

200

Photon energy (meV) 1000 ..-..

S

-

~

c:: .:!:! t.l

E

Bound-to-bound transition

(b) 800 600


<:>

t.l

c

400

<:>

"..::

=I-<

<:> ..t::J

en

200

<

0 20

80

140

200

Photon energy (meV)

.a> 200

E u

Bound-to-continuum transition

(c)

-

'-'

.... 150 c

- - -C1-C3 -----C1-C4 -----Cl-C5

.~

u

S

~

o

100

c

50

""' <

.0

1:\

I" ,: . /'.

J'_"~

o 100

,\

,,\.\

I'

t-

o

- - -C1-C6 -----C1-C7

,, .. ,.

u

o ..-....c..

-C1-C2

.:

..\.

,

"."---

'

~-~. ~_ A.

.:- -.150

-:." ....

•

...

'\.

".

"

-Total ~

---.

-:':»:">: - ~" ". -~- -:-.

200

250

300

350

400

Photon energy (meV) Figure 9.13. The absorption coefficients for int ersubband bound-to-bound tr ansition in a supe rluttice for (a) W = b = 100 A (thick barriers, small miniband widths), and (b) W = tOO t\. b = 20 A (thin barriers. large minibund widths). (e) The absorption coefficient of the in tersubbnn.l bound-to-continuum minibund transitions is shown for W = 40 A and b = 300 A,

'

linewidth r = 40 me V . T he b r o ad spectru m is co nt rib u te d by a large r we ll a s th e tr a nsitions to multip le mi niba n d s. . CI

9.6.3

r

as

Experimental R esults

In Fig . 9.14, we show t he ex pe rime n ta l data for five different samp les b y Levine et al. [22]. The mat erial parameters are listed as follows: 0

Sample

Well Width (A)

Barrier Width (.A.)

x

A

40 40 60

500 500 500 500 50 500

0.26 0.25 0.15 0.26 0.30 0.26

B C

SO

E F

50

1 1.6

0.5 · 0.42 0.4 2

Sample F

Sample E

Samples A , B, C

N D (10 18j cm 3 )

(b)

-

700

2000 E

..--,

5 600

.-

C:l

= 300 K

E

o

1500

I--

500 Z

LL LL

W U

400

I

I

W

o 0 0

1000

I

I I

300

0

1-I

500

I

Q

a:

I I

0 en 100

en

~

I

\

a: o: co

\

<{

~~L_.~_~

0 6

8

I-Q

0

I I

,

<{

W

0 Z

I I

200

LL LL

0

I

Z I--

C:l I--

Z

W

0

-.

roo

T

10

12

WAVELENGTH ).. (urn ) F ig ure 9.U. C:l.l E ne rgy bau d profi le s Iur s.unp lc.. t\ G, C, E ,
II

CWTJCAL PROCE::;SFS iN'SEMICONDUC,'ORS

~:lI'

"11;100

... "'

........

45A 8A35A 35A 30A28A30A ---- - -

Active region - -

~igitally--s­

graded alloy Figure 9.15. A period of a quantum cascade laser [27] using inte rsubband transition between £3 and £2' The barriers are A10Aglno.52As and the wells are In.,..p G a O.53 As materials. The calculated values are £3 - £2 = 295 me V and £2 - E, = 30 meV.

Samples A, B, and C were designed for the bound-to-continuum transition; sample E was designed for bound-to-bound state transition and sample F for bourid-to-quasibound state transition. It can be seen that the bound-to-bound transition (sample) E has a large peak absorption spectrum with a narrow linewidth. Sample F has a narrower spectrum than those of the bound-tocontinuum transitions in samples A, B, and C. These samples result in different responsivities because the carrier collection by the electrodes also play an important role. For example, the respon.. ";ity of sample F is larger than that of sample E because the electrons arriving at the final (quasibound) state can easily be collected compared with those in the final bound state of sample E. 9.6.4

Intersubband Quantum Cascade Laser [27]

Intersubband lasers using a structure such as that shown in Fig. 9.15 as a period have been demonstrated [27] recently. The laser structure was grown with the Alo.4sIno.52AsjGao..nIno.5JAs heterojunctions lattice matched to InP substrate by MBE. There are 25 periods of the active undoped regions, with graded regions consisting of an AlInAsjGalnAs .superlattice with a constant period shorter than the electron thermal de Broglie wavelength. o Electrons are injected through the 45-A AllnAs barrier into the E 3 energy level of the active region, as shown in Fig. 9.15. The reduced spatial overlap of the E J and E, wave functions and the strong coupling to the E I level in an adjacent well- ensure a population inversion betwe e n these states. The laser operates between the E J and £2 levels with an energy difference E J - £2 ;::::; 295 meV or an operation wavclength > 4.2 /-Lill. The applied bias field is around 100 kVjcm and the band diagram is as shown in Fig. 9.15, where the graded region is ncar Hat-band condition. The estimated tunneling

:lxl

time into the first trapezoidal barrier is around 0.2 ps, Therefore, the filling of the state E." is very efficient, while the intersubband optical-phonon-limited relaxation time T 3 2 is estimated to be around 4.3 ps at this bias field with a reduced overlap integral. Since T3'2 is relatively long, the population inversion can be achieved because the lower state oempties with a fast relaxation time (around 0.6 ps) due to the adjacent 28-A GalnAs well with £1 state. The relaxation between the £2 and E 1 states is very fast because the energy separation is equal or close to an optical phonon energy so that scattering occurs with essentially zero momentum transfer. Finally, the tunneling out of the last 30-A barrier is extremely fast (smaller than 0.5 ps), The laser emits photons at a wavelength of 4.2 ,urn with a peak power above 8 mW in pulsed operation. The operation temperature was as high as 88 K.

9.7 GAIN SPECTRUM IN A QUANTUM-WELL LASER "VITH VALENCE-BAND-l\'lIXING EFFECTS In Section 9.4, we present a simplified model for the interband gain of a quantum-well laser based on the parabolic band structures. In quantum wells, valence-band-mixing effects between the heavy-hole and light-hole subbands are important and the subbarid structures can be highly nonparabolic, especially for strained quantum wells, as discussed in Sections 4.8 and 4.9. In this section, we present a general theory for the gain spectrum in quantum wells, taking into account the valence-band-mixing effects [26]. The theory is applicable to both strained and unstr ined quantum-well lasers.

9.7.1

General Formulation for the Gain Spectrum with Valence Band Mixing

Our starting equation follows (9.1.25) for the absorption spectrum with the delta function replaced by a Lorentzian function (9.1.31) to account for scattering broadenings, (x( tuv )

=

C

1 [/(217) ~\'\'IA 1V L..J LJ L.... e . P b a '( 1:.- _ E _ r ) 2 ry

0

n r

t '

(T

k

u

kh

j

b.»

a

fl W

+ ( ['j /- ) 2.

(j'

a -

j") b

\

(9.7.J8)

(9.7.1b)

whe re the quantum numbers are ia:> = ik". 0') for initial states in the valence band and Ib) = Ikb,ry> for final states in the conduction band. The summarions over the spins of the initial (hold stelle::; IT and the final (electron) stales 1/ have been written out e xr licitly inst:=::d of '-~ simple factor of 2. Here

,] = [ , or

~.

for the conduction band,

II

~ind o:

is more complicated. For the

"

- '.•. - , -

- • • _ ; ,. .....

~

• • ':'1.......... , -

..,..... _ . ........... ' - - - , . . - . .

-e-- :

- -

. 'or

,

• • • _ • ....,. -

- . .....

• .',

••-

• • ••

,-.

-~

() V r W A l.

382

}'PO CrSSE~

IN

S EMICOND U~-=TOR o :

formulation u sing the block-cliagonalized Hamiltonian , cr refers to U for the upper Hamiltonian with an eigenfunction in the »rth su b ba n d denoted by elk, oV

'VnU,(kl'r)

fA

=

[g~~ )(k"z)ll) +g;,;)(k p z) !2)J

and (J" = L for the lower Hamiltonian with an eigenfunction subband denoted by

(9.7.2) In

the mth

(9.7.3) The nth conduction su bbarid wave function is denoted by "\V C7) (k' r) -

II

( '

eik',op

=

..fA

A, 'PI/

(z)I S71'/ )

77 = T or 1

( 9 .7 .4 )

Here 15 77 ) ( = u/ r)) denotes the spherically symme tric wave function of the Bloch periodic part of the electron state. We have to label the quantum number k , by (k p m) for the mth subband, where In refers to the quantum number for the z dependence of the wave functions . \Ve label k , as (k /" ri) for the nth conduction s ub ba n d . The matrix element will contain a k (-selection ru 1 • when we take the inner ;_ oduct, k , = k't. Therefore, we can write

a(flw)

=

1 [/(21T) 2 C --"" "" ""Ie· p7)l.TI (!lJnl -fn) 0v (J".7] c: n.c:In c: nm [Ef!Il (T.) _.1:.]2 C k ~ h m ,I. / rt W + (1--'/ .?- ) 2 L' (9.7.5 ) I

wh ere

(9.7.6) is th e energy differen ce between the nth electron subband and the nzth hole subband at k t : Here

2m~,:

(9.7.7)

is taken to be parabolic in the condu ction subbund . The nonp a rabolic valence subband structures art: described by E;:'(I.:. , ) ob ta ine d from the upp er (er = [1) o r th e lower (a = L) H amiltonia n.

9.7.2

Evaluation of the Momentum Matrix Elements

The momentum matrix element

Ie . p;:;;,1 2

is defined as

(9.7.8) with the factor Ok k' dropped. Here we list the definitions of the old and new basis functions and their transformation relations, which have been discussed in Chapter 4. I'

(

Old bases

33) -1fi!(X+iY)i) 2'2

=

~,~) ~ 3 --1) 2 ' 2'

3

;;1(X+iYlL)+ 1

/6- I( X

=

-3\

1

n

2'2/=

l

- iY ) r

~IZj) 2

) + /6 Iz

. ( X - IY ) 1. )

1. )

(9.7.930)

New bases

1) =

1

2)

1

2'2

- a

3-3)

* - -

:2' 2

3 1)

=

3) =

1

33)

a -- -

_.

3

- 1\

f3 '" 2 ' 2· + 13 "2 ' 2

/

3 1\ 3 -1' + f3 - - ) 2'2/ 2' 2

0.* fJ

\ 9.7 .9b )

The matrix element It? ·

,

p~ ; ,~,I-

ca n b e tou nd as shown m the following

I

I

O P";

~C/\L

exam pl e . For TE pol ari zation ( c = i or

=

y) and

1{ ok"k',I (Slpxl X )2 4 ( ¢ n l g~~ )/

+

PR OCE ss r .s TN SEM:CONDU CF )RS Y],

t . cr

=

= U , we obtain

1 +3 ( 1)n g,<;))2 l

~. cos 24> (4),,lg~))(4>n lg,;;))}

(9.7 .10)

where we have used the following relations: Spin t -a

r IP xll )

=

fi

( S t IPx I2 )

=

16 ( Sl pxl X)

r I p .r»

=

V6 ( S 1p xl X )

Ipxl 4 )

=

(S

(S

(S t

( S)PxI X )

f3

f3

r Ip, 12)

(5

r Ip,13)

VI ~ VI

~

-a

.M ( Si p) A()

v-

Spin 1 (S

(5

-

j3*(5 Ip, IZ )

!3 ' (5Ip,IZ ) ( 9 .7 .11a )

.

.'

I lpxl O

-a * =

12

( Sl p) X)

.-- f3 * ( S 1 IP x 13> = ~f6 <5 I p) X >

( 5 I Ip, 12)

~ !3/'f ( 5I p, IZ )

( 5 ll p J ] )

~ !3['f ( 5 1pJZ ) (9 .7 .11b)

'j

7

GAiN SF:;::"

;:'RU~'

VII

n. V;\LFNC ;~-B/'j\j)·.rv1iXl>JLr EFF'::,CTS

J85

Using the axial approximation, the band structures are isotropic in the kr-k v plane; we, therefore, write (9.7.12) The term containing the cos 24> factor in (9.7.10) will not contribute to a(hw) since its integration over ¢ vanishes. Let us pull out the factor I(SI pxlX)!2 in the matrix element, where (9.7.13) is defined in terms of the Kane's parameter P or an energy parameter E p • The expression for a( n. w) can be cast in the form 2

a( hw) = C o-

Lz

X

L I:" f CT

ook t dk t

2rr

0

!l,m

M,C;II( k t )

f/(27T)

') (fc;m

+ (f/2 r

[ E/~~~' ( k t) - h w ] 2

~

-- in)

(9.7.14)

c

Here a factor of 2 for sum over 'T] is explicitly shown, since we find the final expression for the magnitude squared Ie . p~~f is independent of 'T]. TE polarization:

e= x a=

U

a = L

T/\/[ polarization:

e=

2.

u=U u=L 9.7.3

(9.7.15)

(9.7.16)

Final Expression for the Gain Spectrum and Numerical Examples

The optical gain is obtained from the negative of the absorption function g (fUll) = - a.Ul w). Therefore, we have

., 7

'\~ g ( ({)) = C (J ----- \~ c: c:

1<:

,r

/l.lll

dk J.?'Ok----t

'T:'

.

()

I

If"

j'lyIIII!/ (

k ()

-'

f/ (2 'tt )

,.< -- ------..-.-----~-. --,----.. ---- '--::;- (' f r L);~;, ( k!.\ -.. h (r) ] ~ +- ( r /2) - . (

d

I

I

.. -

t"!" '.J

,I

."

'7\J (. 9 . ~/ .1.

-

.... -

• • •-

._ •. ,

."

•.

OPTIC;\L l-'R0(":ES::;f-:~ IN ~'EMICONDUCTORS

386

Notice that the form (9.7.17) or (S'.7.14) is similar to (9.4.9a) using the parabolic band model. For a symmetric quantum well without an applied electric field, we find that the two valence bands with CT = U and (J = L are degenerate. The matrix elements turn out to be the same for (J = U and L. Therefore, the sum over (J can be replaced by a factor 2 and only g;,~) and g;;) have to be calculated with the corresponding energy dispersion relation E,~(k/).

In Fig. 9.16a, we plot the matrix elements 2Jvl(k,)/MJ using Eqs. (9.7.15) and (9.7.16) for the TE polarization and the TM polarization for the first conduction band to the first heavy hole subband (CI-HHl) and to the first ligh t hole sub band (C l-LH 1) transitions in a In 0.53 G a 0.47 As / Inl-XGaxAsI-YPy quantum well lattice matched to InP substrate with a well o width of 60 A. The factor of 2 here accounts for sum over 0- of the hole spins in the valence band (not in the conduction band). The conduction band spin degeneracy of a factor of 2 is always included in the density of states. The barrier In l-xGa x As 1 _y Py has a bandgap wavelength of 1.3 p.m and its parameters can be obtained from Table K.3 in Appendix K. The valence subband structures are plotted in Fig. 9.16b, with HHI as the top valence subband. The nonparabolic behavior is clearly seen. The gain spectra for both TE and TM polarizations are plotted in Fig. 9.17 for two carrier concentrations n. The surface carrier concentration is n s = nls, and the gain coefficient in a period of a total width L T = L z + L b is proportional to 1/L T instead of 1/ L z in (9.7.17). Here L b is the thickness of the barrier. We can see that for a given carrier density, the TE gain coefficient is larger than that of the TM gain coefficient because the holes will occupy the lowest subband which is heavy hole in nature, and the conduction-to-heavy hole dipole matrix is mostly TE polarized as can be seen from Fig. 9.16a. Similar behaviors for the GaAsjAJxGal_xAs quantum-well lasers can be 'found in Ref. 26. 9.7.4

I

Spontaneous Emission Spectrum and Radiative Current Density

By comparing (9.2.33) with (9.2.34), the spontaneous emission rate per unit volume per unit energy interval (s - 'ern - 3eV -1), taking into account valence-band-mixing effects in quantum wells, can be written as

. r SPOil ( E

=

It w )

which is similar to g( w) except for the prcfactor ('bTJfl; E 2/ h\·2) and the Ferrni-vDiruc factors f/ '( 1 -- f/ r , ,, ) . The total spontaneous emission rate per

-

• . • • ~:,-

' _•• '

__

"

3 ~7

(a) '"0

Co)

l-<

~

=

2

~,......,.....,

\

\

0'" tI)

..... 1.6 ~

Q)

E

Q)

,

CJ -LH1(TM)

',/ ,

CI-HHl (TE)

"

1.2

"\ ,

Q)

~

'C ..... 0.8 ~ E "C 0.4

.-Q)

E l-<

---

,/

""

0 0.01

0

0

Z

.J. -- -----

,/

N

~

,, ' "" , >,

CI-HHl(TM)

0.02

0.03

0.04

0.05 0.06

Parallel wave vector (1/A)

(b)

0 .--..

;, Q)

-20

E

<;:»

>.

0.0

-40

LHI

l-<

Q)

:: Q)

"C

:: eo: .c .c

·60 -80

::l

.-

('.J '

-100

,

,

r

,

!

0.01 0.02 0.03 0.04 0.05 0.06 Parallel wave vector (11A)

0

Q

Figure 9. t6. (a) The average of the square of the momentum matrix normalized by the bulk, matrix element for TE and TM polari z ations for the' transition between the first conduction' subbund (Cl ) and the first heavy-hole s u b ba nd (I-IHl) or the first light-hole subbarid CLHl), (h) The valence subbarid structures of the In o s,Gu l l ~7As/ In 1- ,CIa ,As I - v PI' quantum well (L" = 60 A) lattice-matched to an InP substrate:-. - · . . .

unit volume (s-lcm - 3) is to su m up the average of three polarizations (2 x TE + TM) / J and then integra te over the emission spectrum to obtain

R .xp ~~, J{ --,. "PDn(, trw). cI( tiw)

(9.7.L9)

- II

/~. rudi uti ve cur rc nt den sity \ A / em 2 ) for a sc m icoriductor lu se r is defined a s

•

./ ra d

-I.,IR - ~I ; . ~ t)

II

(9.7.20)

(a)

700

---e

--.... (J

600

TE

500

'-'

.-= ~

(J

400 300

c.= ..... 200 0 ~

(J

.-= ~

o

100 0 -100 750

800

850

900

950

Photon energy (meV) (b)

700

---E

--.... (J

600

Tl\Il

500

n= 6 X 1 0

18

em

-3

'-'

.-= ~

(J

400 300

c.: ..... 200 0 ~

(J

.-= ~

~

100 0 -100 750

800

850

900

950

Photon energy (me V) Figure 9.17. The linear gain versus the photon energy for (a) TE and (b) TM polarization at two different carrier concentrations.

where q = 1.6 X 10- 19 C is a unit charge and d is the thickness of the active region where an injection current provides the electrons and holes for radiative recombination.

PROBLEMS 9,.1

In classical physics, the equation of motion for a charge e in the presence of an electric field E and a magnetic flux density B is described by the Lorenz force equation d --p:::..= .F elf

==

c>(E +

Y

>< B)

where p = mv . The magnetic nux density 13 and the electric field

HI

389

classical electromagnetics are expressed in terms of the vector potential A(r, t ) and the scalar potential qJ(r, t ) (see Chapter 5)

vx

B =

and

A

E = -V¢ -

CIA ~

at

Show that using the Hamiltonian in the presence of the electromagnetic field H

=

1 2 - ( p - eA) + e d: 2m

and the equations for the Hamiltonian

d

aH

-r·=

dt

I

Bp,

d

ctt Pi =

aH Br,1

leads to the classical Lorenz force equation. If we choose Coulomb gauge, we have V . A = 0, and ¢ is zero since p = for the optical field. The periodic potential VCr) is the background static potential which is not coupled to the vector potential A for the optical field and is added to the Hamiltonian for completeness. Therefore,

°

1 2 H= - ( p - eA) + VCr) 2m

9.2

(a) For a semiconductor laser operating at 0.8 J..L ill with an optical • output power of 10 mW coming from a 20-,um x l-,um cross-sectional area, estimate the electric field strength E using S = wA2ok o p/2J..L = E 2n r/2,u oc, where n r = 3.4. Calculate k o p and A o· (b) Repeat (a) for a 1.55-J..Lm laser wavelength for the same power. Assume that the other parameters are unchanged. ' I

9.3

Derive the relation (9.2.37) between the spontaneous emission rate per unit volume per unit energy interval rSpon(E) and the gain coefficient g( E). Check the dimensions on both sides of (9.2.37).

9.4

Derive the Bernard-Duraffourg population inversion condition. /

9.5

Fe and F;. for the bulk structure at zero temperature in terms of the injection carrier densities nand p, respectively. (b) Derive an expression for the gain spectrum g( h co ) at T = 0 K.

9.6

Usc the material parameters in Appendix K for GaAs materials, except that the band gap at zero temperature is assumed to be 1.522 eV. Neglect the light-hole hand for simplicity.

(a) Find the quasi-Fermi levels

r I

.'

OPTI< 'AL PROCESSES IN SEMICONDUCTORS

390

(£1) Find the numerical values for the quasi-Fermi levels for the

electrons and holes in a bulk semiconductor in the presence of carrier injection at zero temperature assuming n = p = 5 X 10 l8 cm- 3 . (b) Calculate and plot the gain spectrum for part (a). 9.7

Repeat Problem 9.6 for a quantum-well structure with an effective well width of 120 A.Assume an infinite well model and a constant momentum matrix (independent of k) for simplicity. Label the major transition energies in your plots.

9.8

The E:k relations of a semiconductor material with a nonparabolicity conduction band can be described as

Conduction band:

Valence band:

EJk)

(~J2

__ h2k6 2m~

ko

where k o and (3 are positive material parameters describing the nonparabolicity. (a) Derive an expression for the interband absorption spectrum a(w) for a bulk structure at zero temperature. ASSL':l1e a constant momentum matrix. (b) Plot the absorption spectrum a( hw) vs. the normalized energy (hw - Eg)/(h2k5/2m) for (i) f3 = 0 and (ii) f3 = O.lm;/m r ,. where m , is the reduced effective mass (l/m r = l/m:' + 1/m7). 9.9

Consider a GaAs quantum well with an effective well width L = 100 A (assuming an infinite well model). The electron and hole energies are

'Lm".e ~

where

The other parameters are E; = 1.424 cV; 1.0.540 X 10 -]-.; .T . s = 0.6502 X 10 - [5 e V . n , = 3.68, and ten = 8.854 X W-- 12 F/ rn..

nl 5,

o =, 9.1 q

=

X

la- 3 1 kg,

1.00218 X 10 -

tl= I\!

C,

PROBLEMS

(n) Calculate the subbarid energies, Een(O) and EIIII/(O), for the first two electron subbands and hole subbands, respectively. (b) Using the normalized wave functions and the subband energies of the first two electron and hole subbands, calculate the absorption spectrum due to the interband transitions within the infinite well model. Assume that the matrix element Ie . Peel is a constant. (The overlap integrals between the electron wave functions In(z) and the hole wave functions g,,/z) are assumed to satisfy the relation ffn(z)gn/z) dz = 1 if n = m, and a if n =1= m.) (c) Assume that Ie· pcul = m oE p / 6, and E p = 25.7 eV for GaAs. Calculate the magnitudes of the absorption coefficients (1/ ern) for part (b) at (D Izw = 1.43 eV, (ii) luo = 1.55 eV, and (iii) h co = 1.75 eV.

Estimate the constants Co = 7Te2/(nrCEonI6w), the momentum matrix element Ie' PCLl = 1.5M; = 1.5(rn oE p/6), the reduced density of states p;D = m r/(7Th 2L z ) , and the absorption coefficient (TE polarization) O'.O(tlW)o for a GaAs quantum well with an effective well width L z = 100 A near the absorption edge. (b) Repeat (a) for an Ino.53Ga0.47As quantum well with L, = 100 A near the absorption edge. Note E/lno.53Gao.47As) = 0.75 eV.

9.10 (a)

9.11

(a) Plot the optical matrix elements for both TE and TM polarizations (see Table 9.1) vs. the transverse wave number k ( for a GaAs o quantum well with an effective well width 1 = 100 A. (b) Compare the optical matrix elements in (a) with those of another GaAs quantum well with a smaller effective well width L; = 40 A.

9.12

If a Gaussian line-shape function is used in the expressions for optical absorptions, what is the complete expression for the Gaussian function to replace the delta function o(Ec - E 1, - hw)?

9.13

Compare the interband and the intersubband absorp,tion coefficients in a semiconductor quantum-well structure. Discuss the optical matrix elements, the density of states and the absorption spectra.

9.14

(a)

To calculate the intersubband absorption in' the zz-type modulation-~oped GaAs / AlO.3GaO,7As quantum well with a well width 100 A, we use the following procedures: (i) The ground-state energy E I for the first conduction subband for a finite barrier model can be calculated numerically. We can then define an effective well width L e ll such that ) , f1- ( 'tr ' E··'1 = ----,' --. .2 in '; L d"f )

; I

392

OPTICAL PRC,( ESS;'::S iN SEMICONDUCTORS

ture. (ij) Using the infinite well model with an effective well width Left" from part (i), calculate the second subband energy E 2 and the intersubband dipole moment

where the integration is over the width L eff . (iii) Calculate the absorption coefficient a for a doping density N D = 3 X 10l7/ cm3, assuming that only the first subband is occupied. Assume that the refractive n ; = 3.6, T = 77 K, and the linewidth is 30 me V. (b) Repeat part (a) for an Ina S3GaO 47As/ InP quantum-welt structure with a well width 100 A 'and the same doping density. Use the parameters such as band gaps and the electron effective masses from the Table on the Physical Properties of GalnAs Alloys in Appendix K. Assume ts E; = 40%!lE g • Compare the results with the GaAs/Al o.3Ga o.7As system and comment on the differences.

REFERENCES 1. M. Plane}. Verh. deut . Phys. Gesellsch. 2, 202 and 237 (1900). 2. A. Einstein, Ann. Phys . 17, 132 (1905); "Zur Quantentheorie der StrahJung (On the quantum theory of radiation)," Phys . Z. 18, 121-128 (1917). 3. W. Heitler, The Quantum Theory of Radiation, Clarendon, Oxford, UK, 1954. 4. R. Loudon, The Quantum Theory of Light, 2d ed., Clarendon, Oxford, UK, 1986. S. H. Haken, Light, Vol. 1, Wa res , Photons, Atoms, p. 204, Chapter 7, NorthHolland, Amsterdam, 1986. 6. G. H. B. Thompson, Physics of Semiconductor Laser Deuices, Chapter 2, Wiley, New York, 1980. 7. H. C. Casey, Jr., and M. B. Panish, Heterostructure Lasers, Part A: Fundamental Principles, Chapter 3, Academic, Orlando, FL, 1978. 8. G. Lasher and F. Stern, "Spontaneous and stimulated recombination radiation in; semiconductors," Phys. s«, 133, A553-A563 (1964). 9. S. L. Chuang, J. O''Gorman, and A. F. J. Levi, "Amplified spontaneous emission and carrier pinning in laser diodes," IEEE 1. Quantum Electron. 29, 1631-1639 ( 19(3). lD. H. C. Casey, Jr., and F. Stern, "Concentration-dependent absorption and spontaneous emission of heavily doped GaAs." 1. Appl. Phys. 47, 631-643 (976). l l. M. G. A. Bernard and G. Dur affourg, "Laser conditions in semiconductors," Phys. Status Sulidi 1, 699-'Jl)3 (1961). 12. A. Yariv, Quantum ELectronics, 3d e d.. \Vi1cy, New York, 1989.

"

;<.EFERENCES

.393

13. E. O. Kane, "Band structure of indium antimonide," 1. Phys. Chem. Solids 1, 249-261 (957) 14. M. Yarnanishi and J. Suemune, "Comment on polarization dependent momentum matrix elements in quantum well lasers," Ipn . 1. .Appl. Phys, 23, L35-L36 (1984). 15. M. Asada, A. Kameyama, and Y. Sucmatsu, "Gain and intervalence band absorption in quantum-well lasers," IEEE 1. Quantum Electron. QE-20, 745-753 (1984).

16. R. H. Yan, S. W. Corzine, L. A. Coldren, and 1. Suemune, "Corrections to the expression for gain in GaAs," IEEE J. Quantum Electron. 26, 213-216 (1990). 17. H. Kobayashi, H. Iwamura, T. Saku, and K. Otsuka, "Polarization-dependent gain-current relationship in GaAs-AIGaAs MQW laser diodes," Electron. Lett. 19, 166-168 (983). 18. M. Yamada, S. Ogita, M. Yamagishi, K. Tabata, N. Nakaya, M. Asada, and Y. Suematsu, "Polarization-dependent gain in GaAsjAlGaAs multiquantum-well lasers: theory and experiment," Appl. Phys, Lett. 45, 324-325 (1984). 19. E. Zielinski, F. Keppler, S. Hausser, M. H. Pilkuhn, R. Sauer, and W. T. Tsang, "Optical gain and loss processes in GalnAs / InP MQW-laser structures," IEEE J. Quantum Electron. 25, 1407-1416 (1989). 20. E. Zielinski, H. Schweizer, S. Hausser, R. Stuber, M. H. Pilkuhn, and G. Weimann, "Systematics of laser operation in GaAs j AlGaAs multiquantum well hetero structures," IEEE 1. Quantum Electron. QE-23, 969-976 (1987). 21. L. C. West and S. J. Eglash, "First observation of an extremely large dipole infrared transition within the conduction band of a GaAs quantum well," Appl. Phys. Lett. 46,1156-1158 (1983). 22. For a review, see B. F. Levine, "Quantum-well infrared photr 'etectors," 1. Appl. Phys 74, R1-R81 (1993). 23. M. O. Manasreh, Ed., Semiconductor Quantum Wells and Superlattices for LongWauelength. Infrared Detectors, Artech House, Norwood, MA, 1993. 24. D. Ahn andS. L. Chuang, "Calculation of linear and nonlinear intersubband optical absorption in a quantum well model with an applied electric field," IEEE J. Quantum' Electron. QE~23, 2196-2204 (987). 25. S. L. Chuang, M. S. C. Luo, S. Schmitt-Rink, and A. Pinczuk, "Many-body effects on intersubband transitions in semiconductor quantum-well structures," Phys. Rec. B 46, 1897-1899 (1992). 26. D. Ahn and S. L. Chuang, "Optical gain and gain suppression of quantum-well lasers with valence band mixing," IEEE J. Quantum Electron. 26, 13-24 (1990). 27. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinoson, and A. Y. Cho, ':Quantum cascade laser," Science 264, 553-556 (994).

II

10 . Semiconductor Lasers Semiconductor lasers are important optoelectric devices for opticalcomrnunication systems. For a historical account of the inventions and development of semiconductor lasers, see Refs. 1-3. The first semiconductor lasers were fabricated in 1962 using homojunctions [4-7]. These lasers had high thresh old current density (e.g., 19 000 Ay cm") and operated at cryogenic temperatures. The estimated current density [2] for room temperature operation would be 50 000 A / ern 2 • The concept of heterojunction semiconductor lasers was realized in 1969-1970 with a low threshold current density (1600 A /cm 2 ) operating at room temperature [S-11]. These double-heterostructure diode lasers provide both carrier and optical confinements, which improve the efficiency for stimulated emission, and make applications to optical communication systems practical as transmitters. In the late 1970s, the concept of quan tum-well structures for semiconductor lasers was proposed and realized expe rim e n ta lly [12-16]. The threshold current dens. .; was reduced [16] to about 500 Ay cm '. The rec. .iction of the confinement region for the electron-hole pairs to a tiny volume in the quantum-well region improves the laser performance significantly. The quan- . tum-well structures provide a wavelength tun ability by varying the well width .' and the barrier heigh t. These id evices are also excellent practical examples for textbook problems on basic quantum mechanics, that is, the particle in a box model. In the late 1980s, strained quantum-well lasers were proposed to improve the laser performance [17, l'S]. Superior diode lasers using strained quantum wells reduced [19] the threshold current density to 65 A /cm 2 at room temperature . Further reduction in threshold current density using strain effects has been possibl e' [20]. The threshold current density of semiconductor lasers has improved by approximately one order of magnitude smaller per decade since their invention in 1962 . The improvements resulted from the development of novel ideas and a realizable technology such as liquid-phase epitaxy (LPE), molecular-beam epit axy (1V1BE ), or metal-organic chemicalvapor deposition (MOCYD) [21]. In the 1990s, re search on strained q uantum-well lasers continued. with remarkable progress toward high-perfo rmance diode lasers. Quantum-wire semiconductor lasers ha ve be en realized , The potential improvement using quantum-wire and quantum-d ot se m ico nd uc to r lasers has been under investiY) I -,

0.1

DOUBLE i:-IETEFOJU! ieTION SCr'-·J!CON ')UC

f(;}~.

LASEHS

395

gation both theoretically and experimentally [22-25]. Novel structures such as surface-emitting lasers [26-28] and microdisk lasers [29, 30] have also been realized. The microdisk lasers using a small disk with a radius R of the order 10 J.Lil1 and a thickness of 0.6 J.Lm show interesting resonator physics with gallery-whispering-mode lasing characteristics. Many of these structures also make research on the cavity quantum electrodynamics practical [31]. In this chapter, we study the basic semiconductor lasers and their physical principles. Various important designs and their models arc discussed. Since the research on semiconductor lasers and technology is so rapid, these physical principles and theoretical models will provide a good tool to design diode lasers with the desired characteristics, or to set up a foundation to design novel laser structures with superior performance.

10.1

ro.i.r

DOUBLE HETEROJUNCTION SEMICONDUCTOR LASERS Energy Band Diagram and Carrier Injection

Let us first consider a P-InP/p-Inl __ xGaxAsl_yPy/N-InP double heterojunction [32, 33] structure. Before the formation of the heterojunctions, the energy band diagram is shown schematically in Fig. 10.1a, where the energy levels are all measured from the vacuum level. At thermal equilibrium (Fig. lO.Ib), the Fermi level is a constant across the three regions. The band diagram still contains the features of the bulk properties in the regions away from the heterojunctions, assuming that the central region i~ wide enough. The built-i: potentials are determined by the Fermi levels:

where (10.1.3) With an applied voltage Va' the injection current is assumed to be I, and the current density is I -_.

1

(10.1A)

i·d. where

H'

is the width of [he active region in the lateral direction and L is the

cavity length of the diode laser. The carr icr

II

C'Jlll:C~iltration

in the active region

,

, ".,,~

... , ... ....,

... . -.-" .' . . ........-:- .... .... , , ~ , .,

,

S SHJ CONDlICTOR LASERS

(a ) - -

f-------1- ---- --f ---~~~~um

Ecp -----'------Ecp

- - - - - - - - . E fp Ef P

- - - - - - - - - - -

tAEy

Ey p P-In P

- --

p-In GaAs P

-

-

Ey N

N-In P

(b)

P

p

N

6E:t

------+-------E Ef------------------ -------------- --- -E

CN

f

-t-6E ~

E

v

yp

- - - - - Ey N (c)

P

Ecp--F-N-eX-)-.. .. . ~_{-~c

p

--- J>l f----

, ./ "

F

Fp - - ""- - - - - - - - - - - - - - - - e- - - - - - - - -

~AEy

Eyp

~

N

-7~:

T

- --

qV

_ .,. ,,' .

~

--

'Fp(x)

- - - -- - E y N

Figure 10.1. Ene rgy band di agr am for a P-lnP I p-InGaAsP I N -l nP dou ble heterojun ction str uc tu re (<1) before cont nct, (b ) after co n tact at th erm a l equilibrium (V = 0 ), and (c) under a forward bias (V > 0).

is determined by th e rate equ a tio n: () n dt

= D\, 2(! +

.I

t -

q(

R(n)

(10.1.5)

where the first term on the right-hand side ac co un ts for carrier diffusion, the second term is due to the curie r injection into the active region with a . .

tGl

DOLBLE f-iETFRO]'

jf-':' ,! I()''J" SE:rvnCOf'([jU~TO:ZLA3ERS

397

thickness d, and the last term, Ri n), accounts for the carrier recombinations due to both radiative and nonradiative processes. The unit charge q is 1.6 X 10 -)9 C. The carrier diffusion is a consequence of intraband scattering, which is important for gain-guided structures. For an index-guided structure, we can assume for simplicity that the diffusion effect is negligible, and obtain J - = R(n) qd

(10.1.6)

at steady state. The recombination rate is given by R( n)

A/lrn

=

+ Bn 2 +. Cn 3 +

RstNp h

( 10.1.7)

where the first term is due to nonradiative processes, Bn 2 is due to the spontaneous radiative recombination rate, Cn 3 accounts for nonradiative Auger recombinations, and the last term accounts for stimulated recombination that leads to emission of light and it is proportional to the photon density N p h ' The Auger term is important only for long-wavelength semiconductor lasers. We have assumed that n = p in the active region, i.e., the doping concentration in the active region is small. Otherwise, we use Bnp instead of Bn 2 , and Cni p or Cnp"; depending on the type of Auger processes. B may depend on the carrier concentration:

(10.1.8) The coeffici.. it for the stimulated emission rate is

( 10.1.9) where L'g is the group velocity LJ g = C / n g , n s is the (group) refractive index, C is the speed (:11' light in free space, and gt.n) is the gain coefficient. Below or near threshold, we can ignore RstNp h since the photon density is small, and write n R( n) (10.1.10) T e ( n) where

TJ!l )

1 -

..L1 11 r

+ Bn + Cn 2

(10.1.11)

Given the injection current density' J, the carrier concentration is determined

by J

(10.1.12)

(Jet

I I

-.

~.'

-,

~""""''i'''

-..,....-;-"~--.-~~....

~

..

""'''''l''''~''''-':;'---''''''''''

••••

'<EMICUNDUl.. TOR LASERS

393 '

if we know the coefficients Ann B, and C for the laser structure. Sometimes the carrier lifetime T e is taken as an input parameter and one determines n approximately by n = JTe/qd. Knowing n in the active region, the quasiFermi level in the active region is determined by

(10.1.13a)

(10.1.13b) and the hole concentration in the active region is determined by the charge neutrality condition:

(10.1.14) where the ionized acceptor and donor concentrations are

N;;

NA

1 + 4 exp [( E A -~~~-

-

Fp )

/

kBT ]

ND

~-----=---

1 + 2 exp [ (F:l - ED) / k B T ]

(10.1.15)

(10.1.16)

E A and ED are the acceptor and donor energy levels, and NA and ND are the

acceptor and donor concentrations, respectively. The quasi-Fermi level in the active region for the holes is then determined by

(10.1.17a)

(l0.1.17b) From the electron and hole concentrations, nand p, or the quasi-Fermi levels, F,1 and Fp , we can calculate the gain directly using the band structure. Typically, a linear rela tion for the peak gain coefficient vs. the carrier density such as

(10.1.18) for a bulk semiconductor laser is obtained over a region of carrier concentration, where a is the differential gain, and Il l r is a transparency concentration when the gain is zero (Fig. 10.2). For quantum-well lasers, an empirical formula using a logarithmic dependence for the gain vs. carrier density or

(a)

(b)

g(tlCO)

, g(n)

= a(n -

nu-)

Increasing current injection ft

---t--o<---------..o n rr

Figure 10.2. (a) The gain spectrum for different carrier den sities, the carrier densit y 11.

!I 2

>

Ill;

(b ) the peak gain

Y$ .

gain vs. injection current density is usually used. This is discussed further in Sections 10.3 and lOA. 10.1.2

Threshold Condition

The threshold condition is determined by gth f

=

a;

+ am

lY;=a o(1-f)+a g f a

= In

1 1 -In --2L R 1R 2

( 10.1.19) (10.1.20) (10.1.21)

where I' is the optical confinement factor, a i is the intrinsic loss due to absorptions inside the guide; a s and outside a o, and am accounts for the transmissions (outputs) at the two end mirrors (facets) . With an increase in the injection current density J, the gain will increase due to the increase of the carrier concentration n . When the threshold condition is reached, the carrier concentration n will be pinned at the threshold value nth since the gain is pinned at the threshold gain grh = (a i -+- Cl. m ) / I"; therefore, nth = n t r + (a; + (Yn)/(fa) and

(10.1.22) (10.1.23) Below the threshold conditi on . th e o u t pu t light con sists mainly of spontaneous emission and its ma gnitude is governed by e.. ' . A further increase ill the inj ection current density .' above thresh old leads ' to th e light emission

! I

4l.10

SENLCOfiDUCTOll. LASE.RS

through the stimulated emission process:

1= qd(Anrrl th

=

vgg(n)

=

CJZ~il) + qdRstNp h

+ qdi 'g g til N p h

= Ith

where R s t

+- BIlZh +

(10.1.24)

ugg t h since n is pinned at

nth'

Therefore,

( 10.1.25) The photon lifetime

is defined as

Tp

1

( 10.1.26) which accounts for the loss rate of the photons absorptions and transmissions. We obtain

111

the laser cavity due to

(10.1.27)

10.1.3

Output Power and Differential Quantum Efficiency

The optical output power d eterrnin ed by p

= out

.

Pout

vs. the injection current (L-I curve)

on r

[energy photon oj ( effective.volume ?f-the a photon .J density optical mode. \

IS

1(escape rate) of photons

( 10.1.28) where L'ga m is the escape rate of photons. Here w is the width and effective thickness of the optical mode (d o p / = ell f). Therefore

h co tI

am - -- -( J - l rl1 )

am

+ C'!i

«.; is the

(10.1.29)

The above relation assumes that the internal quantum efficiency 7]i is one , where 17 i is defined as the percentage of the injected carriers that contribute

DOULI' E lL2TEROrJ)' ~ CTI\jr' ; 3Uvl i C0 ND UCTOR

10.1

j

A :5ER::

401

to the radiative recombinations:

(10.1.30)

'rJj-

Generally, TJj <_1, and Eq. 00.1.29) should be modified as h co

am - - - T Ji( I - I t h ) q a m + «.

(10.1.31)

The external differential qu antum efficiency TJ e is defined as dPOtl1/dI

TJ e

=

hw /q

am

= Yl j - - -

am

+ «.

In(1 /R)

( 10.1.32a)

The inverse of the external differential quantum efficiency is - J _

n,

-1

-TJi

L] [1 + a i1n(1/R)

(10 .1.32b)

A pl ot of TJ e- vs. the cavity length L shows a linear relationsh .,-' with the intercept TJi- 1 at L = 0, as shown in Fig. 10.3. In real devices, leakag e current exists. The effect of leakage current I L can be taken into account in .' I

( 10. 1.33) wh ere I,..j

1S

the cur-rent injected int o the active r egion. Therefore, I t h is

Inverse external quantum efficiency

• -1 11 J'

'----

.

-------:~ L

Ca vity leng th

Fi gure J 0.3 . A p in t o f th e inv erse ex te rn a l quantum e ffi cien cy 1/ ) / ~ \IS . the la se r c av ity length L. The interce p t w ith the ve rt ica l ax is grv cs the inv e rs e int ri nsi c qu an t um effi cien cy 1/ ·.,." .

I I

SEM1COND~CTOR

LASeRS

modified to be

(10.1.34) ,

Also, the leakage current I L may also be increased with an increase in I. The output optical power Pout vs. the injection current I relation can be modified to be (10.1.35)

where Do JL accounts for any additional increase in the leakage current due to the increase of I . A typical output power vs. injection current relation is shown in Fig. lOA. Below the threshold injection current , the output light intensity is negligibly small. Above threshold , the output power is linearly increasing until saturation effects occur. There are three possible reasons for saturation [33]: 1. The leakage current increases with the injection current J, i.e., 6.IL =1= O. 2. The threshold current I t h may also depend on the injection current J due to junction heating. The increase in temperature reduces the recombination lifetime Tt!' For example, the Auger recombination rate increases when the temperature is raised. 3. The internal absorption a int increases with an increase in the injection current J.

Linear

-->-r Ith Figure lOA. c urv e) .

A typical diod e laser output power vs. injectio n current density relation (L - 1

le 1

DOUBLE HETEf:ZOJU1"ICTIOr-,

~ ~:r/:;(,ONDUCTOlZ

LASJRS

403

The output power vs. the injection current becomes a nonlinear dependence when saturation occurs. 10.1.4

Leakage Current [34-36]

The leakage current can be obtained from the minority carrier concentration. For example, the hole concentration Pi x ) in the N region satisfies -1 a a -P(x) ---jp+G(x) q ax at

P - Po

(10.1.36a)

Tp

(10.1.36b) In the steady state, assuming no generation in the cladding regions, we find the differential equation for P(x):

P - Po ------=0

( 10.1.37)

DpT p

Let us write P as P/",,(x) and Po as PNO for the hole concentrations in the N region. The solution for the hole concentration is of the form P N (x) = C I e -x/L p

+ C 2 e x / L p + P NO

( 10.1.38)

V

where L p = DpTp. Assume that the hole concentration is P N at x = X N = O. The excess carrier concentration is oPN(x) = PN(x) - P NO = P N - PNO at x = O. The other boundary condition, oPN(x), vanishes at the contact x = WN . The hole concentration is found to be

( 10.1.39) The leakage current density at x

x N = 0 + is, therefore,

=

qD p ( PH -- PNO) i., tanh(HfN/L p ) Similarly, the leakage current density flowing into the P region qD N ( NI' - i'f!,())

--------------

L N tanh( H'r/L,.v)

II

( 10.1.40)

IS

•

(10.1:41)

SEMICONDUCTOR LASEB.S

404

where we have used the electron concentration in the P region

(10.1.,42)

and aNp = Np(x) - N po = 0 at x = - WI" where x is measured from the edge of the P - p heterojunction and the electron concentration is Np(x = 0) = N p . The background concentrations P N D and N p o at thermal equilibrium are usually ignored since they are small. The total leakage current is

( 10.1.43)

Note that the minority carrier concentration P N at £' x N or N; at x = -xl' can be obtained using the band bending and (10.1.17) and 00.1.13), respectively. For PN' we have, at x = x N' the boundary of the depletion region near the p - N junction

(10.144 ) as shown in Fig. 10.lc, where V is the bias voltage. Here E C N - F N is determined by the doping concentration in the N region. We then use 00.1.17) to find PN :

(10.1.45a)

_

')

k B T ) 3/2

N v - '-- ( 2 rrh 2

3/2

(m ll h

3/2

+ m Ih

)

(10.1.45b)

Similarly, for N p at the boundary of the depletion region in the P region, we have Fr: - Eel' = qV + F; - Eel'

. q V + (Fe - E ~ p) -- E G I'

(LO .1.46)

where F"r - E v p is determined by the doping in the P region. The electron

405

concentration in the P region N; is given by (10.1.13),

Eel')

FN N p -- N C F 1/2 ( k T B

- N

-

'.

C

c(F,v--Ecp)//\n T

(10.1.47a)

(10.1.47b) The diffusion coefficients D p and D N in the doped wide-gap semiconductors can be taken from the mobility data based on the Einstein relation, e.g., D p = }-LpkT/q, and D N = }-LNkT/q. 10.1.5 Light-Emitting Diode vs. Laser Diode: The Role of Spontaneous Emission vs, Amplified Spontaneous Emission

In Chapter 9, we discuss the spontaneous emission rSpon(hw) from a forward biased p-n junction diode. The typical spontaneous emission spectrum is commonly used in light-emitting diodes (LEOs). The band gap of semiconductors is chosen such that LEDs in the visible region and the infrared region can be designed with high quantum efficiency. For a simple planar structure such as shown [37-39] in Fig. 10.5, the light emission from the top surface of the cladding layer due to a spontaneously emitted photon will be limited by a cone angle of 2e c ' where Bc is the critical angle between the p-type semiconductor and the ai-. Light with an angle of emission outside of this c'e angle will be reflected i.ack to the semiconductor region. Therefore, the g _ometric design such as a hemispherical lens or parabolic lens above the p-n junction diode has been used to optimize the light emission. Other important considerations for visible LEOs include the responsivity of human eyes. "Infrared

au Metal contact

.-----~~~ra"--------,lr---------,:r--p Active re

N

I / Metal contacts

(a)

(b)

Figure 10.5. (a) Schematic diagram of ~I planar LED device to show the effect of total internal refraction. (b) A semiconductor hemisphere geometry for LED operation.

, I

406

SE iviICONDU CTOR LASERS

Metal contact

Figure 10.6. AlG aAs surface -e m itt ing L ED with an att ached fiber to co uple light o u tp ut to a n optical fiber.

LEDs for applications in optical communication are designed such that efficient coupling into optical fibers can be achieved, as shown in Fig. 10.6, with a surface e mi tte r configuration [40]. The double heterojunction AlG aAs j G aA s st ructure also provides the carrier confinement to increase the efficie ncy [41]. F o r a laser diod e , the cavity design and the amplified sp ontaneous emissio n will be important for the formation of laser modes. To understand the physics of the laser diode vs. light-emitting diode operation, we consider an LED that is designed to have the same Fabry-Perot structure as an LD except that its two e nds are antireflection coated. If we measure the spontaneous e mission power spe ctr um from the top or bottom of the diode, called (a) Light Emitting Diode Window Light L w

Facet Light ..........- - - L F

(b) Laser Diode.

L

Facet Light

LF,---"'"'"'--t=======t=l-----.A-LF ' ,,--- Cl ea ved Facet s

~

Figure 10.7. (u ) Th e sppn t~lneou :) e rnission co m ing o ut o f the side (window li ght L ),,) vs, th e a mp lifie d spo n ta neo us e m is s ~()n com ing o ut fro m th e face t (facet ligh t L r ) in an LED s t ruc tu re . (h) A las er di o d e w i t h th e o utput power co mi ng from i.)Q[ h face ts.

[().l

DOl ' B LE HFTE r(()I L!NCT;'J ,\f

:)EM!C~)' T~l

:( lOR LASER:;

the window light L II " it will be a broad spectrum determined by the spontaneous emission rate per unit volume r ~ponUl(tJ) (Fig. 10.7a) [42,43]:

(J 0.1.48) where w is the width, d is the thickness, and L is the cavity length of the index-guided structure. The facet light spectrum LFUIW), taken from a facet of the LED, is the amplified spontaneous emission spectrum [42-45): LF(hw)

=

h co

fL

r SPOl1(flw)eG,.(/uv)z d zwd

z =o

= hwr

Spo n

e GIIU/(v)L (

flw)

[

-

G n ( flw)

1]

wd

(10.1.49)

where GNUzw) = fg - a j is the net modal gain (including all the losses due to absorptions and scatterings in the device). Note that at the transparency wavelength, GJhw) = 0 and LFUzw) = Lw(hw) when no amplification by the gain action exists. With enough carrier injection such that G; > 0, we see

LED T I

800

= 25°C = 50 rnA

900

1000

1100

1200

1300

PHOTON ENERGY (rneV) Figure 10.8. M e a sured spo nt a neo us e miss io n s pec tr um obt ained by co llec ting ph otons thr o ugh ~I win dow in th e s ub s t ra te L ~I a nd tile amplified spo n tane o us e m iss ion s pec tru m obtained by co llec ting p hotons from th e Iace t L' o f ,10 LED. (Aft er Rd. 42 ).

i I

SEMICONDUCTOR LASERS

408

that the photon density in the spectral region where g(hw) is positive will experience amplification , while that outside the positive gain region will experience ab sorption . Since the gain spectrum is narrower than that of the spontaneous emission spectrum [46] the facet light will be narrower than that of the window light. These results [42] are shown in Fig. 10.8, As a matter of fact, a comparison of these two spectra has been used to extract the gain spectrum of a laser diode structure. By measuring the two spectra LF(hw) and L ...,.Chw) at (he same current injection level, we can obtain the gain

(a) 12 LED :J

TE

c.i

'-"

,.-..

EXPERIMENT

T=25°C

,.-..

6

I

~

=SOmA 20mA lOrnA SmA 2mA

~ d

~ -..

a

- 6 800

900

1000

1100

1200

1300

PHOTON ENERGY (meV)

(b) 12 TIffiDRY n = 3.2Ox10 1 8 an- 3

6

2.46><101 8 an- 3 L89xl0 1 8 an- 3 1.41 XI0 1 8 an- 3 O.74x10 1 8 an- 3

o

-6 800

900

1000 .' 1100

1200

1300

PHOTON ENERGY (meV) Figure W.9. Spe ctrum of th e: rat io 0]' fa cet light t'l wi nd ow light of the LED at T = 25'>C at 1 = 50 ,20, 10, S, and 2 mA for TE pol uri zut icn . (a ) Exp e ri me n ta l data a nd ( b ) theoretical re sults are shown with arbitrary ve r tic a l scal e . The co rre spond ing ca r rie r densit ies are indicated in (b). . (Afte r Ref. -U C0 1993 IEEE .'!

409

spectrum. Furthermore, if we take the logarithmic function of the ratio of the two spectra, In] L F(!lw) / LI\(1z (v)], it will be close to that of the gain spectrum, since it is proportional to In[(cG"L - 1)/ GrJ ~ (rg/l - a)L if the overall gain (Gn L » 1) is large enough (Fig. 10.9). By fitting the gain spectrum with a theoretical gain model, we can extract the carrier density n at a given injection current I, as shown in Fig. 10.10. The carrier density vs. the injection current I is a monotonically increasing function of I, as shown in Fig. 10.10a at zs-c and Fig. 10.10b at 5YC.

(a) ,........ ,

M

so

-....... eo

3

0

LED (T=25°C)

----~

~

CI')

2

Z 0

(..W

LD (T=25°C)

~

~

1

I l h=9.7mA

~

u

0

0

10 20 30 40 CURRENT I (m:\)

50

60

(b)

I t h=21.5mA

o

I

o

I

r

I

I ,

10

I

I

I

,i 20

I

,

11,.,. I

30

I

40

r

I

,

I

r

50

60

CURRENT I (rnA)

I

I

Figure 10.Hl. Carrier density versus injection current plotted for both LED and LD at (a) T = 25°C and (h) T~"'" 5SOC. The solid curves are just visual guides. (After Ref 43 (QI

1993 IEEE.)

no 10.1.6

SEMICONDlJCTOl\. LASEPS

Analysis of the Amplified Spontaneous Emission

In a laser diode, the Fabry-Perot cavity provides feedback reflections of light inside the cavity. We can understand the lasing action by looking at the amplifiedspontaneous emission process below threshold. The intensity of a Fabry-Perot mode inside the cavity satisfies [47, 48] d1±(z) dz

± G,J

=

W( hw)

=

±(

z) + W ( ti w )

(I )f3rspon( hw) h co

(10.1.50a) (10.1.50b)

where Gil = fg - CK i is the net modal gain, f3 is a spontaneous coupling factor which accounts for the fraction of the spontaneous emission photons coupled into the lasing mode, and the factor 1/2 accounts for waves propagating in the +z and -z directions. The solutions for the forward and backward propagation light intensities are

(10.1.51 ) The boundary conditions at z = 0, 1+(0) = R1r-(O) and at z = L, 1-( L) R 21+(L) determine the constants flJ+ and 10 , For example,

=

(10.1.52) The output intensity from facet 2 of the laser cavity is [47,48] /2 = (1 - R z ) I +(:2 = L) =

/

(1 -

(eG"L -

l)(R J e C n L + 1)

R))------~ 2C " L 1 - R 2 e JR

W(flw)

G rI ( hw)

(10.1.53)

Note that in the above analysis, only the optical intensity is considered. The phase information is ignored. \Ve can also see that the threshold condition occurs at G

= tr

_I In(_1_J 2L R[R 2

(10.1.54)

where the optical output approaches infinity at the lasing wavelength. In other words, the output intensity L is proportional to the spontaneous emission coupling ~V(t/(~» multiplied by the transmission coefficient and is amplified by various factors in (to.1.53), including the vanishing denominator.

4 1J

1= 15 rnA

L--....-~~--'- - - - - '-_. . -. L~,' I.IIIJuw~ ~.w.I11

. -. . . L- ~~"'-----'-~------'-----'------'

il..>.U.l1I.1.Llo.l

1.44

1.46

1.48

11

1.50

1.52

1.54

Wavelength (urn) 1 = 13 rnA

E

b t)

4)

0.. en

1.44

1.46

1.48

1.50

1.52

1.54

Wavelength (urn)

1=8 rnA E

E t)

4)

0..

en

1.44

1.46

1.48

1.50

1.52

1.54

Wavelength (urn)

1= 6 rnA

E

E t)

4)

0.. Vl

1.44

1.46

1.48

1.50

1.52

1.54

Wavelength (u rn) Figure 10.1 L A m p lifie d S PO fl li.!flCO US e m iss ion spec tra of a la ser diode at different c ur re n t inje ct ion leve ls, 1= 6 , 8, 13, and 15 rn A . The vert ical sca les a re in arbi tr ary unit s and do not h ave the sa me sca le s in differe n t tlg Llres. The th re s ho ld curre nt is 1 ~ . 5 rnA .

II

412

. SEMICONDUCTOR LASERS

In reality, the optical output is not infinity; we may say that the net threshold gain G; approaches (but is slightly smaller than) the mirror transmission losses (I j 2L )In(l j R] R 2 ) · An interesting 1imit occurs when both facets arc antireflection coated, R] = R 2 = O. We find

(10.1.55)

which is the same as the facet light power L F( trw) per unit cross section wd as expected. We also note that the transmissivity T', = 1 - R 2 can also be written as In(ljR z) when R z ~ 1 because In[lj(l - T 2 ) ] = lnt l + T z) = T z = (l - R z )· Therefore, the mirror loss in its distributed form am = (lj2L)ln(JjR 1R z) is used very often. The evolution of the facet light or amplified spontaneous emission from a laser diode as the injection current increases from below-threshold operations (I = 6 and 8 rnA) to near-threshold (r= 13 rnA) and above-threshold operation (I = 15 rnA) is shown in Fig. 10.11. Note that the vertical scales are not the same for the four figures. The lasing action starts to show drastically when the injection current approaches the threshold level 03.5 rrtA). An expansion of the plot for the amplified spontaneous emission spectrum at 8 mA was' shown in Fig. 7.17, where the Fabry-Perot mode o spacing can be seen clearly to be around 7.9 A for the laser with a cavity length L = 370 u.tt». The carrier concentrations n extracted from the gain measurement below and near the threshold can be extracted and plotted as a function of the injection current. Above threshold, the carrier concentration in the laser diode is pinned at the threshold carrier density, since most of the extra. carriers injected contribute to stimulated emission and output optical power. Important issues on the mode fluctuations and temperature dependence are discussed in Refs. 42 and 43.

10.2 GAIN-GUIDED AND INDEX-GUIDED SElVIICONDUCTOR LASERS In this section, we study both gain-guided and index-guided semiconductor lasers. The difference is in the lateral confinements of the optical mode and the carriers. The index-guided semiconductor lasers have been demonstrated to have superior performance to that of the gain-guided geometry. However, the gain-guided structures are still use d because of their ease of fabrication. They are investiga ted here because many interesting physical processes, includ ing carrier injection and diffusion, optical gain, and spatial carrier profiles, can be extracted from these structures.

10.2

G.'\JNGUIDED AND JNDEX-GlJ1DED

:"~EMICONDLJCTOR LASERS

4J ~

ID.2.1 Stripe-Geometry Gain-Guided Semiconductor Lasers l34, 38, 49-57] A stripe-geometry gain-guided semiconductor laser is shown in Fig. 10.12, where the metal contact is the stripe with a width S defined by processing procedure. The injected current] spreads along the lateral (i.e., y) direction. Therefore, the carrier density II distributes along the y direction determined by the current spreading and the lateraldiffusion. The effective width of the carrier density distribution and the gain profile along the lateral direction are therefore wider than the stripe width S. This gain inhomogeneity along the y direction provides the guiding mechanism for the optical mode confinement in the lateral direction. An associated effect is the antiguidance due to the reduction in the refractive index variation induced by the gain profile. In this section, we study an approximate analysis method to model the gain-guided stripe-geometry semiconductor laser. Analysis of the Optical Modal Profile and Complex Propagation Constant. We consider the TE mode with index guidance in the x direction, perpendicular to the p-n junction plane, and with gain guidance along the y direction, which is transverse to the propagation direction. The optical electric field is approximated by (10.2.1) under scalar approximation, where E/x, y, z ) satisfies the wave equation ( 10.2.2)

(b)

y

-If

x ~

----.

.\\ y

ley)

(c)

loe-K(lyl .../

n(y)

Cd)

~) _...//

I

__ -.S.. 2

I

s 2-

-~----~y

/ 1

----+----

2

2

_S

Figure 10.12.

.s

Y

(a) A stripc-gCt,qldr)i g:tin-guidt,d semiconductor las:: r, (bl cross section, (c) distribution of the current density J( Y J, and Cd) the carrier density profile ll( Y).

II

SEMICONDUCTOR LASERS

41-1

where a parabolic p rofile in the active region is assumed because of the approximate parabolic carrier and gain profiles. : d

x>

2 d

S(X,Y)

[x] <

2"

(10.2.3)

d

x <

2

Our goal is to find the electric field profile Ft x )G( y) and the complex propagation constant k , for the inhomogeneous permittivity profile 00.2.3). We can write E( x , y) in terms of an unperturbed part, e (O)( .r ), and a perturbed part, ~ e( x, y) ,

e(X, y)

=

e COl( X)

+ Lis(x , y)

(10.2.4)

where

e(Ol( x)

-

d

SI

x >

S2(0)

Ix l

SI

x< - -2

2 d

<"2

~s(x,

y) -

d

0

x >

_a 2 y 2

[x ] < -

0

x<

2 d

2

d

d 2

(10.2.5) The unperturbed part s(O)(x) is the pe rrruttrvrty profile of a slab optical waveguide; the solu tio n for the optical field was obtained in Chapter 7. Here we subs tit ute 00.2.1) into (10.2.2) and obtain

(10. 2.6) As suming that the index guidance along the x direction provides a good optical confinement, we approximate Fi; x ) by the electric 'p rofile E;O)(x) of th e unperturbed slab waveguide with a propagation constant f3 z: (10 .2 .7a)

(lO.2.7b)

1(J.2 GAIN ·GUIDED AN!.)

'~I';DSX-(;UIDL::;)

SEMICONDCCTOR LASE:{S

'[15

Using the definition /1-? =

.., _. P.-

/-';:

k?-

(10.2.8)

z

in 00.2.6) and 00.2.7), we find

(10.2.9) To solve 00.2.9), we multiply it by the optical intensity profile l(x) (l3z/2wj.L)IE~O)(x)12 and integrate over x from - 0 0 to +00 to obtain

=

(10.2.10) noting that the wave function Fi x ) = E}O)(x) has been normalized . (rOo- ",,l(x) d x = 1) and L1£(x, y) is zero outside the waveguide, which introduces the optical confinement factor I' = flxl:; d/21(x) d x. Equation 00.2.10) is independent of x and its solutions are the Hermite Gaussian functions (see Section 3.3): d 2un ( t ) - - 2-

dt

II n (

t) = H n (

+ (2n + 1 - t 2)u(t) , )

e - t 2/2

=

0

n = 0, 1, 2, 3, ...

( 10.2.11a) (10.2.11b)

where ( 10.2.1Ic) I

If we change the variable from t =

g 1/')-y

y

=

to

t

in 00.2.10),

(..,? ,))/4 y w~jJva-P

( 10.2.12a) (10.2.12b)

we obtain the following equation similar to 00.2.11a)

(lO.2.13)

! I

_'_ .

w:-_. _ _ . ~

.........._ _ ..

"

,~ . . ~

~

. .............._

_

.--:-:-- . .-..:'" .._~ ..... - - , ., -. ..'f"".

_ .. ~ .~ • •• -

~ .. _

· ... _. _ .. . ·......... . o . _ ~

..

~."-

. ~. - .

".""'" . . ..... -.-'

'~ .

0 · ... ••

• .,.. '

• •••

• .• • •• • --... -

. .. .

-

-

--

" -

, ... ~._

---:- ...

~ ••~' _ • .~ .~.

.. . ..

~ ..... . ..,.. ,~~~. ~"':' .-:- ;~. _ ~ ..

SEf.1ICONPUCTOR LASE.RS

41' ,

Therefore , we find G

=

lJ 2 =

H II ( t )e -

r2

/

2 =

H ll (e

/ 2 y )e --t; y

2/

2

g(2n + 1) = (w 2 p, a 2 r ) 1/ \ 2 n + 1)

( lO .2.14a) (10 .2.14b)

The complex propagation constant of the gain-guided laser, k z > is obtained from (10.2.15)

for the nth lateral mode. We note that f3 z depends on the index m of the TE m mode for the slab guid e geometry. Therefore, k z is prescribed by two mode indices, In and n , wh ich are associated with the optical field variation al ong the x and y directions, respectively. The optic al e lectric field is given by

Ey( x, y, z) ~ E;O)( x) H,,[ (k oG) 1/2 [ ~ J1/4 Y]e - k"o(f/,") ' I ' , ' / 2 e'k" (10.2.16)

Using bin omial expansion in (10.2 .15), we have for n

0

=

( 10.2.17)

Because a

=

a , + ia i is complex, the constant phase front is determined by k~ z

- k Oa i

r )1/2v 2

(Co -

---

constant

=

2

(10.2.18)

which is a cylindrical p arabolic surface (F ig. 10.1 3), where k; is the real p art of k=, which is close to 13=· ; The complex refractive index profile in the active regi on is given by £,

nr(y) +irz i( y ) = ( -

Writing the real and imaginary par ts of

L-:

(0)

-

EO

")

:»

] /2

a-yI

(10.2.19)

:/0) as ( 10 .2. 20 )

,

11).2

GAIN-GJiI:ED Al"iD

1l'~DE>~-('JU!DED

SEMICONDUCTOR LAS:":'RS

<1·17

I I I I 1

,

Figure 10.13. [51].

, .-

,,

_

.- .-

The constant phase front of a gain-guided mode is a cyclindrical parabolic surface

we can expand 00.2.19) using the binomial expansion, noting that IC2i(0)1 « C2r(O), and the perturbation la2y ZI « C2,(0) in the region under or dose to the stripe region. We obtain the real and imaginary parts of the index profiles

nrC y) _ (E 2

r(

0) ) [1 _ a; - a; 1/2

y2]

2c 2r(0)

co

( 10.2.21a)

(10.2.21b)

which depend on y with a parabolic profile.

Current Spreading and Carrier Density Profile. To account for the current spreading effect, an approximate formula for the injected current density ley) can be used:

Iyl < 5/2 iyj > 5/2

(10.2.22)

The carrier distribution n( y) along the y direction can be solved from the diffusion equation, using Eq. (2.4.18) with one y) = nt: y) - no = nt; v), Since the active region is undoped, the thermal equilibrium value no is very small,

n

J( y) (10.2.23)

dv '

where L; = VDIlTn is the diffusion length, and D II is the electron diffusion coefficient. The solution for d y) can be put in analytical form. For example, below the stripe, J( y) = ./0.' \-'le. write Il( Y) as the sum of the homogeneous

I I

SEM'CO ,'JDUCTOR 'LASERS -, .'

4t8

solutions and the particular solution: ( 10.2.24) where No = lOTn /(qd). Since the structure is symmetric with respect to y, we have C 1 is, an even function of y. Ou tside the stripe region,

=

C z , that

(10.2.25) The coefficients C 1 and C 3 are then determined by the continuities of n(y) and the diffusion current density along the lateral direction J/ y) a d n / d y at y = 5/2. We find 2C [=

-KL - s/ 2 L n N Il_ 1+KL e 0 II

and

Therefore, n(y)

=

NO[1 -

(~J]

n

. KL e -S/2L n cosh 1 + KL n i.;

5

<

2"

Iyl >

s 2

Iyl

.'

No

1

+ KL n

[-KL n

+

t

(e\-S/2LIICOSh~ + 2L n

'

_KL __n_)e-C1YI-'S /2) /LII 1 -KL n

1 e -K(I.I'I-S/2)] 1 - KL n

(10.2 .27)

If we ignore the current spreading outside of the stripe region, K ~ 00.2.22) so that J(y) = 0 for I yl > .)'/ 2. Equation 00.2.27) reduces to

[I - e

S / 2 L"

n( y) = No

S

sinh ( "7-

_L"

cosh (

Z" )]

"

J e- iY1/ L"

oc In

s

Iyl < 2

S 1"'1 > .I 2

( 10.2.28)

oTHEORETICAL

15

10

505

10

15

POSITION (~m)

Figure 10.14. Theoretical plot of the carrier dens ity profile n( y ) in a gain-guided se m ico nd uctor laser measured by the spont an eous emi ssion intensity . The theoretical model assumes a stri pe width S = 11.7 ~ m and a d iffusion length L II = 3.6 ~m. (After Ref. 54 © 1977 IEEE.)

where No = JOT" /(qd) = GOT n · The above result (10.2 .28) is the same as (2.4.21) in Section 2.4. Previously we had an approximate rel ation between the gain and the carrier density for a bulk semiconductor laser: g ( y)

=

(! [

n ( y) -

fl l r ]

( 10.2.29)

Therefore, the gain profile can be approximated using the expressions 00.2.27) or 00.2.28) for n( y) in 00.2.28) . Since the spontaneous emission spe ctrum depends on the local carrier density ni: y), a measurement of the e missio n profile as a function of y will determine the density profile. Therefore, the carrier density profile can be mapped out from the emission intensity profile, and the results agree very well [50, 54] with th e theoretical curve calculated using (10.2.28), as shown in Fig. 10.14 . . 10.2.2

Index-Guided Semiconductor Lasers [33, 58, 59J

In the gain-guided semiconductor lasers, the carrier spre ading along the lateral direction degrades the laser performance, such as a high threshold cu rre nt density and a lo w differential quantum efficiency. To . improve the laser perform ance . index-guided structure s have been used su ch that a better optical confinement , and therefore. [In improved stimulated emission process, ca n occur in the a ct ive re gion. One exam pl e is a ridge waveguide laser with a we a kly index-guided structure, as :,hi.)'v'y'U in Fig. to.lSa. Another example is :.:n etched-mesa buried he tc: ,,:;tnli.:tUi"c w'ith improved optical and carrier co nfinem e nts in t he acti ve r (::,~::i o n. ~t S s h \ )', o/ 11 in Fig. lO.15b.

I

I

-c-~

...

_~~

.....,...... -'-_

."_,,,:'~_

'" 00.··•• ," _....

,. _ .•-. ....':-"'"

.

~-

... _.........' ....

--.

.~._

,-., _..-.---....

""'.~.~

._,V ...

~.~

_,r_"~",,

........._.' ....., ..... -..-. - '"

SEM1CONDUCTOF LASERS

'"l,20

(a)

P-InGaAsP

(b)

P+-InGaAsP /

I

P-InP

r

dielectric

InGaAsP

N-InP N-InP Substrate

InGaAsP active layer

XL

n-InP Substrate

Y Figure 10.15. (a) A ridge waveguide laser for index guidance [33, 58]. (b) An etched-mesa buried-heterostructure index-guided laser [59].

For a ridge waveguide laser, an effective index profile n eff ( y) along the lateral y direction can be obtained using the effective index method by solving the slab waveguide problem at a fixed cross section defined by a constant y. The guidance condition and the optical field pattern are then obtained from the waveguide theory for the index profile n ef f ( y), following the procedures described (60] in Section 7.5. The effective index method can also be applied to a rectangular dielectric waveguide structure as the active region of the etched-mesa buried-heterostructure laser. We find the optical electric field for the fundamental TE mode (or higher-order TE modes) from (7.5.3): E

=

yEy(x, y) = yF(x, y)G(y)

(10.2.30)

We obtain the optical confinement factor using the definition

f=

f factive region E I":""f":""E

X

H* . i dx d y

X H*

'idxdy

(10.2.31)

Analytical expressions for the slab waveguide geometry are derived in Chapter 7. Here we may use

--E y H*x

Ex H*

( 10.2.32) WJ-L

Since k ~ /

(cJJ-L

is a constant for a given mode, we can simply use , ,.

i"

I.;

y_

.r, y) [ d ._ x d_y __ f r f= -'x- - I r: ( x.ly \ 12 , j -so) _ ''''I D~ {X, y)! d x d y J ) aC[lvc! 1:: y (

..

'GC

(10.2.33)

QUANTUM-WELL L<\SERS

j()]

which is approximated by the product of the two optical confinement factors along the x direction (with a slab geometry in the effective index method) and along the y direction when the field is approximately given by E y = F(x)G( y)

(10.2.34)

It is a good approximation for a strongly index-guided structure and F( x, y) = Fe .r ) if the geometry along the y direction is uniform such that the first part of the wave function F( x, y) in the effective index method described in Section 7.5 can be assumed to be independent of y. Index-guided semiconductor lasers have been shown to exhibit excellent performance including the fundamental mode operation, low threshold, high quantum efficiency and low temperature sensitivity [33]. 10.3

QUANTUM-\VELL LASERS

Quantum-well (OW) structures [12-16,61], as shown in Fig. 10.16, have been used as the active layer of semiconductor laser diodes with reduced threshold (a) Single-Quantum-Well SeparateConfinement Heterostructure

u __Il'----_ --By .>

(b) Multiple-Quantum-Well SeparateConfinement Heterostructure

~~ E

y

(c) Graded-Index Separate-Confinement Heterostructure (GRINSCH)

Figu r« 11). Hi.

~.--

---

Jlf1Jl

Ba nil -gnp profiles for (a) single-

qU:HltUIl1-wcIL (b) rnult iplc-quanturn-we ll, and (e)

""'''-_____ c

.... v

grade d-inde x sepur are-confineme nt heterosrructur e IGRINSCH) semiconductors lasers.

~El'vIICONDUCTOR

422

LASERS

current densities as compared with those for conventional double-heterostructure (OH) semiconductor diode lasers. Research on quantum-well physics and semiconductor lasers has been of great interest recently. For a brief history, sec Ref. 3. Various designs such as single-Quantum-well (SQW), multiple-quantum-well (MQvV) and graded-index separate-confinement heterostructures (GRINSCH) have been used for semiconductor lasers [21]. As we have seen in Chapter 9, quantum-well structures show quantized subbands and steplike densities of states. The density of states for a quasi-twodimensional structure has been used to reduce threshold current density and improve temperature stability. Energy quantization provides another degree of freedom to tune the lasing wavelength by varying the well width and the barrier height. Scaling laws for quantum-well lasers and quantum-wire lasers show significant reduction of threshold current in reduced dimensions [24].

10.3.1

A Simplified Gain Model

The simplest model we consider is the gain spectrum based on (9.4.18) for a finite temperature, assuming a zero scattering liriewidth first [62, 63J: n m i

(10.3.1a) where

(10.3.1b) (l0.3.1c) and

(10.3.1d) is the reduced density of states. The overlap integral between the nth conduction subband and the m th hole subband is usually very close to unity for n = m, and it vanishes if n =1= m because of the even-odd parity consideration. The polarization-dependent momentum matrix element is listed in Table 9.1 for conduction to heavy-hole and light-hole band transitions. The occupation factors for the electrons in the !Z th conduction subbarid and the electrons in the m th hole subband are = ,10) .f', I. / ( E,.[ L

j .rn (E·" L'

{

= tl j-

COl·)

._-

1':-1//1/

.-

.L-J1 1H 1

J u)

F-'/>II

)

--

( 1O.3.2b)

E

(a)

•

•

(b)

r---

Ee --- Eg-----

-4-----+----+--+-.r--.

o

-I-+--,/--,L..../---r-,"t-----~~~~::;~~ ~~: +fh(E) ~ 1 - fv(E) Population inversion in a quantum-well structure, where E g + F,. - F , > iu» > Eel - E h 1 + E~. (b) The product of the conduction band density of states p) E) and the Fermi-Dirac occupation probability fc(E) for the calculation of the electron density II is plotted vs. the energy E in the vertical scale. Similarly, p/ £)/:,(E) = Ph(E)[l - f,.(E)l is plotted vs, E for the energy (E < 0) in the valence band. Assume that the temperature T is 0 K. Figure 10.17.

(1)

Gain occurs when f(~' > f/ n , that is, the population inversion is achieved (Fig. 10.17). It also leads to E g + Fe - F; > h os, where Fe and }~ are the quasi-Fermi levels for electrons and holes, measured from the conduction and valence band edges, respectively. Only those electrons and holes satisfying the k-seleclion rule contribute sig. .ificantly to the gain process. 10.3.2

Determination of Electron and Hole Quasi-Fermi Levels

The quasi-Fermi levels Fe and F; are determined by the carrier concentrations nand p which satisfy the charge neutrality condition 11

For undoped quantum wells,

-+ N;;

N;

= p

+ Nt!;

( 10.3.3)

= 0 and tvr.,,- = 0, we have n = p

and n = jGCdEpe(E)/"r.(E) o :x:

Pes \ ' ~ ~

L.;;

--

l~) ':CIL

n= 1

) .

fJ;,(E)

H (.'~ t:

-x:

~.!..:!-. \' }./" t ~-'I"" :." I~ .. ~ ,.. I

- rn

.~.

I

," ) _. D.

p

= ~

p"

r~

'-

d E Ph( E )[ 1 -

fA E) ]

-- '":1:

IlZ* e 11

tz2

*

nZh 'iT

tl ~~

( 10.3.4)

:-:;Cv!lCONDUCrOR LASERS

(a) T = OK

~Pied ~ valence band ~ (by ~ electrons)

o

Phfh(E)

p(E)f(E)

Area =p

Area

=n

peE feE) (b) T= 300K 77.T7777777/7T,777:77A - - • - - - -

Ph

Eg

Eel

Fc

Figure 10.18. The functions of the products Pe(E)f/E) and p,,(E)}ir(E) = Ph(E)[l - fJE)] V5. the electron energy E for the calculations of the carrier concentrations nand p , respectively, for (a) T = 0 K and (b) T = 300 K. The area under Pee E)J) E) is the electron concentration and the area under PII( E)f,.( E) is the hole concentration .

where Hi:x ) is the Heaviside step function; H( z ) = 1 if x > 0 and H( .r ) = 0 if x < O. In Fig. 10.18 we plot the products pec E) fcC E) and PI/EX1 - !u(E)] vs. the energy E for T = 0 and 300 K, respectively. The areas below these functions give the carrier concentrations: n and p. Alternatively, it is convenient to use the surface electron and surface hole concentration, n s and P S ' respectively:

Ps

=

I~ dEp~.~( E) [1 - !c( E)] -

p~f( E)

'X..

nZ * h -

'il

h2

.

0:

L III

H(E h m

-

E)

= 1

(10.3.S) where we have integrated over the well width such that the well width L . will not appear in the surface concentrations, except for the quantized energy levels which appear as the band edges in the density of states. .

425

10.3.3

Zero-Temperature Gain Spectrum

To understand the gain spectrum, let us consider the gain at T = 0 K. Remember that the model here is oversimplified, since it does not include the effects of scatterings or inhomogeneous broadenings such as those due to well-width fluctuations and the electron-hole Coulomb interactions. Our model here is to show the simplest picture for population inversion for the optical gain process. At T = 0 K, the Fermi-Dirac distributions are step functions. For the electrons in the conduction subband, we have

E Fe

(10.3.6)

The electron concentration is simply the area under the function p/E)!c(E), which is plotted V5. E in Fig. 10.18a:

m*e

L

(Fc

-

E en )

( 10.3.7)

n = occupied subbands only

For a single (ground-state) sub band case,

( 10.3.8) . the location of the quasi-Fermi level i.. linearly proportional to the carrier concentration at zero temperature. Similar results hold for holes:

(10.3.9) For unstrained quantum wells, the topmost subband is a heavy-hole subband. The hole effective mass in the plane of the quantum well is usually smaller than its value in the bulk semiconductors because of its coupling to the light-hole subbands. However, m~ is larger than for most unstrained quantum wells. We find that for n = p,

mrz

m;

(10.3.10) which means that the areas under the function Pee E)!c( E) and Ph(E)j)E) should be equal. Since nIh > rn~, we find E l d -- F; < I':. - Ed' That is, the quasi-Fermi level F, is closer to the hole subbarid edge than the separation of F; and £(.,. The transparency optical energy occurs at h co = Fe - F", where Ie - I. = O. As a matter of fact, t = 1 and Ie = 0 for h co < E!{ -IF;_. -- Fc ' and I, = 0, f" = 1 for h » > E g + r~. - F,.. The gain spectrum is a

~·T

46

+2

g(tlw)/g m

Gain

+1

E~7(o) -1

'JIICO]':DUCTOR LASERS

T=OK

.,

-------_ ... --- ....

E g + Fe -

r,

Absorption

-2

(a)

(b)

(d)

(e)

Figure 10.19. An illustration of optical gain of a quu.uurn well at T = 0 K (a) Normalized gain spectrum, (b) the energy profile in real space, (c) the energy dispersion in k: space, and (d) the Fermi-Dirac function vs. the energy for the electrons in the conduction band and in the valence band,

simple rectangular window, asshown in Fig. 10.19a:

En < h co < E g + Fe - Fe g(flw)

otherwise

(10.3.11)

For the optical energy between the e l-h] transition band edge and the quasi-Fermi level separation !:-',-..: -+- F, the photons experience gain. Otherwise, they experience absorption with J spectrum similar to that of an unpumpcd quan tum well with a stepwise shape. The only modification is ncar the band edge. The occupations of states by electrons and holes in the quantum well are shown in Fig. 1O.19b in real space and in Fig. 10.19c in k

f:,

: 27

space showing the k-seleciion rule. The F e rm i-Dircc distributions for ele ctrons in the conduction band j ) E ) and for e le ct ro ns in the valence band f/E) arc shown in Fig. 10.J9d, where Eb(k) - 1:'a( k ) = h o: has to be satisfied from the k-sclectiol1 rule and the e ne rgy conservation condition , which lead to the population inversion fa ctor .fJ t.-"b) - f) EJ. We see that for E g +- I?-'d - E h 1 < ti.» < E g + Fe -'- Fu, f/E b ) = 1 and fuCEJ = 0; therefore , gain occurs. Otherwise, f e = 0 and f" = I, and absorption occurs, as sh own in Fig. lO.19a. 10.3.4

Finite-Temperature, Gain Spectrum

At a finite temperature, the gain spectrum looks like that shown in Fig. J 0.20a. The Fermi-Dirac distribution will deviate from a sharp step function and is equal to ~ at the quasi-Fermi level. The electron concentration is the area below the function peCE)j/E) , as shown in Fig. 10.18b, which can be

T == 300K

.,,.-----------.,

+2

+1

...

~--~-------.----- .~_ .

Ec1(O)

hI Absorption E g'-l-F e -F v--______ _ '

-1 -2

(a)

-l.

--

E

E

~ Fe I.... :- r --Fe- -- ~~--- -

J:~:

E

--j---t+ - -

-- E

V/ / / /v0>.

Eb

e 1-_e. g

Eel

tv» E

g

I----i--_+_

- ------ p--,EF '_.0 I---_+_-

.

V ---

hl

~

~'/7 ~ (b)

// / / :; (c )

Cd)

li g Ul' 2 LO.20. "\ 11 idealized gai l; s pe ct r u m v I' ~ I '-iU~\ I : ~ I I I-, l "vc:: 11 ~il T '= 3i lU K d :> ~ ~i ming ;l z ~ r, ) liu cwid th : (a l normalized gain spec t ru m . ( hl r h, ~h ll. ,:ni : ~ ti .:nc rsy p ro file ill r c ll s pace , \ ' ': : th:= pa rabol ic b anJ str uc t ur e ill l: S ;J '-l C c~ . ::l Jid ',d ) the Fe lm i- ·[) ,,·,\C di st ril. u ti o us t h orizo n tul scul cs) 'V ~ . energy E (ver tical scule ).

... ..... . . .

,. ~

..

,..

.~

'

.....-_

_~

••_

~

. , /"l'

- • •• -

or

v_._"

,.,

- .. •

~ ,~

.-

~ErvW::ONDTJCTOR

428

LASERS

obtained an a lyt ica lly:

L

n. In(l +

e(F, -EerJ) / kBT)

(10.3.12a)

1/=1

where the prefactor nc

=

(10.3 .12b )

is a characteristic concentration at finite temperature T. Similarly, the hole concentra tio n is DC.

L

P =

n ~.

In (l +

e ( Eh",-Ft.) / kaT)

(10.3 .13a)

m=l

wh ere

nv

m j,kBT =

(10.3 .13b)

rrtl 2L z

which can account for heavy-hole and light-hole subbands, where m accounts for all hol e subbands. The gain spectrum is given by Eq. 00.3.1a), and is plotted in Fig. 10.20a. It starts with a peak value at the transition edge h o: = E/~}(O) = E g + Eel - E 1I1 , where E( =- 0, and decreases to zero xt E ; + F; - F; ., then becomes absorption at higher optical energies. The sharp rise in gain near the band edge compared with the soft increase in a bulk OD) semiconductor is due to the steplike density of states in 2D vs. the slow increase of th e square-root (IE) density of states in 3D. For a single subband pair, n = eland m = hi, we have the gain spectrum

where g l1l is give n in OO.3.1b). The peak gain occurs at E [ = 0 o r h ro

= E/~} (O ) :

( 10. 3. 15a) whe re

J

FC ( IJo.r W

1

--

E h\tc") -- - -- - - -=--- -

( 10.3 .15b)

( 10.3. 15c)

H;J

QUANTUM-WELL

10.3.5

LASi::I~S

42'j

Peak Gain Coefficient Versus the Carrier Density

As shown in Fig. 10.21, when we increase the carrier density, the gain increases. Approximations used in the literature [62] are fcC E( = 0) = 1 e -It/It,_ and !t,(E( = 0) = e -pine. These results are exact expressions if only one conduction and one valence subband are occupied. They are approximately true if we redefine [62]

=

ftc

(10.3.16)

and a similar equation for holes:

l1

l

,

(10.3.17)

-

Therefore, an approximate formula for the peak gain gp as a function of the electron concentration n is given by

(10.3.18) Since fle/n c = m;;/m~ - R is the ratio of the hole and electron effective mass for a single subband occupation, we have

(10.3.19) which can be plotted as a function of

fl.

The transparency carrier density

g(hCD)/grn .

+1

r----------------

n? -

Increasing n _ _ _ _~iJ(f)

-1

LASER~

SEM [CONDUCTOR

43C

I-

o ..... o

Figure 10.22. A plot of th e occupation proba bility of electrons in the conduction band f .:: = 1 - e -n /" and that o f electrons in the valence band f,- = e- Il / Il , - (the probability of holes in the valence band Iii == 1 - f). The interception point where Ie = t; gives the transparency carrier density tl w

<S §

0 .5

o

u,

L

6

8

occurs at gpO , (10.3.20) If R = 1, we find n tr = n:c In 2 . We can also plot t.c : : : 1 - e -n /ll e and Ie : : : e -n jll, vs. the carrier concentration n ; the intersection point gives the transparency density n to which corresponds to g p = 0, as shown in Fig. 10.22. The peak gain g p vs. the carrier density n is plotted in Fig. 10.23. The analytical formula also leads to an expression for the differential gain [62, 64]: •

(10.3.21) Very often, the g)n) vs. n curve is assumed to have a logarithmic shape as .'

(10.3.22)

E

~

1.5

0.

OJ)

c:: '8 eo ~

ro

o

0.. "0

o

.!:::J ...... c-;l

0.5

§ Figure 10.23. Pe a k g~lin coe fficie n t '!!'p normal ized by the ma ximum ga in g"l is plotted as a function of the carrier d e nsity n

o

Z

2

3 4 n/n c

5

6

7

1l,3

QUf\NTUM-V,'FLL lASERS

where gp = go at n = 11o, and gp = 0 at n = noc- J = n t r . This is an approximate formula which can be fitted to the curve in Fig. 10.23 in principle. vVe can also calculate the peak gain directly VS. the current density J, taking into account the nonracliative Auger recombination, the leakage current, and the radiative recombination. We first assume a carrier concentration n, then we calculate the peak gain and

J r ad

=

qL z R sp ( n)

(10.3.23a) (10.3 .23b)

JAug = qLzRAUg(n)

and the leakage current J le a k from the expressions in Section 10.1; then, J = J r a d + JAug + J 1e a k . Here the spontaneous emission rate per unit volume (s -lcm -3) Rsp(n) can be calculated using the formulation with the valenceband mixings discussed in Chapter 9 or simply using the simplified model [65, 66]:

(10.3.24a) Similarly,

(10.3.24b) which can also be calculated directly using the interaction Hamiltonians for Auger processes, i.e., carrier interactions via the Coulomb potential. The finite scattering linewidth can also be taken into account [67--71] by convolving the zero linewidth gain spectrum g(f, ,) in 00.3.14) with a Lorentzian function Li.tu»), i.e., g o<;w( tzw) =

f g( tzw')L(ii

LV

!ut}') d tuo'

-

( 10.3.25a)

where L(x) =

1'/17"

-,~-­

x~

+

(lO.3.25b)

1'2

The procedure will lead to a peak gain vs. the current density relation: g p'

-

u

o p

(J)

(10.3.26)

A common empirical Iormula is again a L)garithmic: rel ation [72~76l: ,. t' J) =

'J.

,-;' p I . : : : ' ()

(J

1-

L'j

h J .

(:

/

(10.3.27)

SEM1 CONDl iCTOR LA SERS

current density, th e a bove rel ation leads to

(10 .3. 28) which is also a logarithmic relation , yet the coefficien ts have to be different from those in 00 .3.22) . 10.3.6

Scaling Laws for Multiple-Quantum-Well (MQW) Lasers

For a quantum-well laser lasing from only the first quantized electron and hole subbands, we use the empirical logarithmic formula for the peak gain-current d ensity relation 00.3 .27) for the rest of th e discussions in this ch apter:

(10.3.29)

where I v.. and g w ar e the inj ect ed current density and the peak gain co efficient of a single- qua ntum-well (SQ W) str uctu re. Such a relation is plotted in F ig. 10.24a. The transparency cu rre nt density occurs at I .; = 1oe - 1. I

(a)

nw = 1

(b)

nW g'IN=n IN go[ln(nW ] W In W Jo)+1J

F igute lO.2.. t (a) Ga in ' s",) vs. in j ec t ion c u rre nt d en sity (J,,) relation fo r ::I Si:1f; k -q lla ilt U!\,·w el ! str uct u re: ( b g a in ( II II.g;,.) 'is . inj eci io n c u r ren t de nsi ty (n".J,) relat ion fo r a m ul rip rc-uu nn turn we ll s t ruct u re wi th nil' we lls.

nw = 1

/ n W J 'JI,'

For an NIQW structure with n I l ' quantum wells and a cavity length L, the required modal gain G th at threshold condition is (10.3.30) where ex is the internal optical loss, f w is the optical confinement factor per well, and R 1 and R 2 are the optical power reflection coefficients at both facets. We should note that gw is a "material gain" for the quantum-well region assuming that all of the electron-photon interactions occur in the well region and g w a 1/L z in the gain model. The optical confinement factor I'w for a single well takes into account the optical modal distribution and only that fraction of the photons inside the well provides gain. A simple approximation is, therefore [77], fw

=

( 10.3.31)

where ~Vmode is the "full width" at nearly half maximum of the intensity profile of the optical mode. For a separate-confinement structure as in Figs. 10.16a and b, we can calculate the optical confinement factor f o for the optical waveguide part, ignoring the well region first, then use (10.3.32) where 1'0 takes into account the refractive index difference of the large optical waveguide for optical confinement, and L::: / }Vmo d c accounts for the carrier confinement and stimulated emission. The modal gain for a single quantum wellshould be G = rJ,.\.g\.\.

(10.3.33) where

.~.

"2 tr 7'11 r -------., I e

":).'l,.I..'(/I-

(10.3.34 )

SEMICONDUCTI)R

434

LA~:ERS

which is also the maximum achievable gain for a single quantum well . For n w quantum wells, the modal gain is approximately I1 w f w g w ' The threshold current density I th for the MQ'VY structure is then I t ll =

n1vl w

(10.3.35)

17

where 17 is the internal quantum efficiency of the injection current. Therefore , we can also plot the material gain (nwg w ) vs. the injection current density (n wIw) using [72-78) (10.3.9):

tl wg w

=

tlwg+n (::~: 1+ 1]

(10.3.36)

assuming that the well-to-wel! coupling can be ignored for simplicity. This relation is plotted in Fig. 10.24b for n w = 1, 2, 3, and 4. For n w = 1, as shown in Fig. 10.~4a, the intersection with the horizontal axis is 1,\. = Ioe - 1 and gw = O. When J.... = 1 0 , we have gw = go' For n ; = 2, we firrd that 2g w = 0 occurs at (2I w ) = 2Ioe - 1; therefore, the horizontal intersection is shifted to the right by a factor of 2. At 2Iw = 21 o, we obtain the vertical axis (2g w ) = (2g o ), which is twice that for Il w = 1. Therefore, the gain-current density curve for n w = 2 starts at twice the transparency current density of that for J1 w = 1, and increases by a factor of two faster than that for n . . . = 1. We obtain the threshold current density by [72-76) substituting 00.3.29) into 00.3.35):

(10.3.37) If we take the logarithmic function of Ilh, we see ,

(10.3.38)

where we define an optimum cavity length parameter L opt

1 1

== -')

(1 1

ln. - - - \. fl,vf,,·ga ) R I Rz I

.

(10.3 .39)

with a dimension of length . Therefo re , in Jll: varies linearly with 1/ L. Such a dependence has been plotted for va rio us numbers of quantum wells as sh own in Fig. 10.25a. • •

10.3

QUA iTUM-WELL LASERS

(a)

1nJ th

435

-----,>1..

_...1..-

L (b)

I

th

Figure 10.25. (a) Theoretical curves for the logarithm of the threshold current density, InUt h ) using (10.3.38), vs, the inverse cavity length for lasers with 1, 2, and 3 Quantum wells. (b) A plot of the threshold current I t h VS. the cavity length L for n w = 1 using (l0.3.40).

> L

_---L-_----I.. _ _---L-_----'

For real device measurements, the threshold current I tb (= IthwL) measured, where w is the width of the stripe. We have

IS

.'

The above expression for I t h contains a prefactor increasing linearly with L, and an argument in the exponential function that decreases with L, resulting in a minimum of the threshold current occurring at L = L o p t as shown in Fig. 10.25b. The minimum threshold current, Itrr: ll1 , is then given Py

(10.3.41)

If we assume that the loss coefficient a and other parameters such as f w ' 1 o, '7, and go are independent of the number of wells, we find an optimum Dumber of wells, /'lopp such that f t h in 00.3.40) is minimized (31t h /8fl w = 0):

, r Cl' +

n onl

'"

:" -_..~._-t"

'" g U

lL

J --<.-

'" ,!.

L

(1

In ---D

R

\j \1

. n I \ 2 )

( 10.3.42)

... ;--:>•• _" .••••-•.

~-'"""7--->;.,~-.-

'v-,_.

-,

_ . '.

""".~--:--:-~,.....----.,...-.~.

_ ••.. "'.....

.-" ......... _ .....- . _ .. _""

"','

_R •

...,..,.

~------r::--.-."

...- ....... """';'

-~":".'.':'7'""""'..,--:>""'"'-

.'.,

''',' ,~.

~

~',""'.,

-,.,•.... -

-_.~ -~,

._- _. ,

'~10~~_'~'

""\/'<"'1"' ...r-"'-' ' . ,

SEt'.; lCONDUCTOR L/.SERS

4:;6 lc1foLm)

(a)

NEu <,

<1. ~

...c

~

1000

800 700 600

n·3 ."

0-= 0A

A

2~/

~ • .I'

300

~

E...

. ....

.s: ....

-;;;PA.:-8

500 -

t:

"0

200

GaA~17nmvNa.nGaQ:r3A5(5nml

""£ 400

8

500

200 0

10

20

30 40

50

60

-.L LnR-ltem l ) lc

(b)

50

StrIpe width -4}Om


E 4030-

~

20--

\

0--

n -3

~:::

:::J

U

IQ"'~.~·~ -o~

o o

I

I

I

I

I

I

100 200 3CO 400 500 600 700 800 900 Cavity Len~ th C)Jom)

Figure 10.26. Experimental results showing (a) the linear behavior of the In(Jlh) vs. the inverse cavity length IlL following 00.3.38) for lasers with 1, 2, and 3 quantum wells, and (b) the threshold current llh as a function of the cavity length L. A minimum value of the threshold current exists at an optimum cavity length L OP I based on 00.3.40) and 00.3.41). (After Ref. 73 ;D 1988 IEEE.)

10.3.7

Comparison With Experimental Data [73]

The above simple relations between the threshold current density J t ll and the cavity length L or the number of quantum wells file' have been demonstrated experimentally. In Fig. 10.26, we show the experimental data [73] for (a) broad area contact and (b) ridge waveguide GaAs / Alo.22Gao.nAs separateconfinement-heterostructure (SCH) quantum-well lasers. The undopecl GaAs

' ' '.. -:- ....

""'_

,

o

0

quantum wells are 70 A in width ane! the Al o.22Ga o. 7/'lAs barriers are 50 A wide. The linear relation between In(.I'h) and the inverse cavity length II L is clearly shown in Fig. 10.26a for three numbers of quantum wells, /lw = l , 2, and 3. The threshold current l[h is also shown to have a minimum at an optimum cavity length in Fig. 10.26b, in agreement with the analytical relations 00.3.40) and 00.3.41). The stripe widths are 4.0, 4.5, and 3.5 fLm for n.; = 1,2, and 3, respectively in Fig. 10.26b.

10.4

I

STRAINED QUANTUM-WELL LASERS

In recent years, strain effects in semiconductors have been proposed and explored intensively for optoelectronic device applications [79, 80]. The tunability of strained materials makes them attractive for designing semiconductor lasers, electrooptic modulators, and photodctectors operating at a desired wavelength. The concept of band structure engineering [17, 18, 81] is further explored in strained semiconductors beyond the previous applications using quantum wells by controlling the well width and the barrier height. Using strained material systems such as In1 __ xGaxAs/lnP, In1_xGaxAsl Inl_XGaxAsl_yPy, and Inl_xGaxAs/GaAs quantum wells, a wide tunable range of wavelength is possible. Furthermore, the band structure including the band gap, the effective masses, and therefore the density of states can be engineered. It has been proposed that by using a compressively strained quantum-well structure, semiconductor lasers with a lower threshold current density can be achieved because of the reduced; n-plane heavy-hole effective mass [17, 18, 81]. A reduced threshold carrier density also reduces the nonradiative Auger recombination, one of the limiting factors for long-wave length (1.55-fLrn) semiconductor lasers. Experimentally, high-performance strained quantum-well lasers [82, 83], such as low-threshold current density [19, 20, 84--89], high-power output [90-93], and high-temperature operation [94-97]' have been demonstrated with remarkable results. Theoretical models using both simplified parabolic band structures and realistic band structures of strained quantum wells have also been used to investigate the optical gains in these semiconductor lasers [98-105J. To understand the strained quantum-well systems, we will use the 111 1 _.lG~l .r As I InPmaterials as an illustration. The ternary In L _xGa xAs alloy has a lattice constant a(x), which is a linear interpolation of the lattice 1:1Sl.;:ll1 (S of GaAs and I nAs. u( Ga As), and «(In.As): t."(

u(

.r )

.c.,

_':n(G:iAsy +

(1 -.-

:r)a(InAs)

(10A.i)

,

SEMICONDUCTOR LASERS

438

o

The lattice constant of the InP substrate a o is 5.8688 A. When a thin In l-xGa xAs layer is grown on the InP substrate, at x = 0.468 = 0.47, the lattice constant of Ino.53Gao.47As is the same as that of InP. It is the lattice-matched condition. lfa(x) > a 0 (i.e., x < 0.47), an elastic strain occurs which will be compressive in the plane of the layer resulting in a tensile strain in the direction perpendicular to the interface. We call this case biaxial compression, or just compressive strain. Similarly, if a(x) < ao(InP) (i.e., x > 0.47), the strain in the plane of the layer is tensile and there is a compressive strain in the perpendicular direction. We usually call this case biaxial tension, or tensile strain. Theoretical work [106, 107] shows that beyond a critical layer thickness, large densities of misfit dislocations are formed to accommodate the strain. Therefore, the strained layer dimension is always kept below the critical thickness for optoelectronic material applications.

10.4.1 The Influence of the Effective Mass on Gain and Transparency Carrier Density A band structure calculation shows that the effective mass of the heavy-hole band along the parallel plane of a quantum well has a reduction in its effective mass. Therefore, the density of states of the heavy-hole subband is reduced. This helps to reduce the transparency carrier density required for population inversion, as shown in Fig. 10.27. The Bernard-Duraffourg population inversion condition [108] requires the separation of the quasi-Fermi levels Fe - Fe + E g > hco > E g + Eel - E lzl (= E;fl} For conventional semiconductors, the heavy-hole effective mass is always several times that of the electron in the conduction band. The quasi-Fermi level of the hole is usually above the valence band edge instead of being below the band edge, while the conduction band degenerates easily with electron population. Since a large separation of the quasi-Fermi levels F; - F; + E g > E;ff (the effective band gap of the quantum well) is required, an ideal situation will be a symmetric band structure configuration such that nlh = m;, as shown in Fig. 10.27b. The population inversion condition F, - F( + E s = E;ff can then be satisfied easily at a smaller carrier density, as shown. If we take the simplified model for the peak gain [62, 64] from the previous section, 'then

(10.4.2) where R =

mt/ m:

is the ratio of the heavy hole and the cond uction band

E

- m·e (b) m·h-

E

~

'.

E e r -- ---

m*e

Pe = -n1i 2 L

- --- --- -- Fe = Eel - - -

z

E g 1-----+----;u-Pe(E)fc(E)

f----~k

.....I<:""""'I'N.------- F v

I-r----+---__. Ph (E)fv(E)

= Eh 1- - - -

o

m*h

PhC=~L tin z

Populaion inversion in a quantum-well structure with (<1) rnj, > m: and (b)

Figure 10.27. mi,=m;.

electron effective mass. In Fig. lO.28a we plot fe = 1 -" e
L

•

t.

~40

SEMICONDUCTOR LASEf\.S (a)

'-

9u

C<:l <.;...:<

0.5

'§ c..>

u,

0

2

0

6

4

8

10

n/n c (b) ~

E

u -.... ...-. '---" 0..

00

Figure 10.28. (a) Occupation probabilities fc= 1 - e- II / llc and i. = e-n/(RII,l are plot-

00

~

ted vs. the normalized carrier concentration for two heavy-hole-to-electron effective mass ratios R (= n1'h/m;) = 5 and R=l. (b) Peak gain coefficient vs. the normalized carrier density n / n e for R = 5 and R = l.

«l OJ

0.-

II / lie

10.4.2

gm

I::

'C;

0

2

4

6

8

10

n/n c

Strain Effects on the Band-Edge Energies

For In 1-xGa x As quantum-well layers grown on InP substrate, we have the in-plane strain [109, 110J

do - a( x) E=E-tx=c y y -

(10.4.3)

where an is the lattice constant of the InP. For a compressive strain, (lex) > au, therefore, the in-plane strain E is negative. On the other hand, the in-plane strain E is positive for a tensile strain since a(x) < aQ' The strain in the perpendicular direction is rr .J

-·2-----E 1 - {T

(10.4.4)

where o is Poisson's ratio and C l l , e 12 arc the clastic stiffness constants. For most of the III - V compound semiconductors. (J =:: 1/3 or C l2 :::=: O.5C II'

::U

STRA1NFJ) QU. ,NTUH.-V/ELL L:'.SERS

The band gap of an unstrained Inl_'xGa,t As at 300 K is given by

Eg(x)

0.324 + D.7x + D.4x 2 (cV)

=

(1004.5)

For a strained In 1 _ xGa xAs layer, the conduction band edge is shifted by

BE,=a,(E

xx

+

Eyy+E

~2a,(1 ~ ~::)E

oo )

(10.4.6)

and the valence subbands are shifted by

QE

OE H H

=

-PE

P

=

- au ( E x x

E

-

+

Q E = - b2 (E .oCt +

OE L H e yy

+

e yy -

=

-PE

e z z) =

2 ez

J

-

+Q

E

2 a (,' 1 -

(

e12] e

C 11

l2

C = - b (1 + 2 C -.

]

e

( 10.4.7)

11

Therefore, we have the new band edges:

E;

=

E HH

=

Eg(x) + oEc 8E H H = -PE

E LH

=

8E LH

=

-

QE

-P + Q E

(1004.8)

E

The effective band gaps are

(10A.9a)

(l0.4.9b) where the hydrostatic deformation potential a (e V) has been defined: a =

Q

c .-

a L.

As can be seen from Table 10.1, we have Q c < 0 and a c > 0, using our definitions in this section. Therefore, we have the energy shift for compressive strain (with the notation for the trace of the strain matrix Tr(E) = F .r «. + f;~ v \: ."r-

t:

_~ .:

),

II (

b

C" )

+ 2 C' ;~

If>

0

i.e., the conduction band edg':,; is shifted upward by an amount DEc, and the

SEMICONDUCTOR LASE1<.S

442

Physica) Quantitles of a Strained Semiconductor [109,110]

Table 10.1

Physical Quantity In-plane strain e =

EXX

=

t: yy

=

ao-a(x)

Cp e = - 2 --c .L C II

Conduction-band deformation pot ential a c (e V) Valence-band deformation potential au (eV) a = a c - au (eV) Shear deformation potential b (e V) DEc = acCc xx + C yy + c zz ) PE = -a v(c .u+C yy+c z) b Qf; = - '2(s.O" + En - 2 82')

Compressive

Tension

Negative

Positive

Positive

Negative

Negative Positive Negative Negative Positive Negative Positive Negative Negative

Positive

8E H H = -PE - Q £ DE L H = < P, + Q E

Heavy-hole band-edge sh ift (e V) Light-hole band-edge shift ( e V) Effective band gaps (e V)

EC -HH. = E x(x) E C - LH = Eg(x)

+ DEc + P; + Q " + [5 E c + P/i - Q £

valence band edge is shifted downward by the magnitude of a u Tr(~), then split upward by b[l + 2(C 12 /C 11 )]£ for the heavy hole and downward by b[l + 2(C 12/ C ll)]e for the light hole, as shown in Fig. 10.29. For tensile strain, the in-plane strain e is positive since a(x) is smaller than the lattice constant of the substrate a o' Therefore, the direction. of the band edge shifts are opposite to those of the compressive strain. (b) No strain (a) Compressive strain . a(x) > a o a(x).= a o s, DEc = a c T feE) > 0

--------,l s,

t

(c) Tensile strain a(x) < a o

VL-?_E.....;;c_<_O

r

_

Ec

Eg(x)

LH > 0 -- - - -- ---- -- ------- -:;;::

E, .......---:....------L..I oE HH > 0 EHH

oE HH < 0

ELH ----;----- ------- oE U -l < C Figure 10.29. Condu ction an d val en ce band ed ges for se m ico n d u ctors under (3) a co m p re ss ive s tra in, (b ) no s tra in, a nd (c) a te nxil« s train . The fun cti on E/x) is the band gap of th e unstrained In l - rG n.yAs .illoy. Fu r examp le. E../ x ) C~ () .~ 2-+ -I- 0.7x + OAx 2e at 300 K. The band-edge shifts are S Ec = 2 (1 ("(1 - C 12 /C Il ) ~:' , 5 E H .I:l = -- Pc - Qc' (jE L H =
lOA ST:{i'.iNED

QU/\~\TU:v1 'IN

:LL LiSERS

(a) A quantum well under a compressive strain

E

C2 I

Cl

UEc(Ol •

.-z HHI HH2 LHI

(b) An unstrained quantum well

E

C2 Cl

LJ: (c) A quantum well under a tensile strain

E

C2

Cl

U-E-c(O-)

.-

t-----:::O""';::---I!I-k X.

LHI HHl HH2 Figure 1,0.30. Band-edge profiles in real space along the growth (z ) direction and the Quantized subband dispersions in k space along the k ; direction (perpendicular to the growth direction) for a quantum well with (a) a compressive strain, (b) no strain, and (c) a tensile strain.

10.4.3

Band Structure of a Strained Quantum ","'eU

For a quantum-well structure such as an In I _.~ Ga x As layer sandwiched between InP barriers, the band structures are shown in Fig. 10.30 for (a) a compressive strain (x < 0.468), (b) no strain (x = 0.468), and (c) a tensile strain (x > 0.468). The left-hand side shows the quantum-well band structures in real space vs. position along the growth (z ) direction. The right-hand side shows the quantized subband dispersions in momentum space along the parallel (k) direction in the plane of the layer. These dispersion curves show the modification of the effective masses or the densities of states due to both

SEMiCONDucrOR LASERS

444

the quantization and strain effects. More precise valence band structures can be found in Section 4.9. Example: Gain Spectrum of a Strained Quantum Well Using the formulation for the gain spectrum in a strained quantum well, we plot in Fig. 10.31 the gain spectrum for light with a TE (x or 9) or TM (i) polarization for the three cases in Fig. 10.30. We consider an In1_xGaxAs quantum-well system with In l_xGa xAs l-yPy barriers (band-gap wavelength Au = 1.3 j.Lm) latticematched to InP substrate . Electronic and optical properties of similar quantum-well structures were shown in Figs. 4.22a-d, Fig. 9.16, and Fig. 9.17. As shown in Fig. 10.31a for compressive strain (quantum-well gallium mole fraction x = OA1, well width = 45 A) and Fig. lO.31b for the lattice-matched o case (x = OA7, L:; = 60 A), the gain of the TE polarization is always larger than that of the TM polarization, because the top valence subband for these two cases is always heavy hole in nature and the optical matrix element for Cl- HH 1 transition prefers TE polarization. The injected holes populate mostly the ground (HH1) subband . On the other hand , Fig. lO.31c shows-that the TM polarization ma y be dominant for the tensile strain case (x = 0.53, o and L ;; = 15 A). The well widths are chosen such that all the lowest band-edge transition wavelengths are close to 1.55 I-Lm.o The gains were calculated for the sam e total width L y = L z + L b = 200 A in a unit period, where L; is the barrier width. The same surface carrier concentration 11 5 = ni. , = 3 X 10l ~/cm2 is used in all three structures. •

If we recall the parabolic band model (k, = 0) momentum-matrix elements are

lTI

Chapter 9, the band-edge

.-

TE polarization

IX"

• IM cr- .Li l: I~

= I"y' M

c r-h h

12

=

~_,jY. ~lb~

(10A.10a) (10A.I0b)

TM polarization . (10A .10c)

(lOA .10d )

wh ere i\1 u is th e optical mom entum matrix element of the bulk semiconductor, which is related to th e energy param eter E p for the matrix element by /'viZ = m oE[' / 6, and Eo is usually tabul ated in databooks (see Appendix K). .. In practice, the matrix e Ierne n ts depend on the transverse wave vector k , and arc very close to the above va lues at the bund edge where k , = 0, then vary

.

44 5

700

r-'~--r-...,......,......~..---.. ~-.~.~~.....--,---.---r-~~

Compressive

600 500 4 00 300 200

TM

100

o -100 750

."

"

»> .... ,

-~-----

800

850

,,

,

900

950

Photon energy (e V) (a)

700 ,---..

500

.......

c::

400

u

300

-

Unstrained

600

E u ......

TE

<;:»

.-;.;:: Q.)


o

u

.-o c::

C';l

I

200

I

100

o - 100 750

- ....

I

----

-""

.,

,'TM I

800

850

900

9 50

Photon energy (e V) (b)

700 ,---..

E

u

Tensile

600

," ,

500

' --"

.......

400

u

300

cQ.)

;.;:: Q.) o u c .~

o

" ,,, ,

I

200 100

o -100 750

,,

II

',TM \

/

\ \ \

\

'\

I

\

\

\ \

800

850

900

950

Photon energy (e V) Figure 10.31. Gain spectra fo r both TE a nd TM po lariza tion s o f (a) a co m pressive s tra in (x = 0 .41. we ll wid th = 45 A), (b) a u un str airi e d ( l = 0.47, L ~ = 6i)A), a nd (c) a tensile strain Cr = 0 .53. a nd L ~ = 11 5 /\) l !ll . . \lj aJ\s /lnl _ J·i a ~ A sl .r f\ ( ba rri e r bandgap wa vel e ngth A,., = 1.3 u m ) qu a nturu-wcli la se r. T he se ma te ri a l ,pi n coetlicie nt s we re cal cul at ed for the sa me period L r = 200 A a nd s u r fa ce cnr r ic r co nc e ntra tion 11., = n l., , = :'0 x 10 t Zi cm '2 in all thre e s t r u c t u re s.

SEMI CONDl'CTOl~

445

LASERS

-,

'\" . . . _-"'* -- - - -ll_T_~ _----

----

a

o

o

oe-_

--<,

o

0.02

0.04 0.06 k t (A)

--.... -TE

0.08

0.1 2

Figure 10.32. Normalized momentum matrix e le me n t 2IMn m(k c)1 vector using (9 .7.15) and (9.7.16) for co m p ressive (dots), unstrained (solid triangles) strained quantum-well lasers. Here M"11I refers to valence subband (HHI fur co m p res sive and unstrained and LHl for

/Ml vs . the transverse k , (solid squ ares) , and tensile n = Cl and m is the top tensile case ).

as k t increases, as shown in Fig. 10.32, using the valence-band mixing model in Section 9.7 and ignoring the spin-orbit split-off band. Laser actions with the above polarization characteristics, TE for compressive strain and latticematched Quantum-well lasers and TM for tensile strain quantum-well lasers, have been reported [111J. These phenomena also agree with those in conventional externally stressed or thermally stressed diode lasers [112, ] 13]. In Fig. 10.33, we plot the optical modal gain vs. the surface carrier concentration lls = nL :!, where L, is the well width for three cases: (a) compressive, (b) lattice-matched, and (c) tensile strain. For long-wavelength semiconductors, the well width L z has to be adjusted such that lasing action at 1.55 ,urn can be achieved . The design usually requires L, (compressive) < L.- (lattice-matched) < L. (tensile). As discussed in Section 10.3, it is the modal gain fg that is important in determining the threshold condition, and ~

(10A.11)

Figure If).33. Modal gain coefficient vs. th e s u rface carrier co n ce n t ra tio n for compre ssive strain (dots), unstrained (squares). and ten sile str a in ( t r in ng ic s) In1_ xGaxAs / 1111 _ rG a xA ~~ 1_ )·P,. quan tum-well la sers with th e band -edge trun sition wave le ng ths near L.55 ,urn . Th e so lid curv es are TE polarization and the d ash ed cu rve is TM polariza tion .

Compressive ,

Unstrained

1

2'

3

N s (x 10 1 2 fern 2 )

4

llJ,<1

STiZ/\lNEl QUANTUrv! ·WEl'~ jj'SFRS

4c!-7

where L, will cancel with the factor OIL z ) in g. Therefore, it is more useful to compare the modal gain fg for three different strains if the well width L, varies. We see that the compressive strain quantum-well laser has the smallest transparency carrier density at which g = O. However, it also saturates faster than the other two cases. On the other hand, the tensile strain case (TM polarization) has a larger transparency current density, yet it increases faster, implying a larger differential gain. We can also plot the radiative current density

(10.4.12)

where the well width factor L z is canceled, since rSpon(hw) a. 1/L z from (9.7.18), and lrain) is a two-dimension current density with a dimension (Ay'cm"). A common approximation is

R sp (n)

where n s

=

nls , and B'

=

=

Bn 2

=

B'n s2

( 10.4.13)

B/L;. Therefore (10.4.14)

with B, = B/ L=. A plot of lrad vs. lls for three different strain conditions is shown in Fig. 10.34 for comparison using 00.4.12) with the full valence-band mixing model. We see that the quadratic dependence 00.4.14) is a very good approximation.

120

...

~

"

s::

90

u

< '--' "'C

60

<:':l

1..0

100-0:

30

or

0.0

0.9

l.G

_..

~j

N s (x .10 1. I c rn~) 1

'1

3.6

Figure 10.34. Radiative current density Jc~J (,\/c01 2) VS. the surface carrier concentration i-';._ (J /cm 2 ) for compressive strain (dots), unstrained (squares), and tensile strain (tr iangles) quanturu-well lasers.

Se:MJCONDUCTOR LASERS

4...J.8

Tensile

Figure 10.35. Modal gain G = rg vs. the radiative current density Ifact (A/cm 2 ) for compressive strain (dots), unstrairied (squares), and tensile strain (triangles) quantum-well lasers. The solid curves are TE polarization and the dashed curve is TM polarization.

10.4.4

30

60 90 J rad (A fern].)

120

G-J Relation

The peak optical gain can also be plotted vs. the radiative current density J r ad directly for three strains, as shown in Fig. 10.35. We see that the compressive strain case has the lowest transparency current density and its gain saturates faster as the current density is increased. In real devices, the injected current density has other losses, such as nonradiative Auger recombination and intervalence band absorptions, which have to be added to the horizontal axis to obtain the true G-J relation. If these loss current densities are less sensitive to strains, we would expect the threshold gain G l h = r gIll to determine the threshold current density by the intersection of the G-J curve with a horizontal line G = G th . We see that for a small G t h the compressive strain laser will have the smallest threshold current density. On the other hand, if G th is large, the tensile strain laser may have the smallest threshold current density. These G-J curves provide good design rules for strained quantum-well lasers. An empirical G-J relation using the logarithmic dependence

(10.4 .15)

as discussed in Section 10.3 has also been used to study strained quantum-well lasers with very good agreement with experimental data [75, 76, 114]. The major equations are the same as those discussed in Section 10.3 and are not repeated here. The same analysis using 00.3.35)-00.3.42) has been applied to study strained quantum-well lasers. Advanced issues such as Auger recombination rates [l1S-U8], temperature sensitivity [119], and high-speed carrier transport and capture in strained quantum wells [120, 121] are II nder intensive investigation. Many-body effects [122, 123] due to carrier-carrier Coulomb interactions on the gain spectrum have been investigated. Many of these require more theoretical and experimental work to fully understand the physics of strained quantum-well lasers.

449

10.5

COUPLED LASER ARRAYS

Coherent high-power semiconductor laser arrays [124 -137] have been of considerable interest recently. Shown in Fig. 10.36 are two examples of gain-guided and buried-ridge semiconductor laser arrays [125 , 130]. The concept of coupled laser arrays is very similar to that of the antenna arrays. Since the amount of power of a single element semiconductor laser is limited because of its small size in the radiation aperture (facet), a coupled array is expected to deliver a higher power with a narrower radiation pattern (i.e., a

x

(a)

Metal contact Insulating proton implant

P~GaAs Contact P-AlxGal_xAs cladding

GaAs active la e

z

Y

N

-AlxGa] -x As cladding n-GaAs SUbstrate

(b)

Metal COntact P-InGaAsp P-InP .: InGaAsP (

. active layer)

N-InP

_ _ _ _ _ _ I~

+

n - InP sUbstrate

Figure ll.l.J 6. (
SEMICONDUCTOR LASERS

450

better directivity) if the elements are properly excited. The challenging research issues in laser arrays are how to excite the desired (fundamental) mode without wasting optical powers to different sidelobes. In pririciple, for a given semiconductor laser array, the optical modes should be calculated [134J by solving Maxwell 's equations for a given gain and index profile, which is determined by the injection and coupling conditions. Here we present a supermode analysis for its simplicity and ease of understanding the operation of laser arrays. The supermode analysis for a coupled laser array follows closely the coupled-mode theory [126]. The optical electric field can be written as a linear combination of the field of each element, N

E(x, y, z)

=

L

an(z)E(n\x, y)

(10.5.1 )

n=l

where E(n)(x, y) refers to the optical electric field in the 11 th element before the coupling effect is taken into account. The coupling effort is included in the coupling coefficient K assuming only nearby coupling. 10.5.1

Solution of the Coupled-Mode Equations

The coupled-mode equations for N identical equally spaced waveguides, assuming only adjacent waveguide coupling with a coupling coefficient K, can be written as .

and (3 is the propagation constant of the; individual waveguide mode. The general solutions for the normalized eigenvectors A(!)( z ) and their corresponding eigenvalues {3 f for an arbitraryV are

[ = J,2, ... ,N

(10.5.4)

10.5 CO UPLEJ! .A SE~ AR fj ·.. ··/ S .

451

where

f3

=

f3

f 'IT -I- 2 K cos ( -

f

aU)

N

\

=

II

+-

1

R; (

n e ,TT" N+-l

sin

N+-1

(10 .5.5)

1

J

n

=

1,2, ... ,N

(10.:5 .6)

The order of f3 t is chosen such that f3 1 > f3 2 > f3 3 . , . > (3 N- To prove this, we start with the general form 00.5.4) and substitute it into the differential

Eq. 00.5 .2): {3

K

K

f3

0

K

K

al

a1

az

({ z

u3

a

f3

K

K

f3

= f3 f

( 10.5 .7a)

Cl 3

aN

aN

The above eigeriequation can be put in the form ( f3 - f3 t ) j K

1

1

(f3 - f3 t ) j K 1

1

at

0

az

0

0

( f3 - f3 t ) j K

0

-

a3

1 0

1

( f3 - f3 t) /K

0

aN

( 10.5.7b) We us e the property o f the .N X N determinant, D N( 2 cos 8) , which is a polynomial o f or de r N with an argument 2 cos f:): 2 cos fJ 1

D N(2 cos fJ)

=

1 2 cos

e

0

1

-

det 1

0

2 cos 1

f:)

1 2 cos

+- 1) e] sin e

sin [( N

e (10.5.8)

whic h o beys the recu rsion re lati on

whe re x = cos 0 . This r ecu rsi on relu ti o o ca n be. proved easily usin g the

452

definition of the N X N determinant and expanding it in terms of the two elements in the first row (or column). The above recursion relation (l0.5.9) is satisfied by the Nth-order Chebyshev polynomial [138] U; (cos 8), or sine N + 1)61 DN(2cos 8) = - - - - sin 8

(10.5.10)

We define 2 cos 8

=

{3-{3e K

or

{3 f'

(3 - 2 K cos 8

=

(10 .5.11)

The eigenvalue equation can have nontrivial solutions only if the determinant IS zero: sine N + 1)8

=0

sin 8

(10.5.12)

Therefore,

(N + 1)8

m7T

=

m

t, t =

We can choose m = N + 1 -

8f =

t N+17T.

{3 t =

f3 + 2 K cos 8 r ,

=

1,2,3, ... ,N

(10.5.13)

1,2, ... , N, such that

8=(TT-(Jt)

t=

1, 2, 3, ... , N

(10.5.14)

and f3 1, {32' {33, ... , {3 N are in descending order for K > O. The eigenvectors can be obtained by substituting (3 f into OO.5.7b): - 2 a 1 cos 8t

a IT _ {

I -

aN _ I -

+ a2

=

0

2 a IT cos 8f +_ a n + 2 a N cos 61! - 0

n=2 ,3, ... ,N-1 (10.5 .15)

0

I =

The above difference equation can be solved assuming the solution a IT of the form a /I

=

(IT

Therefore , 1 - - .2 cos r

f) f

+.

r =

or r = expt + j 61 /). The general solution for

0

0"

( LO.5 .16) IS

.

therefore exp( + in8 p) or

1\).::

COl) Pj .E D

A. <: 2.. ~

t. •

» F RA Y S

453

exp( - in8 (' ), or their linear combinations:

(10.5.17) Since a o = 0, we find A

=

O. Therefore, an

B sin nBt

=

The amplitude B can be found from the normalization condition: N

L n=

lanl 2

=

(10.5.18)

1

]

Since N

N

L

sin ' net

1 - cos 2nB{

L

=

2

n = ]

n=l

N

- -

2

N - -

2

.-

>e(£

-

e'2u8,

n=l

1

cos[(N + l)ee]sin(Net')

--

2 sin 8r

N+1

(10.5.19)

2

where er = t 7T / ( N + 1) has been used, we find the normalization constant B = fi / i N + 1 . \Ve conclude that the N eigenvectors with corresponding eigenvalues {3f are described by (10.5.4)-00.5.6). Example

1

As an example, for N

1/6l 13 /2 1

13 /3 1/3 / 2

f'

1/6

-

5, we have the following five eigenvectors:

=

1 0 -1 -1

1

'/3

a -1

M A, r Y.) r

1 '2

0

1

with corresponding eigenvalues, /3 /

f3 +

1

1

1

f3

-+- K,

t

-1 0

1 -1

1/6 1

'/3

-13 /

2

1/3

(10.5.20)

-/3 /2 1/6

= 1,2, ... ,5): {3,

(3 _.- K,

f3

··· / 3 K

Plots of the near-field patter ns usin g (t\L.5.I) assumin g that each elemen t E(i,(x, y) is that of a sing le wa veguide are shown in Fig. l(J.37.

:)UyIJO'NDUCTOR

---+---'''''''-l

i I

LhS~RS

f = 1

I

!

i ···--I---r--···· ;

i

.f.=3

f=4

f=5

#1 Figure 10.37.

#2 #3 #4 #5 Center of waveguide

Plots of the superrnodes using 00.5.1) and 00.5.20) for N

= 5.

As we can see in Fig. 10.37, the highest-order mode t = 5 has alternating amplitudes with zero crossings between the elements. This out-of-phase mode tends to dominate the lasing behavior of the laser ar-nys because the overlap of the optical field with the gain profile is maximized for this mode. When the optical field goes to zero between adjacent elements, the mode experiences a minimum of optical absorption between the emitters. Therefore.rthe laser threshold gain is minimized for the out-of-phase mode and it will lase first with enough injection current. Lasing of this out-of-phase mode is undesirable since it gives double sidelobes in the radiation pattern. Various schemes have been used to improve the laser array performance [133, 134]. 10.5.2

Far-Field Radiation Pattern

To calculate the far-field radiation pattern, recall from Chapter 5 that the optical electric field under far-field approximation is given by the Fourier transform of the aperture field, - ik e ik r

E( r)

-----p -1-1Tr

'x"

"

,.. Jr t.1x'L!y'e- .,." ·!·' wl 1

J •

s(r')

where the equivalent magnetic surface current density

IS

(10.5.21)

related to the

aperture field by

IYI s

2/1 X E A ( T')

=

-

-

- 2£ X EA(r')

(10.5 .22)

If EA(r') = yEA(x', y') for TE modes in the waveguides, we have

M, f X

=

x=

2xEA sin 8 sin cf> ( - £ )

+ cos 8 Y

(10.5.23 )

Considering the observation point in the y-z plane, (cf> =

'7T'

/2), we have

E(8) a (-sin8£ + COseY)!!dxdye-ik.rEA(r) The aperture field is given by 00.5.1), noting that E(n)(r) where E (l) is the optical aperture field of the first clement:

( 10.5.24)

E (1)(r - nd),

=

( 10.5.25) n

Therefore, the far-field radiation pattern is N

IE( 8)1

~ nY;:l a~,t) e-;k -nd

2

2

If

2

dre -lk -'£(I)(r)

~--~--~-

Using k . d = kd sin be obtained from

~-

Unit patten,

Array pattern

I-

---

(10.5.26)

e, the array pattern of the t th supermode, A tun,

can

2

N '"' a( f)e-inkdsinfJ

L..J

Il

1/=1

N

1

2( N + 1) 1

2(N +

N

L n

s»

n

=]

sin [( N/ 2) (ef

- -

1

, t'

n

( --1) --

.

L

e ifl(fJr-k d sinO ) -

--

2

e ~' in(el+kd s in Ii ) ,

kd sin

sini( 81.' - kd sin e)

sin [( h '/ 2 )( f) t + kd sin -- -- sin·~ ( 0 t + kd sin

e)]

-

.

o)J f

(/i-I

(

J1

II. 5 , ')"7 _I )

SFMJCONL'UCTOR lA.SERS

456

(a)

o 0 0 0 0 0 0 0 0 o·

-0 0 0 0 0 0 0 0 0 O'

(b)

•

l'\~uj"e liJ •.:"'!,~L

ra) "!'!:,:::, !·n:!~.:.r,);.,l ..!~· {'f ~llr' to :::i:~~~Ir .

L~n-~i~n~(lli. 'l!"C:.l'/.

th) 1'1-:.::

>.

:')r":,

.~i'! It.::) f;;'

~:"". !1~·ii ;:\1:2 Llr-f"i·:::,~'~ 0:~~:"'i)-~ ',',J:

.r :::;

J~~, ~,

... , tt) nt' ~I

tho,,' t .-: :·:~·'·~V nlc<1·-.:.'; \:,·\rt,,~;, r~c':'

457

In Fig. 10.383 the magnitudes of the 10 eigenvectors A( t)( z ) for t = ],2, ... ,10 of a ten-clement array are shown (from Ref. 126). Each mode is labeled by t. For each cigenmode t, the magnitudes a~f), n = 1,2, ... ,10 are represented by vertical bars. The corresponding far-field patterns are shown in Fig. 10.38b. We can see that for the fundamental (t = 1) and the highest (t = 10) order modes, the amplitudes are not uniform throughout the elements. To improve the uniformity of the near-field profile, the two outermost waveguides are chosen [128] to have a smaller spacing to increase the coupling coefficient by a Ii factor, with all the other guides equally spaced. The coupling matrix becomes

M=

f3

IiK

0

0

IiK o

f3

K

0

K

f3

K

0

o

K

o

o (10.5.28)

f3

K

K

f3

o fiK

fiK f3

The lowest and the highest supermodes have a uniform near-field intensity envelope and the injected charges will be utilized more efficiently. These supermodes should be relatively stable with increased pumping above threshold. Another design is to use a Y-junction waveguide [129] str .ture such that all waveguide modes will lase in phase with equal amplitudes. Using closely spaced antiguided waveguides [133, 134], a new class of phase-locked diode laser arrays has been developed. Fundamental array-mode operation in a diffraction-limit-beam pattern is obtained up to 200 rnW. The out-of-phase mode is also shown to lase with 110 mW per uncoated facet. The threshold currents of these antiguided arrays have been shown to be in the 270- to 320-mA range with an external quantum efficiency near 30 to 35% and cw operation up to 200 m W, As pointed out before, the real optical modal profiles in the laser arrays using gain-guided [132] or antigu ided [134-136] structures have to be calculated properly to compare with experimental data. The supermode analysis is presented in this section only to provide some basic understanding of the laser array characteristics.

10.6

DISTRIBUTED FEEDBACK LASERS [33,77, 139-141J

J n Se ction 8.6 we discussed distributed feedback (DFB) structures and studied the electric field (mel the rcflecr ion coefficien r. The reflection CUe i11cicnt depends 011 the coupling coctficicnr K and the de tuning 0 of the

.'

"P-"'-'''''''1'~,~"",----

__

,~~

_,

_~

.....

~

...-:"""......,__ .. .......... ~

--.-,.~

..

--;~._-

_ _ . __ •__ •.

~

., ••..__ ..........

~

~

..

~._,..,.,...~_, _~_.---,.~

..... -.- ." ......... ...,... ~

~

..

-

'·fF'

.~.-

- _ •• -

,~

•.•.....,.

~

. . • .- • ..,•. -

SETvIICOl'tDUCTOR L,A')ERS

propagation constant f3 from the Bragg wave number tTl/A, where t refers to the order of the grating mode used in the OFB process. We can see that the periodic structures provide an optical frequency selection property. In Fabry-Perot-type semiconductor lasers, there are many longitudinal modes within the gain spectrum, which is generally broad. Although there are some side-mode suppressions in cw operation of the semiconductor lasers above threshold, the powers of the side modes increase rapidly when the laser diode is pulsed at high bit rates. In optical communication systems with common chromatic dispersion in optical fibers, these undesirable side-mode power partitions limit the band width of the information transmission rate. Therefore, a mode selectivity provided by periodic structures such as distributed feedback lasers or distributed Bragg reflectors are attractive for applications for single longitudinal-mode operation of diode lasers. A schematic diagram of a distributed feedback (OFB) semiconductor laser is illustrated in Fig. 10.39, where a grating above the active region provides the optical distributed feedback coupling. Using the coupled-mode theory in Section 8.6, we summarize the results for the optical electric field in a distributed feedback structure:

(10.6.1) where Vex) = E~TEu)(X) is the mode pattern of the TEo mode. f3 0 = tTl/A is the Bragg wave vector for the t th order grating. The amplitudes A(z) and B( z ) of the forward and backward propagating waves satisfy the coupledmode equation (8.7.18);

[A(Zl]

d dz B(z)

.[D.f3 = 1

K ba

K ][A(Z)] ab

-D.f3

B(z)

( 10.6.2)

In general, K a b and K b ll are complex for a lossy or gain structure. Only for a lossless structure,

Kb a

=

-

K a*b

(10.6.3)

Define (10.6.4)

I p

Figure 10..39. A schematic dingrarn for a distributed feedback tDFBJ semiconductor laser, where a periodic grating :,[ructurc above the active region provides til.: optical distributed feedback process.

-----------i

N

--DFB structure Active region

., ... _- _ ...... • ..,.,' ~

10.6.1

Solutlnn of th e Coupled-Mode Equations in DF'B Structures

The general solution to the coupled-mode equations is expressed as the sum of the two eigenmodes:

[A(Z)]

[A+] . e B+

B( z)

+

iq z

[A-] eB-'

iq z

(10.6.5)

where B+ A+

-

K ba

tl{3 - q

for the e + iq z eigenmode (lO.6.6a)

~---

-K a b

tl{3

+

l]

and

B-

-A-- -

tl{3

+

K ba

q

. (~r th e e -

~-- - --

-K a b

tl{3 - q

iq z

eigenmode (lO.6.6b)

We can define the DFB "reflection" coefficient for the" +" eigenrnode (10.6 .7a) and the" -" eigenmode ( lO.6.7h) Write the general solution 00.6.5) as A( z) =A+ e lq ;: +

B( z)

rlll(q)B -e-iq .~

= fp(q)A+e i q ;:

+

B-e- iq =

(10.6.8)

The boundary conditions at two sharp ends of the DFB structure are 1. At z

=

0, ACO)

=

r 1 B(O), therefore (1 - r l rp ) A

2. At - = L .. B (' L)

= r, .

'-

-t

+ ( I'm

-

r 1 ) B _. = 0

(10.6.9)

A ( L). t h us

. (10.6 .1J)

In general, 1', and 1'2 are complex. For nontrivial solutions of A + and B-, we must have the deterrninantal equation satisfied: ( 1 - r 1 rp ) (J -- r 2 rIII )

+ (r 2 - rp ) ( r 11/

I'

1

)

e i 2q L = 0

(10.6 .11)

or

(10.6.12) Notice tha t the gain of the structure, in general, should be incorporated into 6.{3, K tl b , and K b a :

(10.6.13) where (j = {3TE u - Po is the de tuning parameter, G = fg is the net modal gain taking into account the optical confinement factor r and the net material gain g. If we further assume that the lossless (or no-gain) condition for the coupling coefficient

K ba

=

-K*ub

is approximately valid, then

(10.6.14) where K = K a b has been used, and the effect of gain is introduced through the term G /2 in the detuning D.p. The eigenequation 00.6.12) should be solved numerically, general. for the threshold gain g and the lasing mode spectrum (or detuning) (j zx: koll - {3o, since PTE ~ kon.

in

(J

I

10.6.2

Laser Oscillation Conditions

If there is no distributed feedback structure, K the eigene quation for the Fabry-Perot lasers,

=

0, rill

= rp =

0, we recover

(10 .6.15) where q ~ [j - i g / 2 . If there are no "harp bou nclaries at z = 0 and z = L such that r\ = 1"2 = 0, we have simply (10 .6 .16)

. Therefore, r)q) and l',)q) s im p ly c!I,: t

;J;;

re fle c tio n coefficients for the

461

forward and backward propagating waves. The above equation simplifies to . K b a = - K*a b ) ( assunllng 2

.'} IKI el~qL = 1 (q + b:.fJ)2

_ _~ _ _

q

=

(10.6.17)

i tl{3 tan ql:

(10.6.18)

The above result can also be obtained from the denominator of the reflection coefficient for the DFB structure (8.6 .37), b:.{3 sinh SL

+ is cosh SL

=

0

(10.6.19)

since the laser oscillation condition is satisfied without an external injected light. Noting that q = is, we obtain (10.6.18) immediately. To analyze the solutions for the oscillation condition, we write

b:.{3L = -iqL cot(qL)

( 10.6.20)

or

GL oL - i - 2

( GLl2 - -iV. ~ ~oL - iT

') {(bI: -

(KLr cot

GL\2

iT'

(KL)2 ( 10.6.21)

..-

The above equation has to be solved for (oL, GL) simultaneously for each given KL. In the high gain (weak coupling) limit, GL »KL, we have q e:::::: 6.f3; therefore, (10.6.17) becomes I

I

( 10.6.22) We compare 'th e magnitude and the phase of the above equation . We obtain [33]

(10.6 .23a) and (I ) (5 L

= I, f)!

-

--

\:2

11

+

. 'I " .

taIl '- J (

~~ ) G

(lO.6.23b)

Once the threshold modal gain G is obtained from 00.6.23a) for <'l given K

\

SEMICONDTJCT,)R LASERS

462

and 8, the threshold current density can be estimated from the model for the gain-current (G-J) relation for a given semiconductor laser structure as an active double-heterostructure or a quantum-well structure. Note that the reflection coefficient for a DBR structure with a finite length L is obtained from 00.6.8) (10.6.24) Noting that B(L)

=

0, we have B-= -rpA + e i2 q L , or

rA1 - e i2 q L )

reo) -

( 10.6.25a)

1 - r In r p e i2 q L j

K* sin qL

(10 .6.25b)

- iD.{3 sin qL + q cos qL -K* sinh SL

(10.6.25c)

6.{3 sinh SL + is cosh SL

for q = is in a passive lossless structure. The last expression OO.6.25c) is the same as that derived in (8.6.37). "Ve can see that, in general, I'(O) is a complex quantity,

f(O) = Ir ( O) l ei 4> 10.6.3

( 10.6.26)

Distributed Bragg Reftector (DBR) Semiconductor Laser

In Fig. 10.40 we show a distributed Bragg reflector (DBR) semiconductor laser. The frequency selectivity is provided by the reflection coefficients r 1

~

DBR re ion

I

PUmp region

CIl

\Ill

ca DBR

)10

re ion

~

p Active re ion

N I

•

--r-Ll~C',,z:,______4~~L,~ ~ -

I

:

[1

r~

I

z=o Figure WAD.

Z :::

L

Schematic diagram o fu d istributed Brag,g reflector (DBR) semiconductor laser.

JilJ,

C1STRIBUTED FSJ:'DB,\CK L!\S; RS .

.

and ' : at both ends, and the active region is the same as a Fabry-Perot edge emitting laser. As a matter of fact, a DBR laser is just a Fabry-Perot laser with a mirror reflectivity varying with the wavelength. The lasing threshold occurs at the wavelength for which the overalI reflectivity is maximum. The analysis is very similar to that of a Fabry-Perot laser except that r 1 and r 2 are complex numbers and are wavelength dependent. The lasting condition occurs at r 1 r 2 e i 2 {3 L

1

--

(10.6.27)

where r I is the reflection coefficient of the electric field looking into the left DBR reflector (with a length L 1 ) at z = 0, and t : is that looking into the right DBR reflector at z = L. The propagation constant is

f3

=

f3 0

1

-

-

2

(fg - a)

( 10.6.28)

where f3 0 is the propagation constant, I' is the optical confinement factor, and ex is the total loss in the guide and in the substrate: (10.6.29) Note that

(10.6.30) We obtain fg th = ex

+

2f3 o L +,,c/>l

1 In (

2L

+

¢2

=

1

)

( 10.6.31)

Zrn-tt

( 10.6.32)

R 1R 2

The reflection coefficient for a DBR region with a length L obtained in (lO.6.25c):

I

has been

(10.6.33) It should be noted that for an abrupt junction between the active region and the DBR region, the reflection coefficient ,'1 (and r'2) has to be multiplied by an effective coupling coefficient Co (with I( '0 \ ~ 1) to account for the imperfect power transmission across the juncti on. that is, r! -_.~ COr l , and the . threshold condition 00.6.31) becomes (139;

~EMICONDUCTOR

46'-i

fg th

=

a

+

1 --In

2L

1

~

RJ?'2 C O

LASERS

(10.6.34)

From the threshold gain 00.6.31) or 00.6.34), the threshold current density JIll is then obtainable from the gain-current model for a given diode laser structure. Stable single-mode operation of semiconductor lasers using DFB and DBR structures has. been achieved. The temperature stability of these structures has also been demonstrated. In a conventional Fabry-Perot semiconductor laser, the lasing wavelength as a function of temperature is dominated by the temperature coefficient of the peak gain wavelength dAp/dT, which is approximately 0.5 nm y degree for long-wavelength InGaAsP/ InP double-heterostructure lasers [139]. On the other hand, for DBR and DFB lasers, the lasing wavelength is dominated by the temperature coefficient of the refractive index, dn/dT, which is approximately 0.1 nrn y' degree. Therefore, a stable fixed-mode operation of a DBR or DFB laser over a wide range of temperature (> 100 degrees) has been achieved [139].

10.7

SlIRFACE-El\tIITfING LASERS

Recently, vertical-cavity surface-emitting lasers (VeSEL) [26-28, 142-147] have been developed with performances comparable to those of conventional Fabry-Perot edge-emitting diode lasers. A few schematic diagrams are shown in Fig. 10.41. The light output is orthogonal to the plane of the wafers. This change in the direction of laser emission drastically changes the design, fabrication scalability, and array configurability of semiconductor lasers. The concept of surface emission from a semiconductor laser dates back to 1965 when Melngailis reported [148] on a vertical cavity structure. Later different groups reported on the grating surface emission [149, 150] and 4SO-corner turning mirrors etched into the device [15;n. Major work on VeSELs was done [147, 152J after 1977. More recently many improvements with significant progress on the performances of the VeSELs have been demonstrated. Because of their short cavity lengths, the 'surfuce-ernitting lasers have inherent single-frequency operation. They have other advantages such as wafer scale probing and ease of fiber coupling. Therefore, VCSELs have potential applications for optical interconnects, optical communication, and optical processing . 10.7.1

Threshold Condition

The threshold analysis of a VeSEL can be modified from that of a DBR sernicond uctor J user wi th a complex propagation constant f3 = f3 r + i( a ['g) /2:

(10:7.1)

465

(n)

Ligh t output

, ."

Jrc.

( b)

(c)

Li ght output .....aI

'ftIJ "

.... ... ."

t

;

Insulator

Metal cont ac t

}

/

}n

p-type doped D BR mirror (Quan tum :VCl~ ~c tive reg Ion

~{

-type dope DBR m irro

~ Substrate

-

~

{ •

....

Light ou tpu t

i

F igu r e to .-H. (a) Sch em a t ic d ia g r am of a s urface -e mi t ting la s e r [1 4 7J. (h) Front nu d (c) bac k su rface-em it t ing la sers using etch ing th rou gh th e ac tive re g ion [ 142].

for the D BR mirro rs a t the to p a nd bot tom of t he vert ica l cavi ty. T he co ndi tio n is

I hre sho ld

( 10.7 .2)

using R [ = It) 2 a nd R 2 = I,. 2 ! 2. H ere the optica l co nfine men t fac to r T ; ~C' C O ll l1 tS fo r bot h th e lo ngit ud in ul nnd t r.insverse confin em ents

( 10 .7 .3 )

SEMICONDUCTOR LASERS

-:j66

where d is the active layer thickness, L is the cavity length, 'Y = 2 if the thin active layer is placed at the maximum of the standing wave, and 'Y = 1 for a thick active layer. The factor d / L accounts for the longitudinal optical confinement, and S is the transverse optical confinement factor. The symbol (t accounts for the optical absorptions inside and outside the active region, plus any diffraction loss due to the mode mismatches when the laser mode propagates to the reflecting mirrors. This diffraction toss depends on the size of the diameter of the active region and the locations of the mirrors.

10.7.2

Carrier Injection and Optical Profile

Depending on the fabrication processes and the structures of the surfaceemitting lasers, the current distribution, the carrier density profile, and the optical mode pattern vary. For example, as shown in Fig. 10.42a, the ion-implanted regions act as insulators and the injected current into the active layer will have a distribution as shown in Fig. l0.42b. The guidance of the surface-emitting laser in the transverse direction is the same as that for the gain-guided stripe geometry diode laser, except that here the injection profile has a circular cross section instead of a long stripe geometry, and the light propagation is along the vertical direction instead of the parallel

(a)

(b) J(r)

10

~",J

{} t",,,"

n-type DBR

s

I

S

Light output

(c)

(d)

nCr)

.:

I I --~---=-»f r-"·'~---+----t-

S

S

IE(r)12

r-::l:r-=---f-----1I----+---=--r

s

s

FigUH 10.-12. ( ~tl Sch e mat ic d; ~i :::r~!111 o f (\ ~~t in' ~ i1i ,J d surface-emitting laser 'with ion-plantation reg ions, (ll ) c urr e n t ~k nsi t y . (.:.:) "':;; ,T ic l den sit y. ,tro d (e1) fun d ame nta] o p tical mo de profile (153).

i~67

direction to the active layer. The carrier density profile (Fig. l0.42c) can be solved from the diffusion equation given the current density profile. The fundamental optical mode pattern is shown in Fig. l0.42d once the gain prof [e is obtained g(r) = g'[n(r) - n t r ]

(10.704a)

for bulk lasers, where g' is the differential gain and tltr is the transparency carrier density. For quantum-well lasers, one may use

ll( r ) ] g ( r) = go {In [ -----;;;:

+ 1}

(10.704b)

where go and no are constant parameters. Approximate analytic formulas can be found in Ref. 153. For the surface-emitting laser in Fig. 10.41 b, the device has been etched all the way through the active region to provide lateral carrier confinement. The top structure has a cylindrical semiconductor region with air as the outside region. The interface scattering and diffraction losses also provide a certain degree of mode selectivity and the fundamental (EH 11' HE lJ or TEM oo) mode of a rectangular or cylindrical waveguide is considered to be very important. In the following, we show the experimental results from Ref. 144. In Fig. 10.43, we show the near-field patterns of a I5-j1.m-square VeSEL with a structure (using a shallow implant range of about 1 u.tt: with a proton dosage of 3 X 1015 j cm 2 at 100 keV) at various current injection levels above tl. eshold. At a small injection current above threshold condition, the fundamental TEM on mode (called the EH oo mode in Chapter 7) lases with a fullwidth-half-maximum of 7.8 j1.m, as shown in Fig. 1043a, and its lasing wavelength is 9500.85 A, as shown in the emi~sion spectrum in Fig. 10.44a. As the current increases, the lasing mode changes to a TEM o 1 mode, shown o in Fig. 1O.43b, with a lasing wavelength 9~05.90o A, shown in Fig. 10.44b, and then shifts to a TEM 10 mode lasing at 9517.40 A, shown in Fig. 10.43c and in Fig. J0.44c over a small range of current. At an even highe r current level, both the TEM 00 and TEM II modes lase with the total near-field pattern shown in Fig. 10.43d, and their emission wavelengths are 9516.92 and 9523.80 o A, respectively, as shown in Fig. 10044d. The mode pattern in Fig. l0.43d can be spectrally resolved due to the wavelength difference of the two modes and is shown in Fig. 10.43e, where the TEM an and TEM II modes are clearly separated in the near-field image. The red shift of the lasing wavelength is caused by the junction heating as the injection current increases. The narrow linewidths of the lasing spectra also indicate that the laser emits a single transverse mode, since the wavelength separations between two udjaceni transverse modes are around J to '2 A for the gain-guided VC:SELs

SEMICONDUCTOR LA:;[ RS (a)

(c)

(b)

(d)

/---------1 10J.UT1

NEAR FIELD DISTANCE (e)

tt

TEM 1 1 TEM co o

->j 10A

r.-

WAVELENGTH

>

The cw near-field patterns of a 15-f.L1l1 square VeSEL .enuumg (a) a TEM()() t Iuudumerunl) mode with a full-widr h-chalf-maximum of 7.8 f.Lol at :1 small current above threshold. (b) a TEM III a nd (c) ~I TEt'v1 11l mode at higher currents. ami (d) both TEt'vl l ll • and TEM II modes at even h ighe r cu rre nts. The muck pattern ill (d) is spectl:~tll: resolved to show the near-field image in (e). where the TC:~/l"'1 and TEM I ! 1l100ieS are clearly separated due to their wavelength dif-rerc'nce:,.,lllw.n ill Fi,;. Ifl.-l--I. (After Ref. 1-1-1 C· Il)l)] IEEE.) Figur-e lO...D.

lC.7

SUI"\FACE-EMJT!"1"lG LA)T_HS

I

(a) TEM 00

J'

I

9500 (b) TIEM01'

-

( J)

rz

",

·r~.

'~.......... ~.

~

I 9505

ID

a::

<

>-

(c) TEM

..

/

.(/"

,

_."

-··-:/ ..r : ....

10

!:: (J)

z

UJ

r-

z

I

9517 (d) TEM 00 and TEM 11

Figure 10.4-4. The lasing Spectra at different current injection levels correspondir , to those in Figs. 10,43a-d. The lasing v.. ,'e-

'-

lengths are (a) 9500.85

I

(c) 95[7.40 A. In (d), the two lasing wavelengths for the TEM oo and TEM 1 1 modes

9520 ~1Ak•

are 9516.92 and 9523.80 A. respectively. (After Ref. 1~-J. ,[: 1991 IEEE.)

r,

WAVELENGTH (A)

10.7.3

A, (b) 9505.90 A, and

Junction Heating

It should be noted that the junction heating is important in surface-emitting lasers. A simplified model to account for the temperature rise due to the injected current from a circular disk above an infinite substrate is [145] PI I.

--

--

P"!Jl

4(T5

.-

( 10.7.5)

where PI I is the total input clccuic power (the cu rr e n to-voltage product), P l is the optical liuht output. (T is the thermal conductivity- of the serniconductor (= 0.45 W/cmoC fur C~lA')). and S is the disk radius, ~p

~

S;::!vllCONDCCTOR LASERS

4 70 Pout

"Tlext(l)

= slope

~

ClJ

~

o

tI

0..

..... ::l 8::l

Figure 10.45. A plot of the light output of a surface-emitting laser vs. the injection current for different temperatures. The slope at a particular current level I is the differential quantum efficiency, which can be negative at a high current level or a high temperature.

10.7.4

o

.:E

Increasing temperature

00

;J

L..--......w..------

~I

Injection current

Optical Output and Difl'erential Quantum Efficiency

The optical output power is

(10.7.6)

where Y] depends on two factors: (1) the injection efficiency accounting for the fraction of injected carriers contributing to the emission process (some of the carriers can recombine in the undoped confinement regions where the carriers do not interact with the optical field), and (2) the optical efficiency accounting for the fraction of generated photons that are transmitted out of the cavity. Note that 'the threshold current depends on the injection current as well as on the junction temperature Tj c l ' Models for the laser output taking into account the temperature heating have also been developed in Refs. 154-156. .' The differential quantum efficiency is then current-dependent:

(10.7.7)

'vYe see that Y]cx/l) can be negative if d fell jdl > 1. The light output Pout VS. the injection current I will have a negative 'Slope in this case, as shown in Fig. 10.45. Experimental data showing these r.e garive external quantum efficiency can he found in Refs. 145, J 46, and 154.

PROBLEMS

47l

PIlOBLEMS 10.1

Plot the energy band diagram for a P-Al 0.3 Ga o. 7As 1 i-GaAs / NAl O.3Ga O.7As double hetcrojunctiori under zero bias with the active GaAs layer thickness d = 1 u. m. Assume that N.4 = I X 10 18 ern -- 3 in the P region and N D = 1 x IOn; em - 3 in the N region. Use the depletion approximation.

10.2

Plot the energy band diagram for a P-InP1 i-InGaAsP1 N-InP double heterostructure under zero bias with the active layer d = 1 u. musing the depletion approximation. Assume that N A = 1 X 10 18 em - 3 in the P region and N D = 1 X 10 18 ern -3 in the N region. The active InGaAsP layer has a band gap energy 1.0 eV and is lattice-matched to InP. Assume that the band edge discontinuity ratio t:.Eclt:.Eu is

601 40. 10.3

A semiconductor laser operating at 1.55 J.Lm has an internal quantum efficiency 77i = 0.75, intrinsic absorption coefficient a i = 8 em - I, and the mirror reflectivity R = 0.3 on both facets. (a) Plot the inverse external quantum efficiency 1/77 e vs. the cavity

length L. (b) Assuming that the threshold current is 3.5 rnA at L = 500 ,um, plot the optical output power vs. the injection current I. Label the slope of the curve. (c) Repeat (b) for I t h = 4 rnA and L = 1000 I-~ m. 10.4

If the mirror reflectivities are R I and R'2 at the two ends of tl semiconductor laser facets, the threshold condition is given by

fg th

=

1 1 a , + ;--1n-"--L R]R Z

(a) Find the expressions for the optical output powers, PI and P 2 ,

from facets 1 and 2. (b) Plot PI YS. R I for 0.1 ~ R l S 1.0 and P".!. vs. R"2 for 0.1 :s; R 2 1.0. You may assume that (Xi = 10 ern -I and L = 500 ,urn. (c) Compare PI with P 2 if

10.5

s

R I = 0.3 and R 1 = 0,5.

(a) Derive the carrier density profile n(y) in 00.2.27) and show that

if we ignore the current spreading outside of the stripe region, n( y) is given by 00.2.28). (b) Plot lI(Y) using 00.2.28) and show that it agrees with the expe rimen tal data in Fig. 10.1 ··L 10.6

Describe the mode patterns ! F:.,') .r , Y )1'- US!.Jl~ (to.? i6) for guided semiconductor laser.

C!

gum-

SErvf:CO;-..JDLCIOR LASERS

'O?

10.7

(a) Explain the physical reasons why In llh of a quantum-well laser increases linearly with the inverse cavity length 1 j L as shown in Fig. 10.25a. What determines the slope of the line? (b) How do you find the cavity length L such that the threshold current lth is minimized? What are the factors determining this minimum threshold current? (c) Find typical nume rical values of some quantum-well lasers and replot Figs. 10.25a and b.

10.8

Plot the gain spectrum using 00.3.14) for a 100-A GaAsjAl o.3Ga ruAs quantum-well laser assuming a parabolic band model and an infinite barrier approximation to calculate the subband energies and wave functions. What is the maximum gain coefficient gm?

10.9

(a) Calculate

o

the parameters Ill' and Ill' for a 100-A GaAsj AJ O.3G3(J.7As quanrum-well laser at T = 300 K. Assume single subband occupation. Find the transparency carrier density n tr for part (a).

10.10

Discuss how the number of quantum wells can be optimized to minimize the threshold curren t density at a given threshold gain value using Fig. 10.24.

10.11

Calculate the conduction, heavy-hole and light-hole band-edge energies for Inl_xGaxAsjln052AJU.4NAs quantum-well structures where In O.5 2 Al o A S As is lattice-matched to the InP substrate, assuming the gallium mole fraction (a) x = 0.37, (b) x = 0.47, and (c) x = 0.57 in the well region. Use the physical parameters in Appendix K.

10.12

Discuss the advantages of compressively strained and tensile strained quantum-well lasers based on the effective mass, the density of states, and theoptieal momentum matrix elements.

10.13

Compare the normalized optical momentum matrix element in Fig. 10.32 with the analytical results in Table 9.1 in Section 9.5 usmg a parabolic band model. I

10.14

Using the coupled-mode equations in Section 10.5 assuming only nearby-element coupling, write the explicit eigenvalues and eigenvectors for (a) three coupled waveguides and (b) four coupled waveguides. Plot the modal profiles of the supermodes similar to those in Fig. 10.37.

10.15

Derive the array pattern 1'0," the /' t h supe rmode, A/Un, in (lO.S.:27).

1O.l6

Derive theoscillation c oridit io n ( I U.b.21) for a DFB laser. Discuss the solution for the phase iJL and the.' g<'lill GL.

10.17

Derive the lasing conditions for thl~: DBR semiconductor laser III Fig. 10.40.

RFFEI·.ENCES

47:

10.18

Compare the threshold condition for a surface-emitting laser with that for a DBR semiconductor laser. What are their differences and similarities'?

10.19

Explain why the optical output power vs. injection current (L-I) curve of a surface-emitting laser has a negative differential quantum efficiency at a high injection level.

10.20

Compare the current density, carrier concentration, and optical modal profiles of a gain-guided surface-emitting laser (Fig. 10.42) with those of the gain-guided stripe-geometry semiconductor laser in Section 10.2 (Fig. 10.12).

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seMICONDUCTOR LASERS

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78. l\iI. Yamada, K.

EIt!C!I"(I!?i: ·S.

T
O~it
arid M. \ dll1.tgi"fJI. "C,t!cLllalil)f) of la"ing gdin and threshold current in ("I/\s-/\IG, rnulti-qua ntum-we ll lasers." Truu.:

IEeE

S.

3d ed.. \\/lIc:y New York. 1900.

}/)/I. £63, 102-LO~ ( 1%5).

47~

SEfvlICONDUCTOR LASERS

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c.

87 . H . Temkin. N. K. Dutta, T. Tanbun-Ek, R. A. Logan, and A. M. Sergent, c , InGaAs / InP quantum well lasers with sub-mA threshold current," Appl. Phys. Lett. 57, 1610-1612 (990).

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»:

I

T. Tsang, M. C. WlI, T. Tanbun-Ek, R . A. Logan, S. N. G. Chu, and A. M. Sergent, "Low-threshold and high-power output J .5-,um InGaAsj InGnAsP separate confinement multiple quantum well laser grown by chemical beam epitaxy." Appl. Phys. Lett. 57,..-2065-2067 (990).

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;'''~

.............. ,...

'.~.

~

..., ....

,~.

';"

... ... '

,

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SEM[CONDUCTOR LASERS

48U

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u-;

Y. Zou, J. S. Osinski, P. Grodzinski, P. D. Dapkus, W. Rideout, W. F. Sharon, and F. D. Crawford, "Effect of Auger recombination and differential gain on the temperature sensitivity of 1.5-,um quantum well lasers," Appl. Phys. Lett. 62, 175-177 (1993).

J L6.

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117. M. C. Wang, K. Kash, C. E. Zah, R. Bhat, and S. L. Chuang, "Measurement of nonradiat ive Auger anel radiative recombination rates in strained-layer quantum-well systems," Appl. Phys, Lett. 62, 166-16b (1993). 118.

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124. D. R. Scifres, \V. Strider, and R. D. Burnham, "High-power coupled-multiplestripe phase-locked injection laser," Appl. Phys. Lett. 34, 259-261 (1979). _ 125. D. R. Scifres, R. D. Burnham, C. Lindstrom, W. Streifer, and T. L. Paoli, ., Phase-locked (GaA\)As laser-emitting 1.5-W cw per mirror," Appl. Phys, Lett. 42, 645-647 (1983). 126. J. K. Butler, D. E. Ackley, and D. Botez, "Coupled-mode analysis of phase-locked injection laser arrays," Appl. Phys, Lett. 44, 293-295, 1984; and "Erratum," 44, p. 935 (1984). 127. S. Wang, J. Z. Wilcox, M. Jansen, and J. J. Yang, "In-phase locking in diffraction-coupled phased-array diode lasers," Appl. Phys. Lett. 48, 1770-1772 (1986) .

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-~>-,p\n,

Eels ..

!!.,,'i,jf-UO'!(

I);' :Halheni(ltico! Functions,

SEMICONDUCTOR LASERS

4R2

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Pj\RT IV Modulation of Light

•

11 Direct Modulation of Semiconductor Lasers For a sem icond uctor laser, the output power of light intensity P increases linearly with the injection current above threshold, as discussed in Section 10.1, Eq . (10.1.31): h co

P

=

qTfi

In(l/R) cr.L + In l/R (1 - 1tlt )

Therefore, for an injection current with a dc component and a small signal ac modulation,

1=10+i(t) we expect the optical output power to have corresponding dc and ac components (Fig. 11.1):

P(t) =Po+p(t) .>

In this chapter, we study the direct current modulation of diode lasers and som e intrinsic effects, such as relaxation oscillations, modulation speed, and the laser linewidth theory. Our goal is to understand how the laser light o utpu t characteristics vary as /we increase th e modulation frequ ency of the injection current.

11.1

RATE EQUATIONS 'AND LINEAR GAIN ANALYSIS

Assume that only one mode is lasing; then the rate equations for the carrier d en sity N (l/cm 3 ) and the photon density S O/cm 3 ) can be written as [1-4]

clN

cit clS dt _

]

N

q,{

T

._- - -- ._- :·, ('u( =' IV) ..{' J

- .r · c

S

'i/ ( •.

tV)) .-- .--- + f3 R sp

>...

( 11.1.1) (11 .1.2)

T{..

487

D]F.E:::T MODULATJO)\; 01~ SE.,liCONDUCTOR LAsERS

~P(t) = Po + pet)

pet) ......


~

o

0..

--8;:::l

P 0..

--

;:::l

o

Time :;> t

i(t)

Figure 11.1. For an injection current J = 10 + i(t), the optical output power is Pi t ) = Po + pet), where (10' Po) is the bias point for the direct current modulation of the semiconductor laser.

where

]

=

q

=

the injection current density (A I ern 2 ) a unit charge 0.6 X 10- 19 Coulomb)

d

=

the thickness of the active region (ern)

T

=

carrier lifetime (s)

L'

= c r n , is the group velocity of light (cm y s) .'

g(N) = the gain coefficient (l/cm)

r Tp

=

optical confinement factor

= photon lifetime (s)

f3 = the spontaneous emission factor

R s p = the spontaneous emission rate per .unit volume (ern -3 s-1) In the first rate equation (I1.1.1) for the carrier density N, the first term J Iqd is the injected number of carriers perunit volume per second. The current density} is the current I divided by rhe cross-section area of carrier injection. The recombination rate N

R( N) -

( 11.1.3) T

11.1

Rid E EOUATrONS AND LINEAR GAIN AI'iAL'(SIS

accounts for the carrier loss due to radiative and nonradiative recombinations. The time constant T generally depends on N, but is assumed to be a constant for simplicity. The term Lg(N)S is the carrier loss due to stimulated emISSIOns. In the second rate equation 01.1.2) for photon density S, the first term rL!g(N)S, is the increasing rate of the number of photons per unit volume due to stimulated emissions. The second term, - S / T p ' is the decreasing rate of the photon density due to absorptions and transmission through the mirrors of the laser cavity,

~

= L'

(a. + ~ln~J R

TIL p

(11.1.4)

where T p has the physical meaning of the average photon lifetime in the cavity. Photons disappear from the cavity via absorption processes or trans~~mission out of the end facets. The last term, {3R sp , is the fraction of spontaneous emission entering the lasing mode, which is generally very small. The spontaneous emission factor {3 can be calculated with a simple analytical expression for index-guided laser modes with a plane phase front (5]. For a gain-guided laser with a cylindrical constant phase front, the spontaneous emission is shown to be enhanced by another factor [6, 7]. It is noted [3] that the optical confinement factor r appears only in 01.1.2). This is because the optical mode extends beyond the active region d by the factor r. The two rate equations ensure that the total carrier and photon numbers are balanced taking into account the difference in the thicknesses (d versus d / I ).

11.1.1

Linea r Gain Theory

I f we assume that the photon density S is not high so that the nonlinear gain saturation effect can be ignored, we have a simplified gain model:

•

(11.1.5 )

where go = g( No), and g' = ()g IaN is the differential gam at N = No. Assume that 1 ( t) = 11)

+ j ( t)

'V( r) = NI;

+

s(r )

+

=

Sll

Il (

S(

where ii.t ) .'ti! J, n nd sl.i) a re small Sigll;tl~ mg dC"\'tdu.;;', III' til)" :F;·j 5'1)' l--:":;fJcctivcly.

t)

t)

l:()mp:~!;::d V.,:i(tl

( 11.1.6) their correspond-

DIRr::cr klODULATICIN OF SEMICONDUCTOR LASERS

490

de Solutions. The photon density So and the carrier density No at steady state are f3R:;p

50 = - - - - - ( 1 /1"p)

~ (:~

No

T

-

(11.1.7)

- r cs 0

(11.1.8)

LgoS o )

Note that for negligible f3R sp , the inverse photon lifetime is approximately 1 -

= fL'go

(11.1.9)

'T p

ac Analysis. The small-signal equations are

jet)

d

-net) dt

d -set) dt

=

=

- '-

qd

net) -

-- -

u[g'Son(t) + gos(t)]

(11.1.10)

'T

set) fv[g'Son(t) + gos(t)] - - ~

(11.1.11)

For a current injection with a microwave modulation angular frequency

(s)

(11.1.12) where jew) is the (complex) phasor of jet), we find the solution by substitutmg

net) = Re[n(w)e- i w t ] set)

Re[s(w) e-i(dt]

=

(11.1.13)

into (11.1.10) and (11.1.11). \Ve find the complex amplitude new) for the carrier density

n( (v)

I,

r-

reg So \

i co --

r ig () -+-

~Jlp.). s( w) (11.1.14)

where (11.1.9) has been u-:..~d. and the cornplex vnagnitude s(UJ) for the

photon density

s«({»)

fvg 'So[j(w)/qd] D( (v)

=

(J 1.1.15)

where the frequency dependent denominator Dt co) is

~

D(w)

iW( ~

- w' -

+

vg'So) + "g'::

(11.1.16)

Its magnitude square is

(11.1.17) Since the above function depends on w 2 , if we let y 2 occurs at (J/Jy)ID(w)1 = 0, we find

w; =

vg So

1_

1. ( 1 + ug'5 0 2 T

-

-

Tp

-

)2

2

= w ,

the mmimum

(11.1.18)

where the last term is usually negligible compared with the first term and

(11.].19)

is used. The relaxation frequency

I,

I,.

= wr /

27T

271 is

( 11.1.20)

A

which is proportional to or the square root of the optical output power since it is linearly proportional to the photon density 5 n . The frequency response function is s ( (u ~

I = .r I'.!? ' So / Wi

.i( w) I

! D ( (v) ! ( 11.1.2 L)

;::O Ir~ E · ..T rv'lnD~JLAT;O:~

-1-92

OF

5;::~''lICONDUCTOR LASERS

which has a fiat respon se at low frequencies 00\\/ pass) and peaks at co = W I" then rolls otT as the fr equency increases further. The bandwidth or 3-dB frequency I3dB (= W J lll3 / 2 7T) occurs at ID(W 3 d S >1 = or

n'w;,

(",juB since ID( (u

=

0)1

=

r '7

(Un' + wjdB (~

+

TpW; ~ 2 w;

(11 .1.22)

w;.

Example The frequency response for a distributed feedback semiconductor laser [8] is shown in Fig. 11.2 for different optical output powers at 5, 10, 15, and 20 mW. The peak respon se occurs at the relaxation oscillation frequency j,. and decreases to - 3 dB at f3dB compared with the normalized respons e at low frequencies (0 d B). T he bandwidth increases as the optical output power PI) increases with a ,;Po dependence, since PI) is proportional to the photon density So, as shown in Fig. 11.3. The 3-dB_!requency I3dB YS. the square root of the output power are shown with the relaxation frequency fro Since the output power Po is proportional to I - I t h , the plot, II" YS. ,; [ - I t h also shows a linear relationship. Therefore, II" can also be plotted vs. ';1 - I t h for semiconductor lasers and a linear relationship is shown [9]. Improvements of the modulation bandwidth using quantum wells and quantum wires or strained quan tum wells have been discussed [4, 10, 11]. l:II

M

12

.-

CD "0

W (J)

z

6

0

Q... (J)

W 0::

0

-

>U

z

w => a w

a:: u,

I

-

-

-

-

I I

--

I

-6

, I

-12

fr '::t 'oJ

I

l

lf3dB

I

1

6

9

FREQUE~jC '{

~2

15

is

(GHz)

Figure 11.2. Smul l s ignal f r·::l!u .:nl.:~ r ::"p01! S<:.' of a ci i:,l ,·i hu kJ fCI.' db 'ld·; luse r u t d ifle rcnt o u tp u t po v.. ers . Tile s ub rno u ru lclilPC::' ,lili r\.· \\ ,l ~, 21)"C'. Th e p'-',lk re s p o nse de te rm in es the rel axa ti on (re qu e ncy t,
4'·:3

20 r------r----.----,...----,.----

16 3 d8 BANDWIDTH-f 3 d8 N

I

~

12

>u z W

::J

o

w

8

0:

---- RELAXATION OSCILLATION FREQUENCY (fr)

LJ,..

4

OL-

o

'--

-'-2

1

.J.-

- ' - -_ _- - - - '

3

4

5

Figure 11.3. The relaxation oscillation frequency and the 3-dB bandwidth of the distributed feedback laser in Fig. 11.2 are plotted as a function of square root of output power P (or Po in the text). (Arter Ref. ~U

11.2 HIGH-SPEED l\;10DULATION HESPONSE \YITH NONLINEAR GAIN SATURATION [12, 13] 11.2.1

Nonlinear Gain Saturation

The gain model g(N) can be taken from ( 11.2.1) where ,go = g(No), s' = (C>g IaN ),v=No IS the differential gam at NI). The factor '1 + ES accounts for nonlinear gain saturation, which is important when the photon density is high. The factor E is called the gain suppression coefficient. The steady-state solution at 1 = 10 is obtained from did t = 0 in the rate equations: JI)

NI)

qd )

8iJ'

-; (I

1' I' ------.-. 1. +1::,)'11

L"'Y ,:.1) S ()

+

-

._.~--

1

T

Sp ..

_-

eo .> \

I

Tp

~

T

l

:'l

( 11.2.2)

PSI' . j

~"I

(

')

1 1..... .J

,.-"

)

" .

."

.,.

~

~_~

"'";"1 '\

~

• • "'""t••

~'" .,

.. tr,;,,-r-::-:'- -

. ' f. ~~" ·~ _

I.JJ I~LC ; ' ~,·C(y)U :..i \T IO N

494

:1·'

,...,.. . ....

..

OF SE .:MCONDUCTOR LASERS

If e =F O~ the general solution of (11.2.3) is

~ r- (~"» - r

5o = 2

e

cg 0 - e (3 R Sf) )

and No is obtained from (11.2.2) using the above 50' Using the linearized expression by substituting (11.1.6) into (11.2.1), we obtain

g(N)

go 1 + £5

=

0

g'n(t) + 1 + e5

( 11.2.5) 0

The ac responses ni t') and s( t) satisfy the following equations: d dt

-net)

jet) net) -d - - - -

=

q

T

[U g '5 0 1

+ £So

1net)

ug o

-

(1

+ cS c.

)2 S

( t ) (11.2.6)

O

and

~s(t) dt

=

(

fug'So ]n(t) 1+E5 0

+

set)

..,s(t) (l+eSof fuga

which can be derived by keeping :the first-order terms of n( t (11.1.1) and (11.1.2) and using 01.2.5). 11.2.2

)

(11.2.7)

and st.t ) in

Sinusoidal Steady-State Solution of the Small-Signal Equations

Using d/dt

~

- iw in 01.2.7) and 01.2.3), we relate s to n

.

-lw(1 + £5 0 [

! +.

E5 0 f3 R sp ] --;; + ~ s( w)

.

=

(fLIg'5 o) n( w)

(11.2.8)

'rYe then use (11.2.6) and find 1

(

-r,

- i w + _.T

cg'5 o I)

+ - - --, tl ( w ) _. 1 + c') o ,

j( (u ) qr.!

vs;

-(---~S'/ s( w) (lJ .2.9) 1+ e

or

· i .2

N )NLINEAR (J /d"r SATUIU\TJON

4';5

Therefore, the small signal photon density function is

s( (1.»)

Define (11.2 .11 )

which is the same as 01.] .19). We obtain the frequency response function for semiconductor lasers 2

s( W) j( W)

(1].2.12)

where a damping factor 'Y is defined:

= rg'

So(1 +

= Kf} +

_ 8-

ug'rr;

J+

1 T

1

(11.2.13) T

Here a K factor (ns) K

=

47T

2

(T!~

-+

~-) rg '

(11.2.14)

I

and W r = 2 »I, have been used. The 3-d B cutoff frequency occurs ,H ( 11.2.15)

S~~MICC·NDUCTORLASERS

DIRECt tvIOijL)rj'\TIO>1 OF

The maximum possible bandwidth occurs when the following condition is satisfied:

(11.2.16) such that the frequcncy response function is a monotonic decreasing function, (11.2.12) a W;/((tJ4 + w;). We then have the maximum relaxation frequency by solving 01.2.16) for I,.: 27Tfi j~,lllax =

(11.2.17)

K

By fitting the frequency response function 01.2.12) to the experimental data, the damping factor y can be found together with the relaxation frequency L. Equation (11.2.13), y = Kj} + 1/ 'T, shows that a linear relation with a slope K holds if the damping factor 'Y is plotted vs. j} and the intercept with the vertical axis gives the inverse carrier lifetime 1/ 'T . Example Figure 11.4 shows the experimental results [14] for two strained quantum-well lasers with (1) a slope K = 0.22 ns for a tensile strain laser with four quantum wells, and a maximum bandwidth j~. max = 27Tfi / K = 40 GHz, and (2) a slope with K = 0.58 ns for a compressive strain laser with four quantum wells and fro max = 15 GHz. Both lines intercept the vertical axis at 1/ 'T ::::::: 5 GHz, and the carrier lifetime Tat threshold is 0.2 ns for both 40

,--------~---------~-~----.

t

--. N

I

30

<.9 '-'" ;=-

20 • 4 MOW tensile strain K = 0.22, l ma x ::; 40 GHz o 4 MQW compressive strain K = 0.58, 'max = 15 GHz

10

o '"o

.L..-.-.

20

.L..-.-._ _ .------JL--.-_

40

60

_----..J

80

-.J

100

(1 (GHz 2 ) r igur e 1l...l. and

1/'

Experimented results ,huwing rh,C' li.ic ar relation between the damping factor 'Y

for two quantum-well Lt~t:J::;, 'Y = f\f,~ -l- liT. The slope: g:"/cs the: K factor (in ns) an d

the: intercept with the y axis gives the inverse cl:'ri::l" liL:tinlC J /T at threshold. (After Rd. 1-.1-.)

IU

SE MICUN D UCT O R Lt\S[.R S;'LCTRAL Ul'IEWlDTH

samples. The difference of the slopes of the two laser structures can be explain ed fr om the differen ce in the differential gain of over a factor of 2, sin ce J} /P o = 3,6 GH z 2/ mvY for the compressive strain la ser and f } jPo = 7.7 GHz 2/ mvV for the tensile strain sample. Ii]

Since the damping f actor depends on th e K factor, relaxation frequency , a n d the inverse carrier lifetime, the experimental data provide very good guidance for the design of high-speed semiconductor lasers . The K factor, K = 41T 2 ( 'T{I + (e/vg'» , can also be used to determine the nonlinear gain suppression coefficient E. Theoretical models and experimental data on strained and unstr airied quantum-well lasers have been presented with interesting results [15-24]' T h e measured nonlinear gain suppression coefficient E ranges from 2 to 13 X 10 - 17 cm ' for InGaAs or InGaAsP materials in the quantum wells [15, 17, 18]. V arious physical mechanisms such as the well-barrier hole burning effects [20], carrier heating and spectral hole burning [21], carrier transport [22, 231, and carrier capture by and e scape from quantum well s [25-27] have been investigated and-shown to affect by varying degrees the high-speed modulation of semiconductor quantum-well lasers. More work is in progress to understand the ultimate limit on the high-speed modulation of semiconductor lasers.

11.3 SElVIlCONDUCTOR LASER SPECTRAL LINEWlDTH AND THE LINEWIDTH ENHANCEMENT FACTOR The spectral properties of semiconductor lasers have been investigated since the early 1980 s. Experiments by Fleming and Mooradian [28] showed that the laser s p ec t ra l linewidth has a Lorentzian shape and the linewidth is inversely .p ro p o rt io n a l to the o p t ica l output power. However, the magnitude of the .linewid t h wa s much larger than they had expected from conventional theories. A m odel proposed b y Henry [29, 30J explained the phenomena by noting th at the se mico nd u cto r la ser is similar to a detune.d o scillator, a n d th ere is a sp ec t r um linewidt h e nh a nce me n t due to the coupling betwe en the amplitud e a nd phase fl uctua t io ns of the optical field . A linewidth enhancement factor 0'<, is introduced

a

() n'jrJ N =

"

_ . ~~

un "j aN

( 11.3 .1)

a nd th e la se r Iin e w idth h as a b roud e nin g enhan cement by an a mo u n t 1 + a ~ . H e re /1 ' nn e! II " a re th e re al and imaginary par ts of the refractive ind ex du e to the ca rr ier injec tio n int o th e active re gio n and N is t he ca rr ie r density, A

more form a l d eri vati o n h ~I S bee u give n by Va hJh and Yar iv [J 11. In this sec tio n, we prese n t th e m udel of Henry.

·198

DIR ECT tv :ODl: .U \ T l , )N O F S Ei'dl CON[ >U ClOR LASERS

11.3.1 Basic E qua tio ns for th e Optical In t ensity and. Phase in the Presence of S po n ta neous Emiss ion

Consider an o p t ica l field F-:; ( z , t ) give n by

E( z ,t) = E ( t ) e ilkz -w, ) E( t) -

(11.3.2)

(i(iT e i .pl l )

(11.3.3)

where [Ct) represents the intensity and ¢(t) th e phase of laser field. We assume [Ct) = E(t)E*Ct) has been normalized such that it represents the average number of photons in the cavity. The time-depend ent magnitude Ei t' ) is a complex phasor , assuming that its time va r ia tion is much slower than the optical fr equ en cy UJ. The phasor E Ct ) is plotted in the complex plane as a vecto r with a magnitude fi(t) a nd a phase ¢ (t) , as s ho wn in Fig . 11.5. The basic assumption is that a random spo n ta ne o us emission alters Ei t ) by /1 E , which adds a unit magnitude (one photon) and a ph a se e, which is random : .

-

...

(11.3.4)

There a re two co n tribu tio ns to th e ph ase change /1¢ :

(11.3.5) wh ere /1¢ ' is du e to the out-of-phase component of /1E , and /14>" is due to th e intensity chan ge which is coupled to the phase change . T o obtain the first contribution, /14> ', we note from Fig. 11.5, that 11 /1¢' ::= sin 8, or sin 8

(11.3 .6)

11 Irn E (t)

"

_~_--t.

F igure J 1.5 .

--::> Re E (t)

A pl o t o n til e co m p le x o p tica l c k,': ric he ll! [:29] EU) ~'"

/f(lT e:w li(p U )J d om ain

s how ing that its m a gn itu d e ..;7 a nd p hase :/) c.ui k : c ha nge d by th e s po n ta ne o us em iss io n of a p ho ton ( mag n itu d e is o ne si nce (he in te nsity ( iu s b ~: e n norma lize d to rep re se nt th e ph oton n umb er i ll the c.ivity) with a p h.rse ch :lIlge j, ,!,' . .

'11.-'

SEM ICONDUCTOP. LASER SPECTRAL L rI';EWID~~H

To obtain the second contribution, 6¢", due to the intensity change, we start from the wave equation 1 a2 2" -2 E,.£(Z, t) c at

where c is speed of light in free space and .», E,. = e j £0 of the semiconductor. We obtain

BE(t)

2iw

-£

c2

= -

at

r

2 (W- E c2

r

-

( 11.3.7)

the relative permittivity

IS

)

k 2 E(t)

( 11.3.8)

neglecting the term B2 Ei t ) jBt 2, since E(t) is a slowly varying function. We can also express Erin terms of the complex refractive index ~

VEr

I

= n

+

(11.3.9)

."

In

and

co

-ji; c

=

k=

w

w

c

c 1

-n' + i-n" co ,

co

-(g - a) 2

-ll

C

--n" c

( 11.3.10)

where g is the gain coefficient, and a is the absorption coefficient of the optical intensity. At threshold, the gain is balanced by the absorption, (g = a), n" = 0, and e , is real. Changes in carrier density N will cause n' and nil to deviate from the threshold values: E,. =

n'

(

= n ,/-

+

A' ts n

+ 2'In

r

+ lun . ")2 A

A "( 1 un

. ) la e

(11.3.11)

where we have defined a linewidth enhancement factor a e as the ratio of the change in the real part of the refractive index to the change in the imaginary part: .1n' at' =

.1 nil

( 1.1.3.12)

Therefore

DE (it .. (;

( ,

'-- 't -,

~~~-

-

) I' (j ,

.- i a ., ) E'(I )

\"l~l") J ..) . .)

f " -::- --C;- ' ~"'I":'

__ ~ .~--: .~",,-.,.,<".,, '_ .•.• ~.-r ....• - .._ . - ..-

. , or" -

..... . . .

500

~,

-':

"'....

,.~'

~

"

_.- "

-

-,,-

-

DIRE CT rviOD Ui A T IU!':

.,'

-

O~:

••.

_

• _

, _ . ..

"

, ..

.'

.... ..

,. ~ '"

'..

t

~

SEMICONDUCTOR LASERS

where the group velocity L: = C / u' has been used , and we ignore th e dispersion effect for simplicity. If we include the effect of the material dispersion , (11.3.8) has to be modifi ed [29J and the result fo r 01.3.13) still holds with the group velocity given by u = cj(n' + W on' jo(u), which is derived usin g k = (UlZ'je and lJ = (okjaw) -l. Substituting E(t) = fi(i)-eirJ>(t) into (11.3.13) and separating the real and imaginary parts, we find 1 dl

- ( 2 dt d¢ -

dt

-

-

g - O' )

2

vI

( 11. 3 .14a )

(g-2 a)

(11.3 .14b)

[)O'e

Therefore, we obtain d¢

2/

dt

(11.3.15)

dt

Initially at t = 0, 1(0) = I + 6.1, and at t die out, 1(00) = 1, and we obtain

=

co ,

the relaxation oscillations

a 6.¢" = .': 6./

21 O'e

2 / (1

rr

+ 2y /

cos 8)

(11.3.16)

which can be derived from the relation among the three s id e s of the triangle in Fig. 11.5. The total phase change is then 6,

«.

= -

2/

+

6.¢"

+

IT/ (sin e + a

1

e

cos 8)

(11.3.17)

The ensembl e average of the spontaneous emission events at a time duration t is contributed from the constant term ( O'ej2 I) multiplied by the total number of the events , R spt:

( 11.3.18) since <s in &) = ( cos () > = O. H ere R "p is the spontaneous e m ISSlOn rate (l js). Equati on ( 1 1.3.18) gi \ e ~ an a ng u lar fr equency s h ift: rl '-

6!u = -- <6(b ) df '

=

a" - H. '2 J sp

( LJ.3.19)

l Jl .•.,~

SE;

lll~ONDUCTOR

,:-,Ot

LASER Sr::,CTRAL LU-':EWIDTH

The total phase fluctuation for Rspt spontaneous events gives the variance:

(11.3.20)

We use an absolute value for It I since (6.4>2) is a positive quantity. Therefore, we found the mean (6.4> > and its variance (~4>2 > as above. 11.3.2

Power Spectrum and Semiconductor Laser Spectral Linewidth

The power spectrum of the laser is the Fourier transform of the correlation function:

YV( w)

f

co

d t ei

(v

E *( t ) E ( 0) >--

t(

-oc,

_ foo

dt

e iw t (

[1( t ) I ( O) r / 2

e-ic.(t»

-cc

:::::: 1(0)

foo

dt

eiwt(e-i..lU)

(11.3.21)

-00

where the small intensity fluctuation is neglected and the amplitude function for the field E(l) = [1(t)]1/2 eiJ>(t) has been used, which does not include the central frequency of the laser, that is, w in (11.3.2), and .'

~4>(t) =

4>( t) - 4>(0)

Since the spontaneous emission events are random, the phase ¢ should have a Gaussian probability distribution function, P(~cP) = a Gaussian function. The ensemble average for a Gaussian distribution is [30, 32] (e-i..l(t»

=

IX d(D..(/.»P(~(p)

e-i..l,/,

-COL

( 11.3.22) Using the result for the variance <::'.(//')

LI1

(11.3.:20), we can define a

coherence time as 1.LL"~ I l {'

)

- -----!\ 41 'i'

( 11 .J .2J )

•

SU2

such that function

DJRi:C'T MODULATlON OF SErvllCONDUCTOR LASERS

<6.cP 2 ) /2

=

It 1/ t The power spectrum (] 1.3.21) gives a Lorentzian C'

(11.3.24 ) with a full width at half-maximum (FWHM) of Dow =

2 _ (1 + a;) R t 21 sp c

or

Do! =

(1 +

an

4TT" 1

R
(11.3.25)

(11.3.26)

The number of photons 1 in the laser cavity (the photon density multiplied by the volume) is related to the optical output power by (I0.1.28)

( 11.3.27) where Ifzw is the total photon energy and va m is the escaping rate of the photon out of the cavity with a length L, where a

rn

1 L

1 R

= -In-

(11.3.28)

and R is the mirror reflectivity at both ends. We find !1f=

(11.3.29) I

which are commonly used 'to explain the spectral linewidth of semiconductor lasers. Alternatively, the spontaneous emission rate (l/s) is related to the gain coefficient (I / cm ), by a dimensionless spontaneous emission factor n~p:

R sp

= ugn

sp

(11.3.30)

where 1

(11.3.31) and t:..F = Fe - F; is the separation of the quasi-Fermi levels between the electron and the hole. Note that the photon number 1 and the spontaneous

ermss io n rate R ~p (l Is) are defined for the whole volume of the active region.

11.3.3

Linewidth Enhancement Factor in Semiconductor Lasers

Experimental data [33] for the laser linewidth 6./ depending on the op tical output power with an inverse law are shown in Fig. 11.6. The surprisingly large linewidth measured in this set of data was explained by Henry [29] using the correction factor (l + a ~), where a e = S. More measurements have been done for various semiconductor lasers including index-guided double-heterostructure, unstrained and strained quantum-well lasers, with reduced linewidth enhancement factor [34, 35] a e . In a semiconductor laser, the injected carrier-ind uced refractive index change is associated with the change in the gain. The linewidth enhancement factor a t can be directly expressed in terms of the differential change of the refractive index per injected carrier vs. the differential gain: dn' ae =

-

200

eln"

47T dn/dN

--

(11.3.32)

A dg/dN

r----,----.---r-""'T'"--r--T----,r--~-_..........

160 .' N

:r

~

120

:z:

~

Cl

~

w

80

Z ..J

SLOPE a 9.28 77 K

40

o

0.3

lNVERSE }<":g Ui' L'

1.1. O,

Se m ico nd uctor

1.~

1.0

PO "HER

L ~ :< f lil,!<:v,idtil ';eiSUS

Z.O

z.~

1 {mW- }

invc rse power

at

three temperatu res ex hib it-

in g. the lin ear beh av ior . ( A l' tc i Ref. 33.) The magn itudc of the lari,:c lincwidth was explain ed [29] ll si r~g th e co rrec tion fucior 1' -1- (t :: with ry, == 5 ut r(Joln temperature.

. ~

-:'

-H-''''r ,.,....

.,"'

~

,

•.

.-~ ... ~.

---

'..

.

12 . - - - - - - - - -

't

l;j'D

10

'-

o

u ~

8-

" 4 MOW tensile strain (In O.3Ga 0.7 As)

o 4 MOW compressive strain (Ino,eGao.2 As)

6

4 2 O'------"'-------....L.---_-...JL-.1400 1450 1500 1550 Wavelength (nm)

----.J

1600

~

Figure 11.7. The linewidth enhancement factor LYe vs. wavelength of two types of strained MQW lasers. The lasing wavelength is indicated by the arrows. (After Ref. 14.)

where we have used dn"/dN = (-1/2)(dg/dN)/(21T/A) in 01.3.10) with w / c = 274/ A, and A as the wavelength in free space. The linewidth enhancement factor varies from about 1.5 to 10 and is dependent on the lasing wavelength. Experimental data of Q'e for two strained quantum-well lasers are shown in Fig. 11.7. The high-speed modulation characteristics of these lasers are shown in Fig. 11.4. The dependence of CI.' <: on the loss or the Fermi levels of semiconductor lasers has also been shown to be important [36]. For strained quantum wells, theoretical analysis [37] shows that Q'e can be reduced to about 1.1 using tensile strains with TM polarization of semiconductor quantum-well lasers. Similar behavior to the spectral broadening in semiconductor lasers also occurs in an intensity modulator using the e lectroabsorption effects [38-40]. This is because a phase modulation due to the change in the refractive index is associated with the change in the absorption coefficient. PROBLEMS 11.1

11.2

Estimate the photon lifetime for a semiconductor laser assuming the following parameters: CY i = 10 cm- 1, refractive index = 3.2, and, cavity Ie ngth L = 200 ,u m. Discuss the effects on photon lifetime if we increase the cavity length or reflectivity by a factor of two. Derive (a) the de solutions for So arid No and (b) the ac solutions for s( t ) and nt; [) in the linear gu in tl! - ory .

•

11.3

.. .,~

Discuss how Olle rnuy improve the relaxation frequency semiconductor laser.

I,.

of a

RLFER Ei\iCES

50)

I 1.4

Derive an expression for the 3-dB angular frequency W 3d B for the small signal frequency response of a semiconductor laser using the linear gain theory.

11.5

Derive the de solutions for So and N u in the nonlinear gain theory for the frequency response of a semiconductor laser.

11.6

Derive the ac solutions for si t) and nCr) in the nonlinear gain theory.

11.7

Discuss the effect of decreasing the damping factor 1'.

11.8

Plot the frequency response curve when the condition 2w~ = 1'2 satisfied.

11.9

Derive 01.2.13) and discuss the approximations used.

11.10

IS

Derive 01.3.13) and 01.3.14) taking into account the material disperSIOn.

11.11

Derive (11.322) for a Gaussian distribution function P(t::.¢).

11.12

Describe the factors determining the spectral linewidth of a semiconductor laser. How may one decrease the semiconductor laser linewidth?

REFERENCES 1. T. Ikegarni and Y. Sue matsu, "Resonance-like characteristics of the direct modu-

lation of a junction laser," Proc. IEEE 55, 122-123 (1967).

2. A. Yariv, Quantum Electronics, 3d ed., Wiley, New York, 1989. 3. K. Lau and A. Yariv, "High-frequency current modulation of semiconductor lasers." Chapter 2, in W. T. Tsang. Vol. Ed., Lightwai:e Communications Technology. Vol. 22, Part B. in R. K. Willardson and A. C. Beer, Eds., Semiconductors and Sernimetals, Academic, New York, 1985. 4. K. Y. Lau, "Ultralow threshold quantum well lasers," Chapter 4, and "Dynamics of quantum well lasers," Chapter 5, in P. S. Zory, Jr., Ed., Quantum Well Lasers, Academic, San Diego. 1993.

). T. P. Lee, C. A. Burrus, J. A. Copeland, A. G. De ntai, and D. Marcuse, "Short-cavity InGaAsP injection lasers: Dependence of mode spectra and singlelongitudinal-mode power on cavity length," IEEE 1. Quantum Electron. QE-18, 1101-1112 (1982). 6. K. Petermann, "Calculated spontaneous emission factor for double-heterostructure injection lasers with guiu-induced w.ivc guiding,' IEEl:-' J. QW/J1twn Electron. QE-15. 560-570 (1 (79). 7. W. St rcifer, D. R. Scifres, and R. D. Bu r ah.rm. "Spontaneous emission factor of nurrow-suip gain-guided di,)(\t' lasers.' Elntron . Lett. 17. 933 (l9R I).

,\ N. K. ouu., S. J. \Vang. ,\. B. Piccirilli. r: F. Karlicck, J1' .. R. L. Brown, M. Washington, U. K.. Ch;I:,rt;llJit~. aile! ,-\.. Gua uck.: "Wide-bandwidth and

DIR:cCf MO)'ULAi I{)N OF SF MICONDUCTOR LASERS

506

high-power InGaAsP distributed feedback lasers," 1. Appl. Phys. 66, 4640-4644 (1989). 9. K.

Uomi,

M.

Aoki, T.

Tsuchiya,

and A.

Takai,

"Dependence

of high-speed

properties on the number of quantum wells in 1.55 fLm InGaAs-InGaAsP MQW A/4-shifted DFB lasers," IEEE.I. QuailtLilIl Electron. 29, 355-360 (1993). LO. Y. Arakawa, K. Vahala, A. Yariv, and K. Lau, "Enhanced modulation bandwidth

of GaAJAs double heterostructure lasers in high magnetic fields: Dynamic re­

sponse with quantum wire effects," Appl. Phys. Lctt. 47, 1142-1144 (1985). 11. Y. Arakawa and A. Yariv, "Quantum well lasers: gain, spectra, dynamics," IEEE

1. Quantum Electron. QE-22, 1887-1899 (1986). 12. R. Olshansky, P. Hill, V. Lanzisera, and W. Powazinik, "Frequency response of 1.3 fLm InGaAsP high speed semiconductor lasers," IEEE 1. Quantum Electroll.

QE-23, 1410-1418 (1987). 13. R. Olshansky, P. Hill, V. Lanzisera, and W. Powazinik, "Universal relationship between resonant frequency and damping rate of 1.3 fLm InGaAsP semiconductor lasers," Appl. Phys. Lett. 50, 653-655 (1987).

14. L. F. Tiemeijer, P. J. A. Thijs, P. J. de Waard, J. J. M. Binsma, and T. V. Dongen, "Dependence of polarization, gain, linewidth enhancement factor, and K factor on

the

sign

of the

strain

of

InGaAs / InP

strained-layer multiquantum

well

lasers," Appl. Phys. Lett. 58, 2738-2740 (199l). IS. T. Fukushima, J. E. Bowers, R. A. Logan, T. Tanbun-Ek, and H. Temkin, "Effect of strain on the resonant frequency and damping factor in InGaAs / InP multiple

quantum well lasers," Appl. Phys. Lett. 58. 1244-1246 (1991). 16. S. D. Offsey, W. J. Schaff, L. F. Lester, L. F. Eastman, and S. K. McKernan, "Strained-layer InGaAs-GaAs-AIGaAs lasers grown by molecular beam epitaxy for high-speed modulation," IEEE J. Quantum Electron. 27,1455-1462 (1991). 17. H. Yasaka, K. Takahata, N. Yamamoto, and M. Naganuma, "Gain saturation coefficients of strained-layer multiple quantum-well distributed feedback lasers,"

IEEE Photoll. Techllol. Lett. 3, 879-882 (991). 18. -J. Zhou, N. Park, J. W. Dawson, K. Vahala, M. A. Newkirk, U. Koren, and B. I.

,

Miller,

"Highly nOl1lkgenerate four-wave

mixing and

gain

nonlinearity in

a

strained multiple-quantum-wcll optical amplifier," Appl. Phys. Lett. 62,2301-2303 (1993).

19. S. R. Chinn, "Measurement of nonlinear gain suppression and four-wave mixing in quantum well lasers." Appl. Phys. Lett. 59, 1673-1675 (l99JJ. 20. W.

Rideout,

W.

F.

Shlrfin,

E.

S.

Koteles,

M.

O.

Vassell,

and

B.

Elman,

"Well-barrier hole burning in quantum well lasers," IEEE Photon. Technol. Lett. 3. 784-786 (1991).

21. A.

Uskov, J.

Mork,

and J.

Mark.

"Theory of short-pulse gain saturation

in

semiconductor laser amplifiers," IEEE Photo/l. Tpchllol. Lett. 4, 443-446 (1992). "

R. Nagarajan, T. Fuku:)hirnd, S. 'vV. Corzine, aDd J. E. Bo\vcrs, "Effects of carrier transport on high-speed C]ucntum \Veil lasers," Appl. Phys. Lett. 59, 1835-1837 (j 9(1).

23. A. P. 'Wright, B. Garrett. G. H. B. ThD:np.,,"', :1lld J. E. A. \Vhitemvay, "Influence of carrier transport un \\-:,',ckf1gth chi;'p (ft" InCi'-t/\s/InC:T:'lJ-'\sP MQ\V lasers," Electroll. LeU. 23,1911-1913 iL9(2).

sm 24. E. Meland , R. Holmstrom, J. Schlafer, R. B. Lauer, and W. Powazinik, " E xtremely high-frequency (24 Gl-Iz) InGaAsP diode lasers with excellent modulation efficiency," Electron. Lett. 26, 1827--1829 (1990). 25. P. VY. M. Blom , J. E. M. Haverkort, P. J. van Hall, and J. H. Wolter, " Ca r d e r - carrie r scattering induced capture in quantum-well lasers," Appl, Phys. Leu . 62, 1490-1492 (1993). 26 . D . Morris, B. Devcaud , A . Regreny, and P. Auvray, "Electron and hole capture in multiple-quantum-well structures," Phys. ReI.'. B 47 , 6819-6822 (1993). 27. S. C. Kan, D. Vassilovski, T. C. Wu , and K. Y . Lau, "Quantum capture limited modulation bandwidth of quantum well , wire, and dot lasers," Appl, Phys. Lett. 62,2307-2309 (1993). 28. M . W. Fleming and A . Mooradian, "Fundamental line broadening of single-mode (GaAl)As diode lasers," Appl. Phys. Lett. 38, 511 (1981). 29 . C. H. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J. Quantum Electron. QE-18, 259-264 (1982). 30 . c. H. Henry, "Spectral properties of semiconductor lasers," Chapter 3, in W. T. Tsang , Vol. Ed., Lightwave Communications Technology, Vol. 22, Part B, in R. K. Willardson and A. C. Beers, Eds., Semiconductors and Semimetals, Academic, New York, 1985. 31. K. Vahala and A . Yariv, "Semiclassical theory of semiconductor lasers, Part I," IEEE J. Quantum Electron. QE-19, 1096-1101 (1983). 32. M. Lax, "Classical noise, V, Noise in self-sustained oscillators," Phys . Rev. 160 , 290-307 (1967). 33. D. Welford and A. Mooradian, "Output power and temperature dependence of the linewidth of single-frequency cw (GaAl)As diode lasers," Appl. Phys . Lett. 40, 865 -867 (1982). 34 . Y. Asai , J. Ohya, and M. Ogura, "Spectral lincwidth and resonant frequency characteristics of 1nGaAsP/ 1nP multiquantum well lasers," IEEE J. Quantum Electron. 25, 662-667 (J 989). .' 35. N. K. Dutta, H . Temkin, T. Tanblln-Ek, and R. Logan, "Linewidth enhancement factor for In GaAs Z Inf' strained quantum well lasers," : Appl. Pliys. Lett. 57, 1390-1391 (1990). 36 . Y. Arakawa and A. Yariv, "Fermi energy dependence of linewidth enhancement factor of GaAIAs buried heterostructure lasers," Appl. Pliys. Leu . 47, 905-907 ( 1985)'37 . Y. Huang. S. Arai , and K. Komori, " T heo re tica lI inew id th enhancement factor of Ga I _. .. In .,.As / GaInAsP/ InP strained-quantum-well structures," IEEE Photon. Tee/fllUt . L eu . 5, 142-145 (1993). 3':-:. F. Koyama and K. 19a, "Frequency chirping of external modulation and its reduction ," Ele ctron. Lett . 2l, 1065-1066 (985) . .N . Y. Noda, M. Suzuki, Y. Kushiro, and S. Akiba, " H igh-s p ee d e lectroab sorption m odulat o r with st r ip -lo ad e d Ga InAsP planar waveguid e ," 1. Lightwtu:e Tcchnol . L:r-.:L l445-l453 (l986). 4(). 1". 1-1. \Vood . •. Multiple quantum well (MO\,V) waveguid e modulators ," J . Li ghtH '(/ CC Tcchnol. 6 , nJ --757 ([':HS).

12 Electrooptic and Acoustooptic Modulators In this chapter, we discuss eleetrooptic effects and modulators. The bulk electrooptic effects are di scussed first, and their applications as amplitude and phase modulators are presented. These devices using waveguide structures are then shown . The basic idea is that the optical refractive index of electrooptic materia Is, such as LiNbO 3' KH 2 P0 4 , or GaAs, and ZnS semiconductors, can be changed by an applied electric field. Therefore, an incident optical field propagating through the crystal with a proper polarization experiences efficient electrooptic effects. The transmitted field changes in either phase or polarization, which can be used in the designs of phase modulators as well as amplitude modulators. We then discuss scattering of light by sound and present a coupled-mode analysis for acoustooptic rncdulators.

12.1 ELECTROOPTIC EFFECTS AND AMPLITUDE MODULATORS [1, 2] To understand the e lectrooptic effects, we consider a crystal described by the constitutive relation associating the displacement vector D to the electric field E by a permittivity tensor E.

(]2 .1.1)

arid the permeabllity tensor € as K:

IS

fLo.

Let us define [he Inverse of the permittivity

K

The index ellipsoid

I~) Y

==

£--1

(12.1.2)

th e cryvt al is descritcd by c, c.

\"'

L'

\-

'"

U ~ !\. i I ~ t·l...'

Ii

c'"

(12.l.3)

12. )

dJ::CTR002T1C L fFEC"T:-; ;\i',Jj t\l\fPU rUDE

i'vIODULAT\)R.~

~09

where Xl = x, J 2 = y, and x 3 = z for convenience. For most crystals, E is symmetric due to the symmetry property of the structure. Therefore, ~ can be cliagonalized to be

= £=

[

E~'

0

(12.1.4)

E }'

0 The coordinate system iIL which £ is diagonalized is called the principal system. In this system, EoK is a diagonal matrix with the diagonal elements equal to the reciprocals of the square of the refractive indices of t he three characteristic polarizations along the direction of the principal axes:

EoK = EO

l/Er

0

0

2 1/n . x

0

0

0

l/E y

0

0

l/n;,

0

0

0

l/E z

0

0

1/11;

(12.1.5)

and the index ellipsoid (12.1.3) is

1 -+-+-Xl7

E

where I'li 12.1.1

=

o ( Ex

y2

')

Xi?

x-

EyE z

n~

Xi7

7

+

n 2y

'1

Z'-

+

fl~

1

( 12.1.6)

I~i/Eo' i = x, y, and z.

Electrooptic Effects

In a linear clcctrooptic material, the index ellipsoid is changed in the presence of an applied electric field F, and K i j becomes K i j + 11 K i } , where the change 11 K i j is linearly proportional to the electric field: 3

Eu

11K,}

=

L

rij k

r,

(12.1.7)

k=J

The linear electrooptic effect is also called the Pockels effect, after. Friedrich Pockels (3] 0865-1913), who described it in 1893. These r i j k coefficients are also called Pockels coefficients. Equation (12.1.7) can also be generalized to include the quadratic e icctrooptic effects. which are usually smaller than the linear effects, -' r) l ~ ' . . cJ ll .:..

EL ECTRO(..?TlC I\ ND :\COUSTOOPTIC

510

MODULATOR~;

However, for materials with centrosvmrn ctry, the index ellipsoid function mu st be an even funct ion of th e applied el ectric field, since it must remain invariant upon the sign reversal of the el ectric field. Therefore, r i j /.; vanishes and the quadratic e1cctrooptic effects dominate. The quadratic ele ctrooptic effect is also called the Kerr effect , after John Kerr [3] (1824-1907), who di scovered the effe ct in 1875. From the symmetry property of the crystal, the following matrix correspondence is defined:

(1 2.1.9)

that is, we have (ij) ~ 1 = 1,2, ... ,6, and ri jl<. = rj i k == r fl" Note that r1k 6 X 3 matrix, and Eq. 02.1.7) can b e rewritten as

to

(~K)]

r

(6.K)z

' 21

(6.K)3

r31

(6.K)4 (6.K) s (6.K )1)

Ii

I'll.

'13

r 33

'41

rn r 32 r42

r5 1

1'5'2

r 53

r6 ]

r 62

r63

" 2J

'"43

[;: ]

IS

a

( 12.1.10)

Depending on the symmetry of the crystal, many of the matrix elements r 1k m ay vanish. We usually refer to some databook [4] or reference tables [5-8] for the crystal symmetry and nonvanishing rJk va lues. A few important electrooptic materials are shown in Table] 2.1, for illustration purposes.

Example The potassium dihydrogen phosphate (KO P or KH 7 PO,) crystal is uniaxial in the absence of an applied field : ...

7

X.:. EO

L « >», IJ

n 20

z-J

y-

-+-

.,

Il ~

+

n J2

1

(12.1.11)

where n .r = Il~. = no' and n z = 11 c : With an applied field, we have r 1k = 0 except for r6J 'and "41 = r 52 · Therefore, Eq. 02.1.10) gives Eo(6.K)4 = r-lIF] , 1;'o( t1 K )) = r 52F2 , and EI)(6.K) t, = r cJ:;F:,. Note that Xl = X, x 2 = y, x J = z , a nd lise the mapping table in ( I? . J ,9 ):

cu :L 1K

i j ' \ i .t"j

= 2i .l i F, v; +-

21':'>2

F\..\.:: + 2 rl ) F

yx

_

'") ') (1 7 .1..1.:-

iJ

wh ere a fa ct o r 2 acco unts for t he syrnme rr ic property of th e matrix il K i j ,

In general, the above index ellipsoid may not have the principal axes along the x, Y, or z directions any more, as will be shown in the following examples. III 12.1.2

Longitudinal Amplitude Modulator

Consider the setup as in Fig. 12.1 with the applied electric field along the propagation direction (z ) of light F = iF_. The index ellipsoid is

1

(12.1.14)

The above equation shows that the principal axes along the x and y directions are rotated because of the cross term , 2r6 3 F;: xy . \Ye have to find Table 12.1

A Few Electrooptic Materials 'With Their Parameters [1,4, 6, 9]

Refractive Index

Point-Group Symmetry

Materi al

no

3m

Li.Nb0 3

2.297

Wavelength Ao(j.Lm)

Ill!

2.208

0.633

Nonzero Electrcoptic Coefficients (10 -12 m/V) "D r~2

1'12

32

Quartz (Si0 2 )

1.544

1.553

0.589

42m

KI-I 2 POol (KDP)

1.5115 1.5074 1.5266 1.5220 1.5079

1.4698 1.4669 J .480 8 1.4773 ] .4683

0.546 0.633 0.5'+6 0.633 0.546

42111 4 2m

1'4 1

" 02

NH"H 2PO .. (ADP) KD zPO"

r oll r~1

1'41 1'4 1 1'41

= 8.6, 1'33 = 30.8 = 1'51 = 2R, r 22 = 3.4 = "61 = -r2.'2 = -"52 = 0.2 = r 2 1 = - I ' l l - 0.93 = r ·.., = 8.77 , r 6J = 10.3 = r ~-,- = 8, "fl.1 = 11 = r 5 '2 = 23.76 , 1'63 = 8.56 = r .;;,., = 23.41, 1'0:' = 7.820 = r Y2 = 8.8. I"bJ - 26.8 =

" 23

~-

(KD '~P)

43m

GaAs

3.60 .3,42

43m

.3.34 J .29

InP

.1.20

43m

43111

~.. I;()

ZnS e /3-Zn S

2 .3() .

_-

-- _. _. _

= fl o =n 0 = !'lo

0.9 1.0 10.6

= II .)

I .Of)

- .

1 . ~)5

II ('

{l ( , , , . J._~.)

=1? = II

O.h

0

- - -- - _ . _ - ~ _

..

~ . _- - - -

1'41 = Tel l

I" - , :> -

= r '-,

=

1" '.1

= 1.1

-

r 'i J

-- 1.5

r 52 = r I1.; = 1.6 - . rj : ~ = . r :. = L.-+5 lJ = 1'5 2 -.. rt> 3 - 1.3

r .j J =

"-I t 1'-11

F-l 1

r~!

= r·, '- = -, = - r J_

r() .~ =

1""\ ). )'

2.0

= 2.1

._--

_.

FLEC~(ROOf'T'C

:512

Passing axis

,\;-iD ACOUSTOUPTJC MODULATCiZ:;

x

x

>

Output

y

,,

Polaroid

Polaroid

+V(t)Figure 12.1. A longitudinal amplitude modulator in which an applied electric field is biased along the direction of optical wave propagation,

the new principal optical axes such that the index ellipsoid can be described + ~ K )ij matrix. A coordinate rotation of 45° on the x-y plane gives

by a diagonal (K

1

X=

Ii

y

- ( -X'

(x' + v')

1

Ii

+ y')

(12.1.15)

Substituting 02.1.15) into (12.1.14), we find 1

( 12.1.16)

We may rewrite the index ellipsoid as ,

,1

)

X -

Y-

,:- + n,

Ii)'

,:-

1

z~

+

( 12.1.17)

1

.-, n~

where n ,x =

1

,

1/" - - - - -

(1 --

r (;:.ll ;, F. )

,.

J ,i'

"'"

1/ o

-

0

+ - ".' Il '-, F 7

I)

~

( l:?.l.lSa)

and, similarly. ; I'

L;

--1I

'2

~ F ' " ,.

( i2.1.18b)

12.1

LLECTEOUPllC

EFf~h(·TS.<\ -I[.

AMPLITUDE MODULA lORS

513

vYe note that in the new coordinate system (x'-y'··-z), the matrices (K j.K)ij and E tj are diagonalizcd: 1/ Il~;

0

0

1/ n'}~

0

0

Eo(K + t1K)

,

0 = E=

°

lin;

Ex

0

0

0

Ey

r

0

0

0

Ez

+

(12.1.19)

,2 , ,2 d E = neEo' 2 F l ' In . where Ex, = nxE or a pane wave propagating z o, E y = nyEo an the + z direction and polarized in the X' direction in a crystal described by diagonal permittivity tensor (12.1.19),

E -- x"'E 0 e i /3 z

( 12.1.20)

It is easy to show (Problem 12.2) from Maxwell's equations that the propagation constant f3 is

(12.1.21 )

where k: larly, for

=

(Ih! /-LuEu

=

27T /

"'0

and

"'0

is the wavelength in free space. Simi-

E -- y"'£ 0 e i /3 z

(12.1.22)

we find the propagation constant is f3 = kn'y. The incident optical field, after passing through the polarizer, can be expressed as E

st: 1I e i k z

=

"I

= X

Eoe ikz + y --e Eo ik~"I

(12.1.23)

12

fi

in free space. Upon hitting the surface at z = 0, the wave is decomposed into two orthonormal polarizations along the x' and y' directions; each satisfies all of the Maxwell's equations independently since both are characteristic polarizations of the crystal. The propagation constants of the .e and Y' components are kn':.' and k,(., respectively. Neglecting the reflections at the surfacex z = 0 arid .2 = the optical field at z = t can he written as

e,

.

E

r :

..... 1:' I ~ 1.:,' "-' ,.. ' ...:...

£J

.l

-

/ j

V.:..,

..' p

v

II, n

"I.

I

,.., I

.L I

"

>

..

.:.

0

__

. . ' .: pI/.:. II ,. I

/:- v

7 -

.

(12.1.24 )

ELEc r RO C) PTIC o'\ L D AC( iU$ r O :) PTIC

514

MODUJ.ATOR~;

Th e transmitted field passing through the se con d pol aroid is th e y cornponentofEin02 .1.24), or y . E, using Jy: J = Ct - Ji) /12 and yJ = ( x + y)/{i:

(12.J.25) The transmitted power intensity divided by the incident power intensity proportional to

1S

(12.1.26) Noting th at F/t) t = vet) is the applied voltage , we defin e ~-r ' the voltag e yielding a ph ase difference of 7T between the two characteristic polarizations, that is, k(n~t - n'y) t = Ti",

(12.1.27) and obta in

( 12.1.28) By vary ing VCr ) = Fz(t) t , the o u t p u t light intensity is modulated. To obtain a linear respons e, V ( l) has to be biased near V-rr /2 wh ere th e transmission factor is 50 % , as s ho w n in F ig. 12.2, i.e .

V( t)

( 12 .1 .29)

and the transmission factor is PI = Sin 2 ( JI Pi 4

I[

+

O 1TV sin w 2V;;m

l-'_o

- :- 1 + sin ( 7 V sin w ) -

V't t

t)

'"

t

] ]

( 11.1.30)

12.\

ELECTROUPTIC

E~FECTS

. 'J P,fP i = sin-

At'i;) A:v!; t.i

~

I I -1_!. I

,ViODUIATORS

5i5

(reV2V~ (t",) I

0.5

n n.r:

-

- - - - - - _. 0.5

_

I

Figure 12.2. Transmission of a light intensity in a longitudinal amplitude electrooptical modulator. The applied voltage V(t) is biased at a de value Vr r / 2 .

We see that for a small input signal zr Va « V7T , and a bias at the 50% point, V7 T / 2 , a linear response can be achieved. However, since Vr, is typically very large, a better way is to add a quarter-wave plate between the electrooptic crystal and the output polaroid, with the two principal axes of the plate along the x' and y' directions such that an extra phase difference of 1T /2 is introduced between the x' and y' components of 02.1.24):

Eo E = V 2 ( _t' e .17T/')- e Isk• n'n r f

+ y' e 'k ' f) I

II,

(12.1.31)

(12.1.32) .f

In this case, the modulat ion voltage is VU) = Va sin No de bias is necessary.

12.1.3

(IJ'/I

t instead of 02.1.29).

Transverse Amplitude Modulator

A transverse amplitude modulator is shown in Fig. 12.3 in which the applied field is biased in a direction perpendicular to the propagation direction of light. The incident optical electric held nfter passing through the polaroid is

E.I

EL FCTROOPTIC A N D ,-\COUST O \"',P f JC MJDULXi"ORS

5 16

AzF z

,z

z

z

z

- - - --

,

y' ",

,,

~x'

,,

,, y' =

Polaroid

t

Polaroid

Figure 12.3. A transverse ampl itude modulator. The bias field iFz is perp endicul ar to th e dire cti on of opt ical wave pr opagation .

propagating in fre e space. Note that the direction of prop agati on ha s been ch os en to be along the y ' di rection of the electrooptic crys tal, which makes an angle 45° with the principal x and y axes of the unbi ased crys tal. After passing through th e crysta l, the x' and z components g ain different p hase s at y' = t:

(1 2.1.34) Here, we still use KDP crystal in our analysis. The tran smitted field throu gh '- he secon d polaroid with the passing axis given by ( - ,i' + Z)/fi is

E = E . (

2

- xA, + z~ ) (

f)

v-

E0

= - ? ( - e ' k fI , r + el k < t ) "I

"

fI

-

Th erefor e, we finel th e transmission factor .'

(1 2.1.36 ) where

(12.1. 37) Here th e time-d ependent Iie lcl f~{t) m o dLll ~lte s th e pha se (p( t) , whi ch deter-

12.2

PHASE MODUwl\.TOR "

5L7

mines the output light intensity a s () function of time. Since F/O = V(t) /d, where d is the thickness of the crystal , we nne! that the phase difference between th e two characte ristic polarizations (£ ' and components) is, from 02.1.37),

z

Vet) d f

(12.1.38)

Again we can define a half-wave voltage V1T as the voltage required to introduce an extra phase shift to 77":

v

=

1T

277"

kn ~"63

(d)

( 12.1.39)

t

vVe can see that the factor d / t can be cho se n to be small; therefore, the half-wave voltage V, is reduced compared with that for a longitudin al amplitude modulator. We write

¢

=

k t' 77" V e t ) - 2 (n o -n ,J + 2 V1T

( 12 .1.40)

The transmission factor is then

(12.1.41) A linear response is obtainable if we choose

kt

71

----;-(11 1) - Il e )

4

( 12:'1.42)

and the input sign a l V( t ) = v~ ) sin (,)"" is small , Vo « V1T , The transmission factor is rhe sa me as (12 .1.32) and is similar to Fig . 12.2.

12.2

12.2.1

Pl:-L-\SE IVIODULATORS

Optical Phase M udula tion

Consider an in cid ent opt ical {k id pr op a guti n g alon g th e z dire ction and passing thro ugh th e pol a r. .lcl a~; s ho« n in Fig. 12.4: l~'

1...,

__ "': / r~ . \ 1:..

c:

-,

L:

11-. :."

') t ) l' 17~._.

ELL~CTI~OUPTIC I\N

SIS

0 ACOUSTOOPTiC MODULATORS

Passing Axis r x -v

Polaroid V(t) Figure 12.4. A longitudinal phase modulator with an applied electric field iF~(t) along the propagation direction of optical field E.

Since the polaroid has been aligned such that the passing axis is along one of the characteristic polarizations of the electrooptic crystal, x', the transmitted field at z = f is simply E I = X"'E oe ikn't .r

(12.2.2)

= x'Eo eikll"t eikn>uJF:(I)f/2

(12.2.3)

Therefore, if we modulate the applied electric field as (12.2.4)

we find (12.2.5)

where r r r'I F. k -263 0 0

l

iJ

(12.2.6)

The electric field in the time domain is

E,(f,L) = Re(E,e-I'AlI) =FEf)cl)s(kn(}t+ osinlrJl/IL -

Wl)

(12.2.7)

The phase of the output optical field is modulated by the factor 8 sin we use the mathernurical identity [10] --;-... ~

t

j'~})

"nt'

C.

(--: i '11 ( '!.J + 7i,l 2. }

wml.

If

. ., "J' (. 1)_ ...c..0

-

_..

_._~

.~~--

-- - -

12.2

PHASE MO j) "l JLA TO i ' S

:5I Y

or, equivalently, .:,t:,

e j,)

si n
"W

} ( 8) e im lD

(J 2 .2 .9)

/II

1/1 ..~ -

' ,I)

Equation (12.2.7) can als o be written as

Et( t,t)

Re(Ete- i
=

=

~ Re [ x'Eo e i(k,, _

Re(i'Eoei(kll ,J - w( )e ioSinw",t)

~~

( -w ,j m

cc

f m ( b) e im w m' ]

xAlE 0

=

( 12.2.10)

'11 =

-

0'0

The output optical field contains, in addition to the fundamental frequency (() with an amplitude 1 oUj )E o, various sideband s with frequencies, co +, (() m ' W + 2(v m , . . . , etc. , with corre sponding amplitudes +1 1( 0 ) £ 0' 1 2( 0 ) E o, ' " . Note that 1 _m(o ) = ( - l) 1m ( 0 ). If the m agnitude 0 = 2.4048, the root of the zeroth-order Be ssel function (1 0 (0) = 0), all of the power in the fundamental frequency w is transferred to the nonze ro-ord er harmonics. Jn

Example LiNb0 3 has a 3m point-group symmetry and from Table 12.1 its 'n matrix has the form [1, 11]

r=

-

0 0 0 0

-'22

'13

'2 2

}"13

0

'3 3

' 51

0

r 51

0 0

1" 22

where

r 12 r 42

= - '22 = I" SI

' 23 =

'u

r; l -

- r 22

0 0

Th e r va lu es a t a wavele ng th Ao = 0.633 /-Lm are r u = 8 .6 X 10 -- 12 rn y V r 'n

= 3 .4 X 10-

l2

my V

1'33

=

30 .8

X 10 -

12

m ,' V

r5 1

=

28.0

X

10 -

12

rn y' V

T he r efractive indices of LiNbO J h ave a un iaxial form: II, .X ~o, ,"Z

,I '

= .II

{I

:;.; '). . . ._ ') ~. / 1 7

"'1 ,' 1, '

,

i

,'

-t

::.

_. -

1/•

e'

-

'-

'") ')O {~ ,~ . ,,,", u

(12.2.11 )

ELECTR OOP'rlC AND ACOVST001'lI C MonLJLAT()R ~

described by C' 0

L (K

+ 0. K ij )

ij

1

XI Xj

i ,i

or using the sy rn rnctry in the

X 2( ~ no

- r 22F 2

r m atrix,

+ r 13F J ) + y 2(~ + no

F 2 + r l3 F 3 J

rn

+ z' (~; + r 33F3 ) + 2yzr S jF 2 + 2 zx r 5l F, - 2xyr"F j

~

1 (12.2.12)

Since " 33 is the large st coefficient , a n applied e le c t ric field alo ng the z d irection will be most e fficie n t for the e lectroopt ic c ontrol. T h e re fo re , for F, = F 2 = 0, the index ellipsoid is given by

or

x2 n 7x-

y ?-

+

1

..,

Il;

(1 2.2 .14a )

w he re the new refractive indice s a re n ,,' -- n,y --

11 0 -

3 .!..2no rUF 3

( 12.2.14b) ( 12 .2.14c)

12.2.2

X-cut LiNbO 3 Crystal

For an Xvcut LiNb0 3 crystal , as s ho w n in Fig. 12.5 a , two elect rod e s are placed symmetrically on both s ides of th e wave g u id es such th at th e bi a s fiel F = i F) is a lo ng the z dir ection a nd th e index ellipsoid is d e scribed by 02. 2.14). An incident optical electric fie ld with TE polarization will transmit as

E -- -;E ()

eikll c Y

-

-

(,, = { )

..!:E

,,, :/,".. /-i(1/2)/lJ' r llF 1 t'

( ) '-

"

,

(1 2 .2 .15<1)

S imil arl y. fo r T M p ol ari z at ion ,

( 12.2 . 15b)

J2..~

PHI\5lE fv'IUDULJ.T01'.S

51\ ~x

(a) X-cut LiNbO 3 phase modulator

z'~

.

,.y

~7~7%. /(ql

I' -----:c:==:;::--.---~-=~ I : '- _~ __ - 'r.,y L~ rf7'..' - -_ _-~ 1

~=JJA

TM

T~>

'------..=.:{-.:----~_ .

F '"' ZF z

. )r.V77~ffiWffi/~

--f"\JL->

"".v~

Incident

Transmitted

light

Electrooptic substrate

light

y

z

(b) Z-cut LiNbO, phase modulator _ __ z

------===__-....

..,\c-----:l2z(ZZ21

~~~~~

I '• I. _ -

t>~v

- -_-(___ ~ V_ ~-

VI

I

~

------~---~-

"-~~~~~~~~~~6::::6:~-'~'" Transmitted light

Incident

Light

Electrooptic substrate

Figure 12.5. Electroopric phase modulator using (a) X-cut LiNb0 3 substrate. where th e electrodes ar e pl aced symmetrically on both sides of the waveguide, such that the bias field is along the z dire ction; and (b) Z-cut LiNbO 3 substrate, where one electrode i::; placed dire ctly above the wavegu ide: such that the bias field in the waveguide is along the positive (or negative) z directi on tor the most efficient phase modulator since fJ3 is the largest electrooptic co e fficie n t.

Therefore , 'IE polarization should be used for most efficient ph ase modulati on since r J 3 > r 13 . 12.2.3

Z-cut LiNbO J Crystal

For a Z-cul LiNbO 3 crystal as shown in Fig. J2.5b, the electrodes are placed such that the wav e g u id e is below one of the two electrodes where the field is perpendicular to the Z-cut surface and :F = iFJ . The optical transmission field will he (TE polarization)

( L2.2.16a)

{ T IVl polar iz ati on)

( 1.2.2.J6b)

LI n d

E=

In thi s c a se . TM p ul~lriz;Jtiull i..., preferred Ior n :'j)S[ effici ent phase m odulat[ \.)I1 . For m or e d isc uss io ns on th-.::· 'polarizut ::))1 ..uul e lec tro d e de sign s, inc] ud ing the TE a nd TM polarizario n conversions. sec Refs . 11 - ·1 6.

ELF CTROOPTIC AND I-\COUST G n l' T IC IvlODULATCIRS

521

12.3

ELECT1l00P'flC EFFECTS iN \YAVEGUIDE DEVICES

In Chapter 8, we discussed optical directional couplers using parallel waveguides. Here we d iscuss briefly the applications of electrooptic effects in waveguide structures . We h ave discussed the use of KDP and LiNb0 3 in electrooptic amplitude and phase modulators. Some of these materials, such as LiNb0 3 , as electrooptic crystals have been used in many commercial devices. Here, we discuss the use of GaAs in electrooptic waveguide devices. For integrated optoelectronics, semiconductor materials, especially III- V compounds, are 'at tractive be cause many active and passive components, such as semiconductor lasers, photodetectors, and field effect transistors, are made of these compound semiconductors. However, considerable research work is still necessary for integrating passive and active devices with desired operation characteristics. Example GaAs at 10.6 ,urn wavelength no = 3.34, r.l1 = r5 2 = JO - L2 m / V. All other r components ar~_zero.

0 0 0 r=

0 0 0 0

0 0

r-') :J_

0

0

r 63

r-l l

0 0

yF2 + iF..",

For a biased field F = iF L +

"63 =

1.6 X

0

0 (12 .3.1)

the new index ellipsoid is

1 (12.3.2) If we choose F I = F 2 = 0, F ~

-,

-,

y-?

xn~

iF." , we then have

=

z-

+ n - + 2r nJF 3 x y + o ~

~

n-o

1

(12.3.3 )

This is simil ar to that of the KDP materials, except th a t the crystal in the absence of fidei is Isotropic. or n = 11(/ Again a rotation of 45° in the .r-y plane gives .r = (x' + yJ) /{2, Y = (-x' + }.I)/v2, L'

X

-:-

,

iI ~

-

-

- -

- -_

-

_

_~

..

.r "-

I'

,

- /2

, -r

- ~ -~ , "

-" - -- - - --

( .1 2 .3

, "/1 -

II ~I'

I.

-'

(J

_ ._-- -- ~

..

_~ - -

.

<1'} , L )

12.3

~::L E CTi< ()() P

' 1.=.' L P

;1. CT S IL V.: \ :E C, UIDE:

D (~·: \, l(T S

52.3

where (l2.3.4b) (12.3.4c)

The previous analysis for longitudinal amplitude modulators for LiNb0 3 applicable to GaAs materials. D

IS

In the following examples, we consider (a) a Mach-Zehnder interferometric waveguide modulator, (b) a directional coupler modulator, and (c) a A,B-phase-reversal directional coupler, as shown in Fig. 12.6. The input (a) A Mach-Zehnder i nte rfe rometric waveguide modulator

Electroopti c Crystal

(b) A directional coupler m odulator

(c) A 6~-phase-reversal directional coupler

"\.l-Figure 12.6.

._

--J

Wu ve gu ide e l c ct roop tic d e vi c es \" ith ,·kc[l"i)l! c de sig ns. (a) A tvLl ch - Z chmk r

iute rfer umctr ic m od ulator. (h) a c! i r ec LlL1IU I cou ple t rnl !d ulatoi', an(i (c) a .:.'l./3 ·pl u s-:-revt,;r.-; al

directi on a l co up ler .

L LECTR OO PT 1C A N :) A',':. -OUSTOOPTI C

M Oi.JULATOR ~; , ,

power Pin is ta ken as 1 in ea ch case. The electrodes a re d e signed s uch th at th e a p p lie d e le c tric field is al o ng the z di rect ion . F = ±zF.. . , a nd the b ias fields in two waveguid es nrc opposit e in sig ns . Therefore , th e difference betwe en the two propag ation factors is approximately

D. f3

= k [n o + 2.1n ~ r 63 F 3 ·)

-

k (no -

1 ) 2n~r F 63

3

= kn ~r6]F3 V ( t)

= kn 3r - do 63

(12.3.5)

where an e ffective width d is d efined for th e e lec t ric field (F 3 ::= V/ d) . The import ant point is that b.{3 a V( t ). We will stu d y the transm ission charact erist ics of th ese d evices as a function of the detunin g factor D.f3 t . 12.3.1

Mach-Zehnder Interferometric Waveguide Modulator

As shown in Fig. 12,6a, a sing le waveguide is branched into two arms for a distance t and combined aga in into one arm a s the outpu t waveguide. The wa vegu ide dimensi on can be chosen to g u id e th e fund ament al mode only.

=>

Constructi ve

......--l-----,r'-------

ou tput

De stru cti ve output

Fj gur~

12 .7. An illtlSl l" 'l ( ~'.l'~ l d tiie Y -j u nc tj '-'!1 Cl l!" !', hcil- Z eh nJ c i i ntrrfe roru. .t nc W ~\\!\:: ~~,ll i,le modu l.u,».

il" L1cl i vl;'

;\:1:1

l! o;:::;lr ll d iv .:

llU tp l: b

in

<1

525 1

0...

15

h

(l.)

2>

0

c,

0.5

........ ::;l

0.. ........ ::;l

0

0

-2

- 1

0

1

~~t In

2

Figure 12.8. The output power from a Mach-Zehnder interferometric waveguide modulator as a function of the mismatch factor t1{3L.

With a proper choice of the polarization of the incident wave and the electrode design, the tr ausmitted intensity is

(12.3.6)

where the output intensity is normalized such that the peak transrrnssion factor is 1 for perfect power transmission. One way to understand this tr an ission behavior is that if the guided modes are in phase at the exit of the I junction (Fig. 12.7), they add up constructively and transmit with the maximum power. If they are out of phase by 180°, they will cancel each other. Anther way to look at this is that if they add up to a first-order mode, it will leak out over a very short distance, since the waveguide is designed to guide the fundamental mode only, resulting in a destructive output. A plot of the outpu t power POllt vs. the mismatch 1:1 {3 L is shown in Fig. 12.8. The interferometric behavior is clearly seen. 12.3.2

Directional Coupler Modulator

For an incident optical beam into waveguide a modulator, the output power is

In

a directional coupler

(12.3.7<1)

'"' \ ( 1.:.'.. .J./tJi ~

lfi

~.

I:::LECTRO DPTIC ;.\ N D I .,CO UST O O PT IC MODUL,,\ TCiRS

5 26

....

,, --,'~ I

p., '"

...... ll,)

~ 0

p.,

0.5

Kf=n

.....

::J

0.. ..... ::J

0

P ~4

·3

-2

-1

0

K £=nI2 Kf=3n12 2 3

4

b. [3 f. In Figure 12.9. The output power from waveguide a as a functi on of f:j.f3L for K f = 3 tr / 2 for a direction al coupler modulator.

7T

/ 2,

7T ,

and

and (12.3.8 )

where the input power is assumed to be 1. Since 6.f3 = f3a - f3 b = kn>63F3 = kn>63V/ d, we plot the output power Pa vs. 6.f3 t . Suppose we design the modulator with a length t such that Pa = 0, and P b = 1, at 6.f3 = 0, i.e., K t = 7T / 2. To switch to Pa = 1, and Po = 0, we require at least 6.f3 t = 13 7T, assuming the field-induced change in the refractive index affects the coupling coefficient negligibly. (Otherwise, we can calculate the field-dependent K and still use th e expressions for Pa and P b in 02.3.7) an d (12.3.8) to find the output : -we rs .) To switch from a cross state to a parall el state , the applied volta ge Las to be large enough such that 6.f3 t = 13 1T is sa tisfi ed . A plot of P, vs. 6.f3 t fo r K t = 7T /2 is shown as the thick so lid curve in Fi g. 12.9. We also plot P; vs. 6.f3 t for K t = 7T, and K t = 37T /2. W e see that complete switching from th e @ state to the 0) state is possible (fo r K t = 7T /2 or 37T / 2). For K t = 'TT , where we start with the B state at 6.{3 t = 0, it is impossibl e to swit ch to the Q9 state simply by ch an ging 6.f3 t a lo ne . This f act ca n also be checked 'wit h the switching diagram in F ig. 8.19. 12.3.3

6.~-Phase-Reversal

Directional Coupler £11-17]

The 6.f3-ph ase-reversal direction al coupler is s ho wn in Fig. 12.6c, and its ana lysis can be found in Section 8.6. The switching diagram is shown in Fig. 8.21. Suppose we sta rt with the parallel state at K t = 7T , 6 t = 0, where U = ( f3a - {31) /2. The output power is .

.' d; t

._

C;)S 2

(~' ) I.

2

( 12. 3. 9)

1

"

0.5 L

o

-4

-3

-2

-1

0

L1~e Figure 12.10. fl.{3 t for K t

=

1

2

3

4

In

The output power Pa of a fl.{3-phase-reversal directional coupler as a function of TT/2 and 'TT,

VVe plot P a vs. D.f3 t as the solid curve in Fig. 12.10 for K K t = 7T /2 (dashed curve) for comparison.

t

= 7T

and also for

12.4 SCATTERING OF LIGHT BY SOUND: RMIAN-NATH AND BRAGG DIFFRACTIONS The refractive index of a medium can be caused by a mechanical strain produced by an acoustic wave; this is called the acoustooptic effect. A sound wave creates a sinusoidal perturbation of the density, or strain or pressure of the m.iterial. The induced change in refractive index can be described as (12.4.1)

with W s = the angular frequency, k , = the wave vector, k , = 2711 As' As wavelength, and V s = W s I k: s is the velocity of sound in the medium. 12.4.1

=

Raman-Nath Diffraction [9]

Here the length of the interaction between the light and the acousti<: wave is small:

kn

t

(12.4.2)

where k: = 2T1 IA Il , n = the refractive index. of the medium, and /\0 = the optical wavelength in free space. This is called the Raman-iNath regime of diffraction (Fig. 12.11), In this case, the thin region in which the acoustic wave propagates acts like :.1 phase grating. and the diffracted lights can go to Q1 a ny different directions determined by the generalized Snell's law for a

ELECTROOPTIC AND /\COLS

b£J

I.

kn

;+-2

+1

2rrlA,

+1 0 8_1 x-direction

>-

'\

t n

n+~n

x=o

+-e~

MODULATor.,)

... k,..

~2AA

Incident

Light

~()CJPTIC

kx

-1

-2

kn Radius = k n

n

x=/!

Thin Rarnari-cNa th diffraction. The interaction length f is very short and the thin like an optical phase grut ing with the period equal to the acoustic wavelength A,.

Figure l2.ll. region

8C1S

grating:

m

=

an integer

(12.4.3) A simple analysis of the diffraction efficiency for this case E = yE/x, z, t) at x = t

IS

to consider

where

(12.4.5) has been used. We then write the field at x = t identity [IO] c'" cu:::,

J)

L n:=

using the mathematical

i "'.1,,/ ( II) e i msb

(12.4.6)

-x

and set (12.4.7)

(12.4.8)

.')29

The eJectric field at x ;' (x 1~y

-

~ t)

11, L , -C

=

t

=

then becomes

E'oei(klll'-UJI) -

;-, L.rn

»«

l'IrIJlll(~" A/'lt) U.

AO

-00

e- i ll1( k ,, ::

- w , t)

,

e illl k ,z e --i(r.v +mw,)1 (12.4.9)

Since for x > t, the electric field has to satisfy the wave equation in the medium described by the refractive index n, we should have the solution of the form 00

E

eik,w/x·-t)+ik""z-iw",1 fll

/71

= -

( 12.4.10)

00

where k zm ( V III

k.K.J?1

-

(

--n c

mk s

=

):

(12.4.JJ)

and (12.4.12)

We note that

Ws

« w; therefore,

W m :::: W,

' n) ;{f;f

and

W',

k X III

=

( -c

r

,

(mk S

(12.4.13)

The diffraction angle of the m th order is therefore (12.4.14)

or sin - (' [' ~nk s] , '. kn

12.4.2

=

sin -

I ['

,

ni

~~_)' n ): s

(12.4.15)

Bragg Diffraction

Whe n the interaction length ( between the c:)tici.l] and acoustic waves is long compared with kl1/k~, vr: h..ve the i3r
:'7.-

.~"

.'1

- _.

.-

~~

EL ECTRoorT C AND ACDUSTCOP'(IC MODUi j ·.TOIZS

2kn sin

e = ks

k d = kj+k s

+----- f

e is long Figure 12.12. Bragg diffraction. When the intera ction length f is long , a particular angle of incidence with one diffracted beam satisfying the Bragg condition 2kll sin e = k , will be observed . (AO + OB= A/n for constructive interference. Therefore, 2A J sin e = A ln.)

Bragg condition : 2 k n sin () = k ,

( 12.4 .16)

where e is the angle of incidence, which is also the angle of diffraction. There is only one diffracted beam determined by the above Bragg condition (see Fig. 12.12 ). Our a na lysis of the Bragg diffr aclion is presented in Section 12-5.

12.5 COUPL i ~D-MODE ANALYSIS FOR BRAGG ACOUSTOOPTIC WAVE COUPLER [1, 2] The analysis for Bragg diffraction can be based on the coupled-mode theory. We start with the Maxwell equations: .'V

x E

\7

x

H

=

-

a at

- j.L H

a

-D

( 12. 5 . 1)

at

where the disp lac ement vector is ( 12.5.2)

and the refracti ve ind ex varia tion is n ( r , Ii

=

!l

-I- J.1I( r,

I)

(1 7_.J- ...)'~ )

' 1-;"" ': - .... ~ . • ',.'

••

53i

Here the background refractive index 11 and the. amplitude of variation Cin are independent of the position and t . Consider a TE polarized wave E = yEv and assume that both the acoustic wave and the optical wave propagate in the x-z plane (r = .:Lt + z2). This solution E = ;",E y (x, Z, t ) satisfies Gauss's law because

o

-

V'D=V'[c on 2 ( r , t ) E ] = Oy[c on 2 ( x , z , t ) E y ( x , z , t) ] =0 (12.5.4) and v . E = 9 . vE/x, z, r ) (12.5.1) and (12.5.2):

=

0 too. The wave equation

IS

derived from

(12.5.5)

( 12.5.6) We assume the incident electric field to be

(12.5.7) and the diffracted electric field to be (12.5.8) The variation of the refractive index can be put in the form 6. n ( r , t) = D. n cos ( k

!:J.n

s •

r - wst )

_ei(k,.- r--uJ.J)

2

D.../l + _o-i(k,.-r-'''-,I) .... 2

(12.5.9)

Then

(12.5.10 )

where the sCClmL1 de riv.u ive uf E/ k\::; been ignor~d. since \\'".: ,l~;::;ume that the amplitude f) 1") i:~ slowly varying compared with the e:l.p(ik/ . r) dependence and T/ is now ~11011:s the direction of :\./. A similar expression holds for \;2Ei/'

ELE CT RO OPT:( ' /-\. 1\: 0 A C O L ) ~TOOPT J C t\IOOULATOR:)

. 1,"

J _, • .

The term containin g t he product of ~11(r, t)E will give ri se to four terms:

(12.5.11 ) and a similar expressi on holds for ~!l(r, t)E d' Notin g that the total electric field E = E, + E d' we compare the terms of the same spatial and time variations and find

(12.5.12) or

(12.5.13) These results are illu strated in Fig. 1.2.13. Equation 02.5 . .12) shows th e . conservations o f momentum and energy for a photon with initial wave vector k , absorbing a phonon with a wave ve ct or k , resulting in a final photon st at e with mom entum Izk d = hk; + Ilk s and energy fIw d = h ca, + flC.lJs- Similarly, 02.5.13) corresponds to the emission of a phonon from the incident photon. Here II is the Pl a nck constant. Al so not ing that k , = ( wi/c ) n, k d = (wd /c)n, we find fr om 02.5.6) a nd 0 2.5.1 0)

aE.

ik.· 'VE. = i k -' I I art I

W ,? ! I

- - -;> .6. nE d ( r ) 2 c-

Since f; is along the direction of k. and r d is along the direction of k take r = xf , and

ri cos () = x

Ttl

cos 0 = x

=kj

+ ks

kd = k j

lOJ = 0\

+ CU s

0)d

kJ

,a)

et ,

we

(12 .5.14)

-

ks

= cui -

COs

( b)

Figure 12.1J . T he di,lgra :11:> to r th e d irtru c tio n uf lig h l nv so und : (a) k rl und (b) k " = k ; - k .\ . (U , i ;:; l V , - (!I . .

= k , + ks'

( 0"

= U)i

+

(') s

L'. 5

BRAG:"'; ACO l) ";TO ;)] 'TIC 'S.\ VE CDl:PLU,S

533

We obtain dEi dx

iK .E[

=

I

2 c cos

I

Kd

Since

W S «(U I' , Wd

we have

W d :::::: W i

K=

(12.5.15a)

e

w(f6.n =

= wand

2c cos

(12.5.15b)

e

K, :::::: K d

= K:

w6.n

2c cos

(12.5.16)

(j

The solutions for the coupled-mode equation given the initial conditions E/O) and £/0) are

E i ( x)

=

EJO)cos Kx + iEAO)sin Kx

EAx)

=

Ed(O)cos Kt + iEJO)sin K"

(12.5.17)

If initially, £iO) = 0, the field amplitudes are

E i ( x)

=

E j ( O)cos K1:

£A x)

=

iE j ( O)sin Kx

(12 .5.18)

The energy IE/U)1 is co upled to IEi X )1 and backward during the interaction as th e optical waves propagate along the x direction , as shown in Fig. 12.14 . We can write that th e diffraction efficiency at a length f! is 2

2

(12.5 .19)

TJ

.....

, ,,

>,

~

J::

-~

I

0.5

,,

,, "

/~"'-IEix)F

I

I

,, .#

0

,

I

•

~-"--"-----'-"'=:'-~--'---''----<--

rr/(2KJ

x/K

Positi on x Fi~uri~ 12.1-t The coup l i ng \ l l' l' ·; ·.', ;:;. i,;' o. betwe en t l . c i l1~'ilk! ,l ac o us toop tic me drum in w hich a SPl l ,1(1 wave P Wr'_~? ~l tl·S.

~1 11 :J cli !ri~IClccl ( 'ptI C I] \\ ~!v ,-'S ill ~In

ELEC T ROOF t'le AND f ,COUSTOOPTIC MODUlATORS

PROBLEMS

12.1

Calculate the voltage parameter V.. = ). 0 / (2 tl ~ 1'63) for the materials an d wavelengths with the nonzero 1"63 coefficients in Table 12.1.

12.2

Show from Maxwell's equations that for a permittivity tensor in the principal axis system,

o Ey

o (a) a plane wave polarized along the principal axis i and propagating along the z direction, E -- i.E 0 e if3 ;:

the propagation constant is f3 = w-{M,e x; (b) a plane wave of the form E = yEo e it3z will have a propagation constant f3 = f.L e y .

wJ

12.3

For a longitudinal amplitude modulator as shown in Fig. 12.1 , (a) if the bias voltage is V(t) = (0.5 + 0.1 s in W I1l t plot the output light intensity as a f unctio n of time. (b) Repeat r:clrt (a) if Vet) = 0.5V7T sin wJlJ .

12.4

Mod ify the design in Fig. 12.1 by adding a quarter-wave p la te such th at the transfer function 0 2.1.32) can be realized with a linear response and the bias voltage vCt) will not require a de bias voltage .

12.5

For the transverse m odulat or shown in Fig. 12.3 , plot the transmission factor PI /Pi vs. time, as suming that

12.6

A quarter-wave prate is ad ded immediately a fter the first polaroid in th e tran sverse amplitude modulator in Fig. 12.3, and the ele c troop tical m aterial is GaAs ' » , = fl o = 3.42), assuming that the wavelength ,~ () is 1.0 f.L 111. The electric field E i is circularly polarized

.'

)v:..,

'1:;'

e,

I

~-=

(

.r" I

,

r

j --

-;-:: ,~'"

I

befo re impi ng ing o n th e GaAs cr ystal.

-~) /E2o e 1'1 . \ " fl .

5::15

(a) Find the electric field E~ at y' = t' in Fig. 12.3. (b) Find the transmitted field E 1 after passing the exit polaroid. (c) Obtain the transmission factor PI/Pi and plot it vs. time for V(t) = (Vrr / 4) sin (U n / · 12.7

Consider a transverse elcctrooptic modular, as shown in Fig. 12.15. The incident electric field is randomly polarized and only half of its power passes through the polaroid. The crystal is a KDP with an ac electric field applied in the z direction, and the refractive index ellipsoid is described by 110 on the x-y plane and n e along the z axis before the ac field is applied. (a) Find the expressions for the optical electric fields Eland E z . (b) Find the expressions for the electric fields E 3 and E 4 after they have been reflected from the perfect mirror. (d Assume that the applied ac electric field across the modulator in this problem is_f/t) = F zu cos wt. Find the ratio of the output optical intensity to the incident optical intensity as a function of time. Use a graphical approach to illustrate your solution assuming that

kf -en -n 2 e

0

)

8

(d) If we have a dc applied field, F, = Eo, find the value Eo such that the .ncident light Pin is completely absorbed by the system. z z

PIN Incident liaht =::) o

---r------+---.~

x'

x'-

'-

e

Figure 12.15.

12.8

(n)

E"")

~

Reflected light

For A Ga As transverse .modulator, derive the index ellipsoid for

F

F j .1.: -+ F 2 .v

=

-I-

p.,,£

(h) If F is along th~ ( I l I) direction. i.e ..

F

( .t =.:.:

-~

0 -T·· ::) }-I)

.-.•- .------------------

"' V/ _J

•

.~

•

' •

•

••

~

'

. "

: .'

,

,

:'\'

-...~ .. :, . , . .

M •

•

•

•• • •

, ••

,: .. ,

~

...--

.

•

•

• • ~ • •• • ~ :' ~

•

ELE'::'TRCl(IPT!< ' !'.N }) ACOlSrO OPTI C MOD',JLA'(Of{S '

design a tran sverse modulator anel calculate the voltage parameter Vor . .

12.9

Discuss the design of a phase m odulato r using Gai\s compared with .th a t for LiNbO J used in th e text.

12.10

For a Ga As phase modulator, compare the longitudinal configuration in Fig. 1.2.4 vs. a possible transverse configuration such that the direction of the applied e le ctric field F is perpendicular to the direction of the optical wave propagation.

12.11

Derive 02.3.5) and (12.3.6).

12.12

(a) Check the output power Pa in Fig. 12 .9 using (12.3.8) for K

Tr / 2. (b) Plot P,

VS.

(!1f3 t / rr ) for K

t

t

=

2Tr.

=

12.13

Plot the output power PIJ VS . 6.f3 t / pler using 02.3.9) for K t = 3 tr / 2.

12.14

Derive (12.4.9)-(12.4.12).

12.15

D erive the coupl ed-mode equations in 02.5.15a) and C12.5.15b).

77" [or a

~f3-phase-reversal

cou-

REFERENCES 1. A. Yariv, Optical Electronics , 3d ed. Holt-Rinehart & Winston , New York, 1985.

2. H . A. Haus, Wan's and Fields ill Optoelectronics, Prentice -Hall, Englewood Cliffs, NJ, 1984. 3. B. E. 'P... Saleh and M. C. Teich. Fundamentals of Photonics, Wiley , New York,

199( 4. S . Ada ch i, Phy sical Prop erties of ]/J-V Semiconduct or Co m p oun ds , 'Wiley, Ne w York, 1992.

5:' K. H. Hellwege , Ed. , Landolt-Bomst ein Nltme;ical Data and Functional R elation ships ill Science and Technology , New Se ries, Group III 17a , Springer , Berlin, 1982; G roups HI -V 22a, Springer, Berlin , 1986. 6. K. Tada and N . Suzu ki . "Lin ear el ectrooptical properti es of IrtP. " Jpn . 1. Appl. Phys . 19,2295-2296 (1980): and N. Su zuk i and K. Tada, "Electro optic properties and Raman sc att erin g in InP," Ipn . J . A pp l . Phys. 23, 291-295 (1984).

7. S. Ad achi an d K. O e , .. Linear e lect ro-o p tic effe cts in z incble ride-typc se rnicon.iuctors: Key properties of inGaAsP rele va nt to device design, " 1. Appl. Ph ys, 56 , 74- 80 (19S·D; and "Qua d rnt ic e leciroo pti c (K err ) effe ct s in z inc ble nd e -typ e sem ico nd ucto rs: Key prope rt ies 1) [' InG a A.'.;P re lc vun t to device design," 1. Appl . Phys . 56 , l499-l5()4 (l 9K4).

:'> . S. Adachi , Prop erties

orln diu rn PJi m p hidc,

Engin eers. London . 1')9 1.

INSPEC, The In sti tute .o f Electrical

PErF,RENCE>;

537

9. A. K. Ghatuk
l l. S. Thaniyavarn, "Optical modulation: Elcctrooptical Devices," Chapter 4 in K Chang, Ed., Handbook ojMicrowat:e and Optical Components, Vol. 4 of Fiber and Electro-Optical Components, Wiley, New York, ~91}1. 12. H. Nishihara, M. Haruria, and T. Suhara, Optical Integrated Circuits, McGraw-Hill, 13. 14. 15. 16.

New York, 1989. T. Tamil', Ed., Guided-Ware Optoelectronics, 2d ed., Springer, Berlin, 1990. R. C. Alferness, "Guided-wave devices for optical communication," IEEE 1. Quantum Electron. QE-17, 946-959 (1981). O. G. Ramer, "Integrated optic electrooptic modulator electrode analysis," IEEE 1. Quantum Electron. QE-18, 386-392 (1982). D. Marcuse, "Optimal electrode design for integrated optics modulators," IEEE 1. Quantum Electron. QE-J.8, 393-398 (l1}82).

17. H. Kogelnik and R. V. Schmidt, "Switched directional couplers with alternating 6.(3," IEEE 1. Quantum Electron. QE-12, 396-401 (1976).

13 Electroabsorption Modulators Electrcabsorption effects near the semiconductor band edges have been an interesting research subject for many years. These include the interband photon-assisted tunneling or Franz-Keldysh effects [1-3J and the exciton absorption effects [4-9]. With the recent development of research in scmiconductor quantum-well structures, optical absorptions in quantum wells have been shown to exhibit a drastic change by an applied electric field [10-13]' Wh i1e previous excitonic electroabsorptioris in bulk semiconductors were mostly observed at low temperatures , sharp excitonic absorption spectra in quantum wells have been observed at room temperature . These so-called quantum-confined Stark effects (QCSE) [11, 12] show a significant amount of change of the absorption coefficient with an applied voltage bias because of the enhanced exciton binding energy in a quasi-two-dimensional structure using quantum wells. The quantum-well barriers confine both the electrons and holes within the wells ; therefore, the exciton binding energy is increased and the exciton is me difficult to ionize . The analytic solutions for pure two-dimensional and three-dimensional hydrogen models in Chapter 3 show that the exciton binding energy of the Is ground state is four times larger in the 20 case than in the 3D case (14]. The sharp excitonic absorption spectrum with a small scattering linewidth shows the possibility of a big change of the absorption" coefficient by an applied voltage bias. The change in the absorption coefficient can be as large as 10 4 em - I in GaAs / Al xGa I-x As quantum wells [10-13] . Interesting quantum-well electroabsorption modulators at room temperature have be en the subject of intensive research recently. In th is chapter we discuss the theory for electroabsorptions with and without excitonic effects. In Section 13.1 we present the effective mass theory for a two-particle system: an electron -hole pair. The general formulation for the optical absorption due to an electron -hole pair is discussed . We show th at a change of variables from the electron and hole position coordinates r e and to thei r differe nce coordinates r = r ", - r h and their center-of-mass coo rd ina tes R lea d to po ssib le an a lytical solutions [4, 5, 8, 9] when the interaction potentia! is due to (l ) an e le ctr ic field only, which leads to electroabsorption effe cts in whi ch a ligh t is incident , or (2) the Coulomb interaction between the electron '.lnd th e hole, which gives the excitonic

'It

5YI

absorption when a light is incident, or (3) both an electric field bias and the exciton effects. Case 0), the Franz-- Keldysh effect, is discussed in Section L3.2. The exciton effects, case (2), are presented in Section 13.3. Both have analytical solutions for direct band-gap semiconductors near thc absorption edge. Case (3) in a quantum well, presen ted in Section ] 3.4, is called the quantum-confined Stark effect. The general solutions are obtained using two methods: One is based on a numerical solution of the Schrodinger equation in the momentum space for an electron-hole pair confined in a quantum well with an applied electric field [15]. The other method is based on a variational method [11, 12], which is commonly used in the literature because of its relative simplicity and accuracy especially for the bound state energy of the Is excitons. Device applications including quantum-well electroabsorption modulators [16, 17] and self-electrooptic effect devices (SEEDs) [18-20] are discussed in Sections 13.5 and 13.6, respectively.

J3.1 GENERAL FORMULATION FOR OPTICAL ABSORPTION DUE TO AN ELECTRON-HOLE PAIR In Chapter 9, we derive the general formula for absorption coefficient in SI units:

n ( It w)

=

r

2 Co V.I

1< fie n, op"

r

e . pi i) 12

0 ( Ef -

e, -

It w ) [ f (EJ -

f ( E f) ] (13.1.1a)

')

nrcEumuw

(13.l.1b)

The absorption coefficient depends on the initial state Ii) with corresponding energy and the final state if) with corresponding energy EJo The summation over the initial and final states taking into account the Fermi occupation factor f( E) of these states gives the overall absorption spectrum. We also note that the delta function accounts for the energy conservation and the matrix element in (13.1. I a) takes into account the momentum conservation automatically, as has been discussed in Chapter 9, where no interaction between the electrons and holes is considered.

r;

Two-Particle "Va ve Function and the Effective Mass Equation

13.1.1

To describe an elccrron-f.olc pair ~t(H';, rhe two-p :.111 icl '2 \N,-lV,~ function rl., can be 1,/;( r .' T.,) for an -: lectron ~\ t p():,i non r,. '.! t1d a hole ;tt uosition . I'

I

54Q

ELEC ,'RO Af: SORPTION MODULATORS

expressed as a linear combination of the direct product of the single (uncorrelated) electron and hole Bloch fu nctions, !/Jedr) and 1/;,. _ k,,(r/), respectively:

L

LA(k e , kfJI!JCk/r ,Jt/J/' --k,,(rh)

he

hi>

(13 ,1.2)

where A(k e , k 1,) represents the amplitude function. Note that the Bloch functions ~l'"kfre) and rjl,"-k,,(r/) contain both th e slowly varying plane-wavelike envelope and fast-varying Bloch periodic functions. In the effective mass approximation for electron and hole pai rs, all enve lop e function ¢(r e , r h ) :is defined as the inverse Fourier transform of the amplitude function A(k e , k /,): ( 13.1.3)

which is the plane-wave expansion of the two-p article wave function, The Fouri er transform of the wave function 1>(re , r h ) is (13.1.4)

The m ajor difference between l/J and

or 1 %,+ t) , have been dropped from the basis functions and only the plane-wave parts are kept. The envelope wave function e:p (re , r h) sa tisfies the effective mass equation

[E g + E c (

-

i \Ie)

-

Ec (

-

i 'l/J

+ V(r e , r hJ]¢(re,r h) = EcP(r e , r h) (13 .1.5)

where we have replaced k e in the dispersion relation E; =:; EJk e ) by the differenti al operator -i vI.' for the r e variables , an d k , in E( . == E ,"(k iJ ) by -- i VI; for the rlr variables. Using the parabolic model, we have Ec(k ,,) = 2 2 2 Jd E.(" (k h ) -- - IJ-?k * II k.. e ,/ 2 tn * I "/r j ? _nl;,. e an The Interaction potential V(r", , r/) may be of the form (1)

YI /il' \ e' 1") I.. --

eF . (, ...-- I: -- r 11 )"

(13 . 1.6 )

It is th e pote ntia l en ergy of ,I Jre e e lc ctron a nd ~l free hol e in the pr esence of a uniform electric field F. This will le ad to th e Franz- K c ldvsh effe ct [1- 3] for the opti cal abs o rp tion , as di ~ cli ss ed in Se c tion 13 .2. The interaction potential

can also be of the form ( 13.1.7)

It is the Coloumb interaction between an electron at

and a hole at rIJ. Here E s is the permittivity of the semiconductor. This potential leads to theexciton effect [4-9] in the optical absorption, which is discussed in Section 13.3. rc

Solution of the Two-Particle Eftective-Mass Equation

13.1.2

In general, for V(r", r,) = V(r e - r,), which depends only on the difference between the electron and hole position vectors, we may change the variables into the difference coordinate and the center-of-mass coordinate system, r arid R, respectively, as shown 111 Fig. 13.1a.

R= where M

=

m: +

m~.

(13.1.8)

!v!

The corresponding Fourier transform variables of r

(a) Real Space

Electron m~

o (b) Momentum Space

Fjgllrr 13.1.

(~l)

An illu~lLjl;(\11
the ditler'.'lh:c coortli n.it« \I/t',r~r"

+

1I1);r,,)i(II<~

Fou rie r trdil:;f."rm

",-:',:i',11"

+ l.'l); )

SP':ICl'.

(ill

rilt:

;'~'

ll».

l,k,('trDII p\~.,\I.'

r ...

[il'

~!IHI

i

',i

h..

',",TII)r

r .. the hole' position vector f:,.

('elltc~r-()~',nnsc

cl;"rdill~'k ';'O'cl"r

"~·;l
H

~~

til i h.,

and R in the mom entum space are (Fig. 13.1 b)

(13.1.9) respectively, whi ch can also be checked using

expj ik : r + iK . R) = expf ik , . r ', + ik , . r,J We can 'also express the above relation s as [II

= R

m "e

-

-r

( 13 .1.10)

K - k

(13.1.11)

M

and k

m *e --K

0-=

+

M

f

nz*

k

k

=

Iz

_h

lYI

From the corresponding differential operators k e =-iVe

k" = -i Vir K = - i

vR

fz 2k h2

h 2K 2

fz 2 k 2

')_nz"*

2lv!

k = -ivr

(13.1.12)

we ob tain tz 2k 2

-

-

c

-j-'

2m':

+

( 13 .1 .13 )

2m r

and

t1 2_

\-,2

2m ~:

c

_ _

-

tz l

--v 2m ~

2

Ir

tz 2 = - -- V2 2M R

-

-

tz 2

Zm ,

V2 r

Here m , is th e reduced effective ma ss, d efined by 11m ,. = (l Im : ) Th erefore , th e effe ctive mass equat ion 0 3. 1.5) becomes

[

-

tz ]. V 2

21\-1

II.

-

~ v} + V ( r) 2m,.

( 13 .1.14)

+

(l lm ~n.

- (E - E )] cP ( R, r) = 0 ( 13 .1.15 ) g

Tile solution to th e above equation GIn the n be obtained using th e met hod of the s ep a r a tio n o f variab les. noting t ha t the R de pe n de nce ' is a s im p le free particle wave fun ctio n. e t..tx- : A'.'' ;:t.h i\

l~

J. to.

~

.. ')

1.

- J(; - (~ \ r )

( 13 .1. 16)

[

~~-'1 Zrn;

-

2

+ VCr) -- &'.-'\dJ(") '

= 0.:

o

(13.1.17)

and the energy [f: is

'l" (-:;"' =

L"

1..~

l~

~g

-

(1'3.1.18)

----

Define the Fourier transform pair as ik'r

¢(r) -- "L. a( k) -e. . = -

a(k)

=

(13.1 .19a)

,(V

k

e

- ik - r

f d r¢(r) -;rr;3

( 13 .1.19b)

'rYe find

(13.1.20)

13.1.3

Optical Matrix Element of the Two-Particle Transition Picture

The optical matrix clement between the ground state (all electrons are in the valence band) and the final state (the electron-hole pair state) is described by [4, S, 13]

e . pli ) L LA *(k~"

k,,) (c,

\' \., A

k' "

(fle i k I1 P' r

-

1' 1 ,

L. L.J k

1

(, l\. <:.

keleik"p 'r

e . I

/ ;.!

) (k ) ct:

' "

e . pIL',-

11:;)

c

0" ,.+1.. " ,1;,."

1; [,

- LA ':' ( k . -- k )

r: . p " , I, 1-: )

(13.1.21)

l...

w he r..: th e lon z ..vavc lcnath (ot' dipol e ') ap oroxi.n ation t •

'-

,~

~

•

•..

.l

\.)~

1

C:-C

0 h as been used .

EU=CTF~Oi\l,SOR P·;'ION

MOD UU\ TORS

Therefore, the k selection rule ke

+

k, = k

op

has been adopted. Comparing (13.1.20) with the definition k;, = 0, we find that the matrix element is


(13.1.22)

= 0

(13.1.3), and using K = k , +

in

e . pii >:: : e . Pc£' LA ,t, ( k , -

k)

k

=

e.

PC! '

La·' (k) k

(13 .1.23) where

13.1.4

w-e' have

assumed that

e. P

CI

'

is independent of k.

Absorption Formula

The absorption coefficient is given by substituting the matrix element 03.1.23) into (13.1.1)

(13.1.24 ) n

where n corresponds to the discrete and continuum states of $(r) satisfying the effective mass equation in the difference coordinate system, 03.1.17). The equation for a Coulomb potential is the Schrodiriger equation for a hydrogen atom, and its solutions for both bound and continuum states have been presented in Chapter 3.

13.1.5 States

Physical Interpretation of 2 I
2

and Relation to Density of

Consider the case of a free electron and a free hole without Coulomb interaction, that is, VCr) = 0 in (U. 1.17). 'With K = 0, we have g' = E- E;;. Therefore, we u::;(; a new energy E measured from the band gap E<; for convenience:

(13.J .25)

54.;

In the discrete picture, we have the wave function and the energy spectrum

(13.1.26) where n == (n.t' ny, 11), k; normalization rule

=

11\.27T/L, k ;

=

n~,27T/L,

k;

= n z 2 71/

if n = n' if n -=I=- n' have been used. We check

qynCO)

=

( 13.1.27)

l/N and

1 =

L and the

-2

3/2

2

mr1 -2

27T ( fz

11vE

=

p:,D( E) (13.1.28)

which is the thr ee-dime.. .onal reduced density of states, where LI/ = [V/27T)"]!d 3k in the discrete picture has been used. (See Problem 13.1 for an alternative approach.)

.-

13.1.6 Optical Absorption Spectrum for Interband Free Electron-Hole Transitions

The optical absorption is given by integrating 03.1.24) (13.1.29a)

where (l3.1.29b)

which gives the absorption cocthcic nt clue [(,

d

free electron and hole. The

ELECTROA R ,ORPTION MODULATORS

546

momentum-matrix element of a bulk semiconductor is " I e'" . P c ( 1"-

mo

6

= M;;/ =

E'' p

(13.1.30)

where the e ne rgy parameter E p (in electron volt) for the matrix element tabulated in Table K.2 in Appendix K.

13.2

1S

FRANZ-KELDYSH EFFECT [1-3, 21-26]

Let us consider the case of a uniform applied elect ric field, VCr) = eF . r. The Schrodinger equation for the wave function cP(r) in the difference coordinate system (13.1.17) (with K = Q) is 2

1

- 11 V'2 + eF· r cP(r) = Ecj>(r) __ ( 2m,. . Assume that the applied field is in the z direction, F solution can be written as

=

iF (Fig -, 13.2a). The

( 13.2.2) where the z-dependent wave function ¢ (z) sa tisfies

( 13.2 .3) a nd the total energy E is related to E; for the z-dependent wave function

1z2 2 :. E = -2m- ,. ( k .\ + k~) + E , J

13.2.1

(13 .2.4 )

-

Solution of the: Schrodinger Equation for

3

Uniform Electric Field

The solution of the Schrodiriger equ ation 0 3.2 .3) with a uniform field can be obtained by a ch ange of va ria ble:

~

('2f7l r eF \ i: 3(! I 7' tz 2 ) ,-

2 = 1 \

-

E=)

-

eF

( 13.2.5)

lJ .?

FRAN Z · KfJ .D YSJ-)

EFF r~ C:T

Therefore ,

(13.2 .6) The Airy functions [27] Ai(Z) or Bi(Z) are the solutions. Since the wave function has to decay as z approaches + co (because of the potential + eFz), the Airy function Ai(Z) has to be chosen. The energy sp ectr um is continuous since the potential is not bounded as z ~ - 00. Therefore , the (real) wave function satisfying the normalization condition

f oo dZcPE)z)cPEzz( z) -

=

8(E z 1

Ez. 2 )

-

(13.2 .7)

00

for a continuum spectrum is

(13 .2.8) To prove that cI>£.C z ) satisfies the normalization condition, we use the integral repr esentation [27] of the Airy function :

(13.2.9) Therefore,

f O': Ai(t

.'

- a[)Ai(t - a:J dr

-x

(13.?10) where the identity

I

cl !

7.

.. .

f .

t' i f ';. i . : ) ! .o=.

:2

tt ;:; (

I\.

-7-

l:'

I

i.~L ECT '~ OA BS O P.PT i ON

MO D ULAT O RS

has b een use d. Us in g t

= ( 217l ,.. h:

eF) z 1/ 3

and

in (13.2.10), we obtain the normalization condition 03.2 .7). 13.2.2

Summation of the Density of States and Absorption Spectrum

Since the qu antum number is determin ed by (k .\., k y , EJ as described above in the wave fun ction (13 .2. 2) a nd th e co rres pon d ing energy spectr u m 03 .2.4), th e su m ove r a ll th e sta tes n for th e absorption spectrum in 03 . 1.24) has to b e repl a ce d by the s um over all th e q ua n tum numbers :

I: -I:I: f dE z n

k, k ,

wher e th e sum ove r th e energy E, is a n in tegral sin ce E ; is a co n ti n uo us spe ctr u m and a delt a func tion norm alized rul e 03.2.7) ha s be en a dop ted . Therefore ,

+

E,

+ £8 -

til" ]

( 13 .2 .12) wh ere Au is given by (13.1.29b ). Since ")

~

L

A «, «,

d 2k

=

2/ (21/ r

I")

-

m -;'J /

7T

tz-

dEl

(13 .2. 13)

w he re £ 1 = t1 2 (k.; + k; J1 2m r = tl 2 k;1 2m r i we ca rry ou t t he integration ove r £ 1 with th e de lta fun cti on a nd o b ta in the exp ression fo r the abso rp t io n coefficient:

L~:?

rR,-\N?,---KELDYSd

!~Ffl~CT

549

Let /-.) 7V) tr er :

hAF

=

(

\

-7---)

1/3

E g - hco

T=

zm ,

t.o F

( 13.2.15)

We find the absorption coefficient

( 13.2.16) where Ai'( y]) is the derivative of Ai( y]) with respect to 7J. It is interesting to show that in the limit when F -~ 0, we have

p~ [rrfilo F fAi 1( T) dT] ~

Jhw - s,

for hw > E g as expected, since a( fzw)

~ ao( F---.O

trw)

1

=

2 l?1,.

A O - - 2 ( ~-2 ) 217 Ii

3/ 2

~_ h o: - E g

(13.2.17)

Notice that the prefactor A 0 depends on the bulk momentum-matrix element Ie . PeLf = /VI;, which can be determined experimentally [24] by fitting the measured absorption spectrum' 'th the above theoretical results with F = 0 and F O. The Franz- Keldys.. absorption spectrum (13.2.16) is plotted schematically in Fig. 13.2b (solid curve) and compared with the zero-field

*

(a)

Ev

Figure 13.2. (;\) Frunz Ke ldysh effect o r phl)tu-as;~i~ill';cl abso r ptto n In d bulk semiconductor with a uniform electric l'tdd bias. (hi .'\l)~urpti'.)n spectrum ki! a n:1ii2 fIeld F,* (l (solid curve), 'Ihe dashed line is the free electron and [Kilt: ubsorpt ion spectrum ',Vii 1;C'L:i an applied electric field.

::L::T j'RU/\BSOR,'TION

550

MODl..JLATORS

spectrum (dashed curve) using (J 3.2.17). It shows the Fr anx-Keldysh oscillation phenomena in the absorption spectrum above the band gap and the exponentially decaying behavior below the band gap.

13.3

EXCITON EFFECT [4-9, 10-15]

When we consider the Coulomb interaction between the electron and the hole

VCr) -

(13.3.1)

the wave function ¢J(d satisfies the Schr6dinger equation for the hydrogen atom

[

~ V2 + Zm ,

-

v(r)]¢(r) = E¢(r)

(13 .3.2)

The solutions ¢(r) for both the three-dimensional (3D) and the two-dimensional (2D) cases have been studied in Section 3.4 and Appendix A and the results are tabulated in Table 3.1

13.3.1

3D Exciton [4, 5]

We use the most general formula (13.1.24) for the absorption coefficient:

(13.3.3a) n

(13.3.3b)

where the summation over n includes both the bound and continuum states. The' wave functions ¢n(r) have been normalized properly for both bound and continuum states as discussed in Sections 13.1, 13.2, and 3.4. 'For bound state contributions, we have the oscillator strength

, TJ

,

a ~ n'

au =

- - - , --, -

the exciton Bohr radius

J .1.:,

EXCITON EFFEC"-

55!

and the exciton binding energy

L

/I

=

Ry =

In r e 4 7

2f1-( 47/E s )

2 =

the exciton Rydberg energy

Therefore,

(13.3.4)

where E

=

hw - E g

(13.3.5)

Ry

is a normalized energy measured from the band gap E s: For the continuum-state contributions, we obtain

(13.3.6)

where the first bracket is AOp;D( E = lu» - E g ) , and the secone! bracket is called the Sommerfeld enhancement factor for the 3D case [4-9, 28]: ( 7/

I y';) e 'TT/ If'-

sinh( tr II~') 2rr1ft 1 - c-::'.T,-//i

As

E -7

C/..!,

(13.3.7)

we finel ,'I (l

--'

..::'---c;---~---~ ( ~/ E ,':11' ..

R y tt ;) ,

-+-

Tf), ( 13.3.8 )

552

t:L=.C iT.O t\[)SO RFT TON MODULATO RS

wh ich app ro ach es the 3D jo int den sity-of-stat e s rin an in terba nd tran sition wi thou t the exci to n effec ts, (fuu - E g ) / R y , p lus a consta n t 7i'. As E ----7 0,

V

( 13 .3 .9)

which gives a finit e valu e in ' contrast to th e va n is hi n g r esult of the interb and a bso rp tio n at lu» = E g • The total a bs orp tio n due to both th e bound and co n tin uu m states is given by

~ ~ [4'iT 0

') . . ,. -'-R Y a:0 _oJ

+

f:

~ 0( tau R- E

g

n =l /1

S JD

Iz W .- E (

+

y

R

S

~ 1 '1 -

1 / tl
\

y

y

If we include the fini te linew id th due to sca tte rings by repl acin g th e d elta fu nction by a Lo re ritzia n fu nction , 8(x) = (Y/ 7T )/( X2 + y2), whe re y is t he h alf-lin ewidt h normalized by R ydbe r g if x is a normalized e nergy, we find

a (tzw )

4L n=l

(13.3.11 )

13.3.2

2D Exci ton [14, 29]

.

.'

The abso rp tion spec tr um for a two-d im en sion al str uc t u re with exciton effects can also be ob ta ine d usi ng (13. 1.24):

(13.3.11)

For bo und s ta te co nt rib u tio ns . we usc

r;'

L

li

_ -

.,

(n - 2"I ) -.

( 13.3 . 13)

n ,:,

EXCITON EFfECT

553

and obtain

(13.3.14)

where e

.

(tzw - Eg)jR y again. The continuum-state contributions give

=

-

(13.3.15)

. . ..

where the two-dimensional Sommerfeld enhancement factor is

5 20 ( £ ) =

2 1+e- 2 rr/

----=

(13.3.16)

.je

The total absorption is the sum of asUzw) and ac(flw):

a( tzw)

=

1 [ + 1] + S( } (13.3.17) i)

L

A 0 2 {OC 4 3 8 e 27TR ya o n=l (n - ~)

s)

2

(n -

Notice that without the Sommerfeld enhancement factor, 5 20 ( £) is set to 1 and a /tzw) = A oj (27TR ya~). If we include the 'finite linewidth effect, we have

A a(tzw) -

-

-

0----=2 ya 27TR o

f

l

4

co

L n=l

.

1

y 3 ---------

7T(n - t) [

c

t +

+

1 1 2] 2

(n - 2)

de'

0 --;;:-

+

1' S 2D ( e')

y2

1 (13.3.18)

(£'_<)2+1'2j

ELECTVUAB~ORPTION

554

MODULATCRS

Table 13.1 Absorption Coefficients due to Exciton Bound and Continuum States 2 ./-E 1 ./- ') I m r e" e = nco - 'g AD = 7Te e' PCl' a = ~ I 7TE s R = 2 () 2 y 2 2 Ry n,CEoWln o m, \ e 2h (47Te s)

21

4)

A

Bound States

Continuum States

Two-Dimensional Exciton ZERO L1NEWlDTH

FINITE LlNEWIDTH

A0 2 'IT" R y a 02

1'" dc' 0

7T

'Y S 2 D ( c ' ) (

, _ e )2 l + '2 e

Three-Dimensional Exciton ZERO LlNEWIDTH

Ao I:=

1

11

(

~

-rra 0 n

3)_1 0(£ + ~) Ry

n

FINITE LlNEWIDTH

AD

L cc

(

n=l

2)1 3

»«:»

3

'Y/

-

rr

n, (e + 1- ) 7

n:

Ao

2

+1'

2

[X; dE' /,,1-;' S3D(e')

27T2Rya~ 0

'IT"

.

(e'-c)2+'Y 2

The above results for both 2D and 3D excitons are summarized in Table 13.1. The absorption spectra for a finite linewidth and zero linewidth are plotted in Fig. 13.3 for comparison .

13.3.3

Experimental Results for 3D and Quasi.2D Excitons

Experimentally, the linewidth y is always finite and increases with temperature. For example, in Fig. 13.4a, we show the absorption spectra [30] of a bulk (3D) GaAs at four different temperatures, T = 21, 90, 186, and 294 K. We can see that the exciton linewidth is broadened with an increasing temperature. The absorption edge has a red shi ft since the GaAs bandgap

555

13.3 EXCITON EFFECT (a) 2D(y= 0) a(fzw) 1s

(b) 2D(y

* 0),

a(fzw)

with exciton /effect

2s k - - - - - - - - - . without exciton effect ~E --4R y g

fzw

(c) 3D(y = 0)

L...-----"~-+-----_..1iw

(d) 3D(y ~ 0)

a(liw)

a(fzw)

1 with exciton s~effect

J

~

..

,,~. '>-.th

, WI

. ff out excIton elect fzw

'--~,-+-+---------,~fzw

E -R E g g

y

Figure 13.3. Absorption spectra for a two-dimensional (2D) exciton with (a) a zero linewidth and (b) a finite linewidth, and a three-dimensional (3D) exciton with (c) a zero linewidth and Cd) a finite linewidth.

decreases with increasing temperature [31]:

,

T+b

(eV)

(13.3.19)

\, \

where Eg(O) = 1.519 eV, a = 5.4~5 X 10- 4 eV/ K, and b = 204 K. For comparison, we show the exciton absorption spectra [32] of an Ino.53Ga0.47As/Ino.52AI0.4sAs' (lattice-matched to InP substrate) quantumwell structure at different temperatures in Fig. 13.4b. The quantum-well structure has a quasi-two-dimensional character since the electron and hole wave functions are confined in the z direction with a finite well width instead of being restricted to the x-y plane as in the" pure" 2D case. It is expected that the binding energy of the Is exciton in the quasi-2D structure will be between the 3D value (,= R y) and 2D value (= 4R). Furthermore, we also see the splitting of the 'heavy-hole (HH) exciton and light-hole (LH) exciton in a quasi-2D structure, while we do not observe the HH and LH exciton splittings in a bulk GaAs sample because of the degeneracy of the HH and LH bands at the zone center of the valence-band structure. The energies and absorption spectra of the HH and LH excitons are further investigated in Section 13.4. I

~·~LECTR~)'\BSORrTION

55G

i·vt{,DUJ.f'.TORS

1.2

-.

1,1

-;

E

..qu 0 .......

'-"

ts

09 0.8

l

.. 0

0.7· . 0.6

I

!

1.42

1.44

1.46

1.48

1.5

1.52

1.54

1.56

Photon energy (eV) (a)

1.8

InGaAs/lnAIAs UNDOPED

1.6

I!

::::>1.4

I .:

uJ

1\::

~ 1.2

-c rn a:

"i~'~/;~ba"""7,'~'"

1.0

~ 0.8

_ I :.

-.,.,......

. '.

m

I;

-c 0.6

I~

0.4

Ij

/'

iE : ~

:I;

.....""'..,.,-.: ...

:1

o o

0.2

0.80

0.90

1.00

c

100. 200

300

T (K)

_ _L-_-lJ

~ - ~::--~:-:---:-~_----l....

0.70

.

QJ

1.10

1.20

1.30

ENERGY (eV)

(b)

.'

Figure 13.4. (a) Band-edge absorption spectra of a bulk GaAs sample at T = 294 K (circles), 186 K (squares), 90 K (triangles), and 21 K (dots). (After Ref. 30.) (b) Absorption spectra of an Ino53Gau..n As/ Ino.52Al0.48As quantum-well sample at T = 300 K (solid curve), 100 K (dashed curve), and 12 K (dotted curve). The insert shows the half-width at half-maximum (HWHM) of I the first absorption peak as a function of temperature. Squares are measured data; curve is calculated. (After Ref. 32 © 1988 IEEE.)

The insert in Fig. 13Ab shows the measured (squares) half-width at half-maximum (HWHM) of the first heavy-hole absorption line as a function of temperature. The solid line is a fit to the expression

r '2 ( =

HWHM) =

[0

r ph

+ exp

nw L o

( kBT

(13.3.20) ) .

1

13A

QUANTUM CONFINEL1 STARK EFFECT (OCSE)

557

where f o = 2.3 me V accounts for the inhomogeneous broadenings such as scatterings by interface roughness and alloy fluctuations, and the second term represents the homogeneous broadening due to InGaAs longitudinal optical (LO) phonon scatterings with nw L O = 35 meV and r p h = 15.3 meV. 13.4

QUANTUM CONFINED STARK EFFECT (QCSE) [11-13,15]

In this section, we consider the exciton absorption in a quantum-well structure in the presence of a uniform applied electric field. The effective mass equation, similar to 03.1.5), can be written as [12] 2

(He - Hh + Eg

-

41T6 s

le

re

-

rh

I] cP(r

e,

rJJ

E(re , r h )

=

(13.4.1)

where

H= e

(13.4.2a) (13 .4.2b)

The electron potential Veere) or the hole potential Vh(r h) may also include the effect of the electric field lelF . r , or -lelF . rho The interaction between the electron and the hole is due to the Coulomb potential. Let us assume that the quantum well is grown along the z axis and the uniform electric field is also applied along the z direction, the elcstron and hole potential can be written as (Fig. 13.5) (13.4 .3a) ,

(a) F

=0

\

(b) F> 0

. Eel hI ~--~-+~--------------------

---...~""---~

Figure 13.5. Quantum-well energy subbands and wave functions (a) in the absence of an applied electric field, and (b) in the presence of an applied electric field.

:)58

ELE(:TROAl'SOR.PTION MODULATORS

and (13 .4 .3b) 13.4.1 Solution of the Electron-Hole Effective-Mass Equation With Excitonic Efl"ects

Using the transformations for the difference coordinate and the center-ofmass coordinate systems for the x and y components, (13.4.4) where M = m; + m'J:, and p = xX + procedures as in (13.1.8) to 03.1 .15)

yy,

we obtain after following similar

Since the dependence on R t comes from only the leading kinetic energy term, the solution can be written as [33] eiK,'R,

~(re' r h ) =

..fA

F(p, »..

Z/J

(13.4.6) .'

where the exciton envelope function F(p, ze'

Zh)

satisfies

(13 .4.7)

The center of mass of the electron-hole pair is thus moving freely with a kinetic energy h 2K/2j(2M). Here, identical relations to (13.1.8) to (13.1.11), for the parallel components of the real-space and momentum-space vectors exist, e.g.,

(13.4.8)

13.4

QUANTUM CONFINED

~TARK

EFFECT CQCSU

5.'19

Note that -H(

h2

ZII) =

-

d2

~- - -

2m~ dz~

+

V (z ) h

h

(13.4.9) which are simply one-dimensional Schrodinger equations for a particle-in-abox model. Let us consider

where In(ze) and gm(zh) are the free-electron and the free-hole wave functions in the absence of any interactions. With the Coulomb interaction term, the solution F(p, Ze' Zh) is more complicated. However, using the completeness properties of the solutions {In(ze)} and {gm(Zh)}, we can expand the exciton envelope function as [12, 15, 19] (13.4.11) n

m

The Fourier transform pairs for the p dependence can be written as

(13.4.12) The envelope function

ri« ze' Zh)

becomes (13.4.13)

The complete envelope function is obtained from (13.4.6), noting that all possible K( can be included:

Comparing (13.4.13) and 03.4.14) with 03.1.2) and 03.1.3), we find that the

ELFCT[.z()r\USGRPTION MODUL,\TORS

original electron-hole pair state can be written as 1,8, 9, 15, 34]

'It(re,r,J

L L

=

ketn

Gllm(kl)~Jnk)re)ljJm-kil/rlr)

(13.4.15a)

k/llnt

eik ., 'p ,.

fA

(13.4 .15b)

fn(ze)ucCre)

e:-ikllt 'PiI

IA

( 13.4.15c)

gm(zh)u u(r h)

13.4.2 Optical Matrix Element for Excitonic Transitions in a Quantum Wen The optical matrix element is obtained using Ii) = [ground state ), IW(r e , r;), and

(fie'

pli) =

LL

LQ;;~n(kt)(ljJflkJr)le . PIl/lm -kJr)

mn k e l

kilt

If > =

= L LC:m(k{)e . Pcl..Inm nm k,

(13.4.16) nm

where lnm = { ':_ oof,7(z)gm(z) dz, and PeL' = (u/r)lpluL,(r). The above matrix elemen t can be expressed al terna tively in terms of F( p, z e> Z h):


pli) =

fex-c-

dzF*(p = O,z,z)v'/ie' P ee

(13.4 .17)

oe

Noting that K{ = kef + kill = 0 from (13.4.16), we substitute the expression (13.4.11) for F(p, ze' z,,) into (13.4.7), and obtain -(E-E8

) ]

( 13.4.18) Multiplying the above equation by f,:( ze) and g,~(z It) and integrating over ze and Zh' we obtain

(13.4.19)

1:<.4

QUANTUM

«xss:

CONFINEDSTAI~KF.FFECT

' 501

where

(13.4 .20) (13.4.21) We have ignored coupling among different subbands so that only n' = nand m' = m are taken into account in (13.4.20). The wave function 4>nm(P) satisfies the normalization condition in a two-dimensional space: (13.4.22 ) This was derived using (13.4.23)

(13.4.24 ) t

\

13.4.3 i, Variational Method for Exciton Problem

.'

Two different methods are commonly used to find the solution for the exciton , equation (13.4.19). The most common approach is a variational method, which is very useful to find the bound state solution. Noting that the Is state solution of ¢(p) behaves like e -p iau or e - pI(2 a o ), the following variational form is assumed in the variational approach [12J:

<4>1 - (h 2/2m r ) \lp2 <4>14»

-

V(p) I4»

(13.4 .25)

where

4>(p)

· ( 13 .4 .26)

' ::L ~:CT 1{ \./ .\ U S O R f' T I O N MODULATOR~

<4>14> > =

which satisfies the normalization condition

1 in 03.4.22). We find

It is convenient to write in terms of the normalized parameters

ao =

(13.4.28)

and the normalized exciton binding energy

E ex

C ex

=

R

(13.4.29)

y

(13.4.30)

where the function G(x) is defined as an integral .'

\ '.

1o dt (t

te- t

00

G(x) =

2

+ X

2

1/2

(13.4.31)

)

which is a smooth monotonic function and can be approximated [55] analytically. It has th~; properties that GW) = 1 and G(oo) = O. Typically, for a quantum-well problem, Ize - zhl is finite and the argument in G(x) is over a finite range. The pure two-dimensional limit can be obtained by ignoring the z dependence and using G(O) = 1, E ~x

Thus we find the minimum at exciton binding energy.

13

=

13 2

= 2 and

-

4{3

B ex

(13.4 .32)

= .- 4 as expected for a pure 2D

13.4

QUANTUM CONF1NED STARK EFFECT (QCSE)

13.4.4

563

Momentum-Space Solution for the Exciton Problem

Alternatively, the real-space exciton differential equation (13.4.19) can also be written in terms of the momentum space integral equation [IS, 35]:

where

A direct numerical solution of the above equation is discussed in Ref. 15. 13.4.5 Optical Absorption Spectrum with Exciton Effects in Quantum Wells

The solution to the eigenvalue equation (13.4.19) is a set of exciton binding energies and corresponding eigenfunctions for Is, 2s, 3s, and continuum states. We denote the quantum number for the exciton state as x. The absorption coefficient for' a quantum-well · structure can be obtained by substituting the matrix element (fie· pli) into (13.1.1)

(13.4.35,) where the exciton transition.energy is (13.4.36a) ,

and the band edge transition energy is en E hm

--

E g + E en

-

E hrn

(13.4.36b)

The exciton binding energy Eel< is a discretized set of Is, 2s, 3s, .. ' . states and continuum-state energies. Assuming that there is no mixing between different subbands, consider only the pair n = Cl and m = HH1, for example. We

. -...-f"'"T:"t'-

'I ~ """'

__

-- -....;...,;,,--:o"""._._ ~ . ,.

• ..:_ - ."!"',- .., ._ -__

'

~

- . -.o)W>


!"".

"...c._,

~ . ~,. -

....,..".•.

~ '~. ~

_~

••'":"'

~ ~ _ ,.-:'I

I;'Y',

.. -

_. ~

••...,.-,.....-: •••-.,'

,, - ~

._

,., . _ . ~4' . ,

__

' _~ . ~

CU : ...:::T ROABSO Rf T IO r-; MODULATORS

564

m a y drop th e sum ma tio n over nm a n d tr eat e ach pai r of nand m independen tly. This ass um p tio n is va lid only if th e subba ri d e nergy differ ence is much la rge r than th e e xcito n binding e nergy . T he a bso rp tio n coeffi ci ent becomes

2 a(llw ) = C o L

L I4
-

liw)

(13 .4.37)

x

Exciton Discrete ( Lsl -S ta t e Contribution. For the Is state , we find

( 13.4.38)

.

-_..

where a finite lin ewidth 2 y h as b e en as sum ed and the delta function is replaced by a Lorentzian function and x = Is state. The matrix e le m e nt Ie' Pal is ob ta ined [15, 36] from (9.5. 20)-(9 .5.23) in Secti on 9.5 .

For TE polarization ( i = i or

y) Heavy- hole exciton Light-hol e ex citon

(13 .4.39)

For TM polarization ( i = i) Heavy- hole exc iton Light-hole exciton where

Ml

(13.4.40)

is the bulk matrix element discussed in Chapter 9: ma

- 6 Ep

( 13.4.41)

Exciton Continuum-State Contributions. For continuum-state contributions, the wave fu n ction !
2"

~ .\

a n d E ex ~

=

2" " -- A ~ ~ k, t: .v

m y;tzr2

Jd E

"{

(13.4.42)

tz 2 k {2I /(7 m r ) = E { . We obtain the absorption coefficient due to ~

......--

~

_ .. __ "'4••• • ••

~ ••

1

13.4

QUANTUM CONFINED STARK LFFECr (QCSS)

the continuum states

where Ex = E~':n + E, and l¢x(O)/2 is usually approximated by the Sommerfeld enhancement factor for a 2D exciton:

(13.4.44 ) where 1

~ So

< 2. For a pure 2D exciton,

So =

2 and

(13.4.45) The matrix-element M(E t ) is defined [15, 37, 38] as Ie . Pcvl 2 = M(E t)Mb2 , which has been derived in Section 9.5 and is tabulated in Table 9.1. TE polarization

Heavy-hole excito. Light-hole exciton

( 13.4.46)

TM polarization

Heavy-hole exciton Light-hole exciton

(13.4.47)

where cos? ()nm = (E en + \EhmD/(Een + IEhml + E). The heavy-hole exciton contribution to the TM polarization is taken to be zero instead of (~)sin2 ()nm because a rigorous valence-band mixing model shows that the heave-hole exciton has a negligible contribution to the TM case [39-43]. Note that k ; = l/a o and

----

(13.4.48)

E L E ::T K OA B ~ O R P - r I O N

56 6

MODULATORS

Total Absorption Spectrum. The complete absorption spectrum can be written as the sum of the bound-state and continuum-st ate contributions [15]:

(13.4.49) Let us look at the band-edge transition energy

E}~;~:

(1304.50) for the electron subband n and the hole subband m in the presence of an applied electric field. Note that th e subband energies are measured from the band edges at the center of the quantum well (positive for electrons and negative for holes). 13.4.6

Perturbation Method

A simple second-order perturbation th eory shows [44] that (see the example in Section 3.5 and its references)

/ 13 .4.51) in an infinite quantum-well model assum ing an effective well width of L efr . Therefore, the band-edge transition energy for the first co nd uc tion and the first heavy-hol e subba n d is determined by

( 13.4 .52) where C 1 = -2.19 X 10- 3 . As an example, for ~ GaAs/AlxGa1_xAs quantum well with an effective well width of 100 A, the above change in the transition energy is E~}(F) - E/~t(F = 0) = -U.S meY at F = 100 "kY/cm assuming that m : = O.0665mo and m~h = O.34m o' The exciton binding energy E c x also depends on the electric field strength. In general, the band-edge

13.4. QUANTUM CONFINED STARK EFFECT (QCSE)

567

a

2 1 ------l..-~C-..---___'_:7"""==----.~-.=.-------~

tu»

Figure 13.6. Exciton absorption spectrum of a Quantum-well structure in the absence of an applied electric field. The contributions due to the discrete and continuum states are shown separately as dashed lines.

shift due to the electric field is quite clear and appears stronger than the binding energy change with the electric field. Variational methods instead of the above perturbation method for the band-edge energies have also been used [44-47].

13.4.7 Exciton Absorption Spectrum and Comparison With Experimental Data

Theoretical absorption spectrum for a single pair of transitions from the first conduction subband to the first heavy-hole subband is shown in Fig. 13.6, where the discrete Is bound state Ex = E i S contributes as a Lorentzian spectrum, and the steplike density of states enhanced by the Sommerfeld factor contributes as the high-energy tail. With an applied electric field, the quantum confined Stark effects can be measured from the shift of the peak absorption coefficient as a function of the applied electric field. Figure 13.7 shows the polarization-dependent optical absorption spectra for (a) TE and (b) TM polarizations [17, 48] with estimated electric fields [13]. Theoretical results using the parabolic band model [15] presented in this section and using a valence-band mixing model [43] have been used to successfully match these experimental data, as shown in Fig. 13.8. To understand these data, we make the following observations; 1. The exciton absorption peak energy depends on the band-edge transitiop energy Ehi(F), which shifts quadratically as a function of the field, minus the amount of the Is state exciton binding energy E 15 , which is about 8 meV for the heavy hole exciton. The transition energies of the heavy-hole and light-hole exciton peaks vs. the applied electric field are shown in Fig. 13.9. The binding energy for a bulk (3D)GaAs R y is

..

'''~''--

;

/

ELECTt{ Or\ BSO RPT IO N MODULATORS

568 4 ___.----

(a) TE

2

-z 0

en en :E

en

I

0

Z

(b) TM

< a:

t-

c

~

I

2

o~~:::;;;=-.--.:::~6

1.42

1.46

~

1.5

PHOTON ENERGY (eV) Figure 13.7. Experimental absorption spectra of a GaAsjAl O.3Ga O.7 A s quantum-well wavegu ide modulator as a funct ion of field : (a ) TE polarization for (i) 0 kY/ em, (ii) 60 kY jem, (iii) 100 kYjem, (iv) 150 k Vy cm. (b) TM polarization for (i) 0 kYjem, (ii) 60 kYjem, (iii) 110 kYjcm, (iv) 150 kYj cm, and (v) 200 kYjcm. (After Ref. 48.) [13,17].

about 4.2 meV and is 4R }' = 16.8 meV for a pure 2D exciton. A quasi-two-dimensional quantum-well structure gives a binding energy corresponding to an " effective dimension" between 2D and 3D. Therefore , the binding energy of the quantum well is somewhere between 2D and 3D. As a matter of fact, since analytical solutions for the hydrogen atom equation in an a-dimensional space have analytical solutions for an integer a such as 2 and 3, the idea is to extend the general result for a given a using the concept of analytical continuation. Using this approach, the oscillator strength for the bound and continuum states can

13A

QUANTUM CONFINED STARK EFFECr (QCSE)

569

(a) TE

C

W

-< N

.J

:E

a:

o z z o

i=

a. a: o en 1'C

<

1420

1440

1460

1480

. PHOTON ENERGY (meV) Figure 13.8. Theoretically calculated absorption spectra for the quantum-well waveguide modulator in Fig. 13.7: (a) TE polarization, (b) TM polarization. (After Ref. 43.)

be obtained using the analytical formulas once the "effective dimension" is determined. The effective dimension can be extracted by comparing the binding energy calculated variationally, as discussed in this section, with the analytical formula for binding energy·[49]. 2. The TE oscillator strength for the heavy-hole exciton is approximately three times that of the light-hole exciton. However, since the light-hole exciton transition energy is already in the continuum states of the heavy-hole transition, the spectrum would not show a 3:1 ratio. 3. The TM polarization spectra show that it is the light-hole exciton transition which is dominant for this polarization as a result of the optical momentum matrix-selection rule.

}.LECTRO ;\BSDRPTION MODULATORS

5"70 1480

'-,--I-.-----r--r'

I

0

1470

:x- - --x

' - - '?+O

-x ,

--->

1460

E ..-

1450

>-

o

a:

W Z UJ

,

'>4-q

(1)

X,

,, '0

14 4 0 1430

•

HH (exp.)

0

LH (exp .)

+

HH (theory)

x

LH (theory)

1420 1410 -30

0

30

60

FIELD

90

120 150

180

(kV/cm)~_

Figure 13.9. Comparison of experiment al and theoretically calculated exciton energies vs. the electric field. (After Ref. 43 .) .

4. At a fixed optical energy tz w of the incident light, the absorption coefficient can change drastically especially when h co is near an exciton peak absorption energy. This enhanced change of the absorption by an applied voltage is discussed further in the next section.

13.5

INTERBAND ELECTROABSORPTION MODULATOR

The electroabsorption modulators can be designed [l O, 50-56] using a waveguide configuration (a) or transverse transmission (or reflection) configuration (b). Consider the modulator structures shown in Figs. 13.10a and b. Suppose the operating optical energy fzwo is chosen near the exciton peak when a voltage V is applied (Fig. 13.10c). The transmission coefficient is proportional to (13 .5.1 ) . For the waveguide modulator, a(V) is the absorption coefficient of the waveguide region multiplied by the optical confinement factor I', and L is the total length of the guide for the waveguide modulator. For the transverse transmission modulator, a(V) is the average absorption coefficient of the multiple-quantum-well region, and L is the total thickness of the MQW region. The couplings or reflections at the facets are ignored here for

13.5 INTERBAND ELECTROABSORPTION MODULATOR

(a)

571

(c) a(1ico)

a(V)

C::::=~====~tt(V) = T(V)Pin

aCO)

noo (b) Transverse transmission modulator

(d)

Vet)

~Pin

~

----,/----L..--'------'---'-----l...---+t

Vet)

-

I

~

~

Pout ~ T(V) Pin

Figure 13.10. (a) A waveguide electroabsorption modulator. (b) A transverse transmission electroabsorption modulator. (c) The absorption coefficient a.(hw) of a quantum-well modulator at two different bias voltages, V and 0, for example. (d) With a bias voltage vet) across the modulator, the transmitted optical power (solid square wave) is modulated. Dashed line is the input power Pin'

convenience. The on/off ratio (or contrast ratio) Ron/off 13.10d) Ron/off

=

Pout (Von p (V out off

= =

IS

defined as (Fig.

T(O) 0) V)- T(V)

(13.5.2a)

or in decibels Ron/off (dB)

= =

T( Von = 0) 10Iog----T( Voff = V) 4.343[ a( V) - a(O) ] L

(13.5 .2b)

Therefore, in principle, the magnitude of the extinction ratio or the on / off ratio can be made as large as possible by increasing the cavity length L. However, we note that the insertion loss Lin is defined as Lin

Pin =

Po ut ( 0) Pin

=

1 - T(O) = I -

e-o:(O)L

(13.5.3)

EJ ECTR(lAHSOEPTION MODULATORS

572

at the transmission (on) state. Since nCO) is always finite , a large cavity length L will decrease the transm issivity T(O) exponentially. We may achieve an

infinite on / oft ratio using an infinitely long cavity and obtain no light transmission even for the on -state. If T(O) -;, 0, the insertion loss approaches 100%. Naturally, this is not desirable and an optimum design, which maximizes the extinction ratio and minimizes the insertion loss, is necessary. Note that Von is set to zero here only for illustration purposes. For high-speed switching, it is usually set a nonzero reverse bias voltage for fast carrier sweep-out of the absorption region. Another useful figure of merit in the design of the electroabsorption modulator is the change in absorption coefficient per unit applied voltage. a ( VoCf )

a ( Von)

-

(13.5.4 )

~V

where the change in voltage ~ V = IVon on / off ratio per unit applied vol tage is Ron /off ----'------ =

~V

4.343

-

V:>ff I.

This occurs because the

[a(V) -a(O)]L

6V

6a = 4 .3436F

(13 .5.5)

where the electric field across the multiple-Quantum-well region is approximately given by F = V/ L, if the built-in voltage Vb i is ignored. (Otherwise 6 F = (V - Vb) / L has to be used .) For more discussions on the figure of merits, such as ~a/(~F)2, C(~V)2, and the optimization of the contrast ratio, drive voltage, bandwidth, and total insertion loss, see Ref. 56.

13.6 SELF-ELECTROOPTIC EFFECT DEVICES (SEEDs) [18-20, SO-56]

Interesting optical switch devices such as the self-electrooptic effect devices have been demonstrated using the quantum confined Stark effects. These devices show optical bistability and consist of a multiquantum-well p-i-n diode structure with different possible loads, such as a resistor (R-SEED), a constant current source, or another p-i-n multiple-Quantum-well diode (Symmetric or S-SEED). In this section, we discuss basic physical principles of an R-SEED. As shown in Fig. 13.11a, the circuit equation is given by Kirchoff's voltage law:

(13.6.1)

13.6

SELl- -ELECTROOPTIC EFFECT DEVICES (SEeDs)

(a)

573

(b)

R

+

~

V o == V + R I R

Output Light Pout

(c)

= T(V) Pin (d)

Pout == T(V) Pin B

H

o

:>P·in Figure 13.11. (a) ' A circuit diagram for a self-electrooptic effect device with a resistor load (R-SEED) in the presence of an incident laser light at the in pu t Pin ' (b) A graphical solution for the photocurrent response JR = S(V)Pin + Jo(V) and the load line Vo = V + RJR,' (c) The transmission reV) of the laser light passing through the MQW p-i-n diode is plotted as a function of the voltage drop V across the diode. (d) The switch diagram for the optical output power vs. the optical input power with the arrows showing the path of switching.

where V is the voltage drop across the diode. The reverse bias volt age V and the reversed current JR are defined as shown in Fig. 13.11a. For an incident .laser light with a power Pin, the Current JR is the sum of the photocurrent S(V)P in and the dark Current JD(V): ! ( 13.6.2) where S( V) is the responsivity of the diode and is defined as TJ int A

S(V) = q - fzw

(13 .6 .3)

where (13.6.4) is the absorbance, that is, the fraction of absorbed power for a unit incident power, assuming a single path absorption. The absorbance can be derived by noting that the reflectivity at the front surface R p , plus' the transmission after

574

ELEC·'i=(OABSORPTION MODULATOFS

passing through the substrate (I - R p )( ] - R,) e -aL, plus the absorbance A, must equal]. At steady state, the voltage drop across the multiple-quantumwell diode V and the current lR is determined by the simultaneous solutions of the above two equations. A graphical illustration of these two equations is plotted in Fig. 13.11b, where the intersection points stand for the possible solutions for I R and V. As can be seen from Fig. 13.11b, when the optical input power Pin is increased, the photocurrent is irtcreased , in general, and the number of intersection points varies. For an optical power equal to P.In, }, we have two intersection points A and E with corresponding voltages VA and VE . Therefore, there are two possible output states: PO U I = T(VA)P in, " which is higher in output power, and POU I = T(VE)P in l' which is lower in output power because of the transmissivity T(VA ) > r(VE ) (Fig. 13.llc). Once we increase the optical input power to Pin, 2' we find that there are three intersection points, B , D, and F, with corresponding voltages VB' VD , and VF . Therefore, the optical output powers through the diode are given by three possible values: Pout = T(VB)Pin, 2 > T(VD)P in, 2 > T(VF)P in, 2 ' Increasing the optical input power to P in , 3 , we have only two intersection points, C and G, with corresponding voltages, Vc and . Ve . Therefore , we obtain the output powers Pout = T(VC )P in, 3 > T(VC )P in , 3 ' The switching curve and the directions are shown in Fig. 13.11d. At a small input power, Pin < Pin, l' the optical transmission power Pout = T( V)?in is monotonically increasing with the input power P in' since the photocurrent is small for a small incident optical power. Therefore, most of the reverse bias voltage is across the diode and V is close to Va (VA < V < Va)' In this range of voltage, the transmission coefficient is rather flat or monotonic. As the input optical power exceeds Pin, l' P in. 2 , and P in, 3 ' photocurrent will appear in the circuit, and the voltage drop across the diode will drop from Vc to Ve , where the transmission will drop from a high value T(Vc ) to-a low value T(Ve ). After Pin> P i n , 3 ' the output power starts to increase again, since Pout = T(V)P in increases as Pin increases. This switch sequence is therefore A ~ B ~ C ~ G ~ H. In the reverse direction, if we switch down the optical input power from a large value of Pin at H, it will go through H ~ G ~ F ~ E ~ A ~ 0, since at point E the voltage has to switch from VE to VA as we decrease the input power Pin ~ Pin i- Therefore, the optical output power will switch from a low state E to a high state A due to the large transmission coefficient T(VA ) > T(VE ) ·

,

Symmetric self-electrooptic effect devices [53] (S-SEEDs) using another p-i-n multiple-Quantum-well diode as the load, and field-effect transistor self;.. electrooptic effect devices (F-SEEDs) [20] have also been demonstrated to show interesting physics and applications. For example, the S-SEED can act as a differential logic gate capable of NOR, OR, NAND, and AND functions. These devices made by maximizing the ratio of the absorption coefficients in the high and low states while minimizing "the change in electric

PROBLEMS

575

field can give nearly optimum performance [53]. .From the I-V curve of a SEED (Figs. 13.11a and b), we see that negative differential conductivity exists. A SEED oscillator [19, 52] can also be designed by a series connection of a SEED and an L-C resonator circuit and the SEED is optically pumped to produce a negative electric conductance in the photocurrent response. For example, oscillators with oscillation frequencies from 8.5 to 110 MHz have been demonstrated [52]. For the 8.5-MHz oscillator; frequency tuning by changing the bias voltage of the SEED has a tuning rate 16.7 kl-Izy V, This frequency tuning is caused by the change in the capacitance in the depletion layer of the SEED with the voltage change. The capacitance can also be changed optically by changing the optical power coupled to the SEED. PROBLEMS 13.1

In this problem, we use an alternative approach to show the physical interpretation of 21
For convenience, we use the normalized distance ! = r/ ao' the normalized wave number K = k r k ; = ka o ' and normalized energy § = E/R y = K 2 , where a o = 47TE s h 2/ m r e 2 is the exciton Bohr radius, and R y = m re 4 / 2 h 2 ( 4 7T E ) 2 is the exciton Rydberg energy. (a) Show that the solution is of the form

where YtmU~, cp) are the spherical harmonics as shown hydrogen atom case. R(!) satisfies

III

the .'

The solution is the .spherical Bessel function

(b) Show that R ld ( ! ) above satisfies the 8(K - K') normalization rule

57(,

EU.::.cCROAf)SORPTIO;\l MODULATORS

(c) Show that in physical units, the wave function can be written as

satisfying the 8(E - E') normalization rule

where E = K 2 R y = tz 2 k 2 j (2 m r ) has been used. (d) The wave function at r = 0 does not vanish only if t = O. The complete wave function is

Show that the wave function at the origin is

which is exactly the 3D reduced density of states, denoted by p:D(E).

13.2

Derive Eq. (13.1.13).

13.3

Plot the Franz-Keldysh absorption spectrum for GaAs with an applied field F = 100 kV/ ern at T = 300 K.

13.4

Compare the oscillation strengths and the exciton binding energies of the bound states of the 2D and 3D excitons.

13.5

Compare the absorption spectra of the continuum-state contributions of the 2D and 3D excitons.

13.6

(a) Show that for interband absorption in a pure two-dimensional structure without exciton effects, the absorption spectrum taking into account the finite linewidth broadening is given by

where E = (flw - Ec)/R y. (b) Carry out the integration analytically and plot the absorption spectrum a( PI w ).

REFERENCES

577

13.7

Derive 03.4.17).

13.8

Show that C(x) defined in 03.4.31) gives C(O) = 1 and C(co) = O.

13.9

Use' the perturbation result (13.4.52) for the band-edge transition energy E~~(F) to compare with the data shown in Fig. 13.7. Estimate the exciton binding energies for the heavy-hole and light-hole excitons separately. Discuss the accuracy of this simple method.

13.10

Compare the advantages and disadvantages of the waveguide modulator vs. the transverse transmission modulator.

13.11

Discuss the physics and operation principles of a SEED.

REFERENCES 1. W. Franz, Z. Naturforsch, IJ"3, 484 (1958).

2. L. V. Keldysh, "The effect of a strong electric field on the optical properties of insulating crystals," Soviet Phys, JETP 34, 788-790 (1958). 3. K. Tharmalingam, "Optical absorption in the presence of a uniform field," Phys. Rev. 130, 2204-2206 (1963~ 4. R. J. Elliot, "Intensity of optical absorption by excitons," Phys. Rev. 108, 1384-1389 (1957). 5. R. J. Elliot, "Theory of Excitons: I," pp. 269-293 in C. G. Kuper and G. D. Whitfield, Eds., Polarons and Excitons, Scottish Universities' Summer School, Plenum, New York, 1962. 6. R. S. Knox, "Theory of Excitons," in Solid State Physics, Suppl. 5, Academic, New York, 1963. 7. J. D. Dow and D. Redfield, "Electroabsorption in semiconductors: The excitonic absorption edge," Phys, ReL'. B 1,3358-3371 (1970). 8. E. J. Johnson, "Absorption near the fundamental edge," Chapter 6, pp. 153-258 in R. K. Williardson and A. C. Beer, Eds., Semiconductors and Semimetals, Vol. 3, Academic, New York, 1967. 9. J. O. Dimmock, "Introduction to the theory of exciton states in semiconductors," Chapter 7, pp. 259-319 in R. K. Williardson and A. C. Beer, Eds., Semiconductors and Semimetals, Vol. 3, Academic, New York, 1967. 10. T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, and W. Wiegmann, "High-speed optical modulation with GaAsjGaAIAs . quantum wells in a p-i-n diode structure," Appl. Phys, Lett. 44, 16-18 (1984). 11. D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, and C. A. Burrus, "Band-edge electroabsorption in quantum well structures: The quantum-confined Stark effect," Phys. Rev. Lett. 53,2173-2176 (1984). 12. D. A. B. Miller, D. S. Chernla, T. C. Darnen, A. C. Gossard,W. Wiegmann, T. H. Wood, and C. A. Burrus, "Electric field dependence of optical absorption near the band gap of quantum well structures," Phys. Rev. B 32, 1043-1060 (1985).

i

I:

i

I i'I~·

578

c LECTi{OABSORPTION MODULATORS

13. S. Schmitt-Rink, D. S. C hcrnla, and D. A. B. Miller, " Li ne ar and nonlinear optical properties of semiconductor quantum wells," Ado, Phys. 38 , 89-188 (1989). 14. M. Shinada and S. Sugano, "Interband opt ical transitions in extremely anisotropic semiconductors, I: Bound and unbound exciton absorption," I , Phys . Soc. Jpn , 21, 1936-1946 (1966) . 15. S. L. Chuang, S. Schmitt-Rink, D. A. B. Miller, and D. S. Chemla, "Exciton Green's-function approach to optical absorption in a quantum well with an applied electric field, " Phys. Re v. B 43, 1500-1509 (1990. 16. T. H . Wood, "Multiple quantum well (MQW) waveguide modulators," l. Lightwaue Technol. 6 , 743-757 (1988). 17. D. A. B. Miller, J. S. Weiner, and D. S. Chemla, "Electric-field dependence of linear optical properties in quantum well structures: Waveguide electroabsorption and sum rules," IEEE l. Quantum Electron. QE-22, 1816-1830 (I986). 18. D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann , T . H . Wood, and C. A. Burrus, " Nove l hybrid optically bistable switch : The quantum well self-electra -optic effect device," Appl. Phys , Lett:.....45 , 13-15 (1984). 19. D . A. B. Miller, D. S. Chemla, T. C. Damen, T . "H . Wood, C. A . Burrus, A. C. Gossard, and W. Wiegmann, " T he quantum well self-electrooptic effect device: Optoelectronic bistability and oscillat ion, and self-linearized modulation, " IEEE l. Quantum Electron. QE-21, 1462-:1476 (1985 ). 20. D. A. B. Miller, M. D. Feuer, T . Y. Chang , S. C. Shunk, J . E. Henry, D. J. Burrows, and D . S. Chemla, "Field-effect transistor self-electrooptic effect device: Integrated photodiode, quantum well modulator and transistor," IEEE Photon. Technol. Lett . 1, 62-64 (1989). 21. D. E. Aspnes, " E lectric-field effects on the dielectric cons tan t of solids," Phys . Rev. 153, 972-982 (967). 22. D. E. Aspnes and N. Bottka, "Electric-field effects on the dielectric function of semiconductors and insulators," pp. 459-543 in R. K. Williardson and A. C. Beer, Eds., Semiconductors and Semimetals, Vol. 9, Modulation Techniques, Academic, New York, 1972. 23. M. Cardona, Modulation Spectroscopy, pp. 165-275 in Solid State Physics, Suppl. 11.~ Academic, New York, 1969. 24. B. R. Bennett and R. A . Soref, " E lectrore fraction and electroabsorption in InP, GaAs, GaSb, InAs, and InSb, " IEEE l . Quantum Electron . QE-23, 2159-2166 .' (1987). 25. D. A. B. Miller, D. S. Chemla, and S. Schmitt-Rink, "Relation between electrcabsorption in bulk semiconductors and in quantum wells : The quantum-confined Franz-Keldysh effect," Phys, Reu. B 33, 6976-6982 (1986 ). :26. H. Shen and F. H . Pollak, "Generalized Franz-Keldysh theory of electromodulation," Phys. Rev. B 42,7097-7102 (1990). 27. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions , Chapter 10, Dover, New York, 1972. , 28. F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids, Pergamon, Oxford , UK, 1975.

REFERENCES

5'!9

29. C. Y. P. Chao and S. L. Chuang, "Analytical and numerical solutions for a two-dimensional exciton in momentum space," Phys. Rev. B 43, 6530-6543 (1991). 30. M. D. Sturge, "Optical absorption of gallium arsenide between 0.6 and 2.75 eV," Phys. Rev. 127, 768-773 (1962). 31. J. S. Blakemore, "Semiconducting and other major properties of gallium arsenide," 1. Appl. Phys. 53, R123~R181 (1982). 32. G. Livescu, D. A. B. Miller, D. S. Chemla, M. Ramaswamy, T. Y. Chang, N. Sauer, A. C. Gossard, and J. H. English, "Free carrier and many-body effects in absorption spectra of modulation-doped quantum wells," IEEE l. Quantum Electron. 24, 1677-1689 (1988). 33. G. Bastard and J. A. Brum, "Electronic states in semiconductor heterostructures," IEEE l. Quantum Electron. QE-22, 1625-1644 (1986). 34. G. D. Sanders and Y. C. Chang, "Theory of photoabsorption in modulation-doped semiconductor quantum wells," Phys. Rev. B 35, 1300-1315 (1987). 35. R. Zimmermann, "On the dynamic Stark effect of excitons, the low field limit," Phys. Status Solidi B 146, 545-554 (1988). 36. Y. Kan, H. Nagai, M. Yamanishi, and 1. Suemune, "Field effects on the refractive index and absorption coefficient in AlGaAs quantum well structures and their feasibility for electrooptic device applications," IEEE 1. Quantum Electron. QE-23, 2167-2180 (1987). 37. M. Yamanishi and 1. Suemune, "Comment on polarization dependent momentum matrix elements in quantum well lasers," lpn. l. Appl. Phys, 23, L35-L36 (1984). 38. M. Asada, A. Kameyama, and Y. Suematsu, "Gain arid intervalence band absorption in quantum-well lasers," IEEE 1. Quantum Electron. QE-20, 745-753 (1984). 39. B. Zhu and K. Huang, "Effect of valence-band hybridization on the exciton spectra in GaAs-Ga I-.x Al x As quantum wells," Phys, Rev. B 36, 8102-8108 (1987). \ 40. G. E. W. Bauer and T. Ando, "Exciton mixing in quantum w~lls," Phys, Rev. B 38, 6015-6030 (1988). 41. H. Chu and Y. C. Chang, "Theory of line shapes df exciton resonances semiconductor super lattices," Phys. ReL'. B 39, 10861-10871 (1989).

10

42. L. C. Andreani and A. Pasquarello, "Accurate theory of excitons GaAs-GaI_xAlxAs quantum wells," Phys, Rev. B 4-2, 8928-8938 (1990).

10

43.

c. Y. P. Chao and S. L. Chuang, "Momentum-space solution of exciton excited states and heavy-hole-light-hole mixing in quantum wells," Phys. Rev. B 48, 8210-8221 (1993).

44. M. Matsuura and T. Kamizato, "Subbands and excitons in a quantum well in an electric field," Phys. Rev. B 33, 8385-8389 (1986). 45. G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, "Variational calculations on a quantum well in an electric field," Phys. Rev. B 28, 3241-3245 (1983). 46. S. Nojima, "Electric field dependence of the exciton binding energy in GaAsj AlxGa1_xAs quantum wells," Phys. Rev. B 37,9087-9088 (1988).

I,

580

ELF,CTROABSORPTION MODULATORS

47. D. Ahn and S. L. Chuang, "Variational calculations or subbands in a quantum well with uniform electric field: Gram-Schmidt' orthogonalization approach," Appl. Phys. Lett. 49, 1450-1452 (1986). 48. J. S. Weiner, D. A. B. Miller, D. S. Chemla, T. C. Damen, C. A. Burrus, T. H. Wood, A. C. Gossard, and W. Wiegmann, "Strong polarization-sensitive electroabsorption in GaAsjAIGaAs quantum well waveguides," Appl. Phys, Lett. 47, 1148-1150 (985). 49. P. Lefebvre, P. Christol, and H. Mathieu, "Unified formulation of excitonic absorption spectra of semiconductor quantum wells, superlattices and quantum wires," Phys, Rev. B 48, 17308-17315 (1993). SO. H. S. Cho and P. R. Prucnal, "Effect of parameter variations on the performance of GaAsjAlGaAs multiple-quantum-well electroabsorption modulators," IEEE J. Quantum Electron. 25, 1682-1690 (1989). 51. G. Lengyel, K. W. Jelley, and R. W. H. Engelmann, "A semi-empirical model for electroabsorption in GaAs / AlGaAs multiple quantum well modulator structures," IEEE J. Quantum Electron. 26, 296-304 (1990). 52. c. R. Giles, T. H. Wood, and C. A. Burrus, "Quantum-well SEED optical oscillators," IEEE J. Quantum Electron. 26, 512-518 (1990). 53. A. L. Lentine, D. A. B. Miller, L. M. F. Chirovsky, and L. A. D'Asaro, "Optimization of absorption in symmetric self-electrooptic effect devices: A system perspective," IEEE J. Quantum Electron. 27, 2431-2439 (1991). 54. P. J. Mares and S. L. Chuang, "Comparison between theory and experiment for InGaAs / InP self-electrooptic effect devices," Appl. Phys, Lett. 61, 1924-1926 (1992). 55. P. J. Mares and S. L. Chuang, "Modeling of self-e lectrooptic-effect devices," J. Appl, Phys. 74, 1388-1397 (1993). 56. M. K. Chin and W. S. C. Chang, "Theoretical design optimization of multiplequantum-well electroabsorption waveguide modulators," IEEE J. Quantum Electron. 29, 2476-2488, 1993.

PARTV Detection of Light

.-

,I

14 Photodetectors Photodetectors play important roles in optical communication systems. The major physical mechanism of photodetectors is the absorption of photons, which changes the electric properties of the electronic system, such as the generation of a photocurrent in a photoconductor or a photovoltage in a photovoltaic detector. The performance of a photodetector depends on the optical absorption process, the carrier transport, and the interaction with the circuit system. For intrinsic optical absorptions, such as interband processes in a direct semiconductor, the general theory for the absorption spectrum has been presented in Chapter 9. The interband absorption creates electron-hole pairs. The carrier transport of these electrons and holes after generation depends on the design of the photodetectors. In this chapter, we study photoconductors, photodiodes using p-n junctions and p-i-n structures, avalanche photodiodes, and intersubband quantum-well photodetectors. Our focus is on the understanding of the physical processes of the carrier generation and transport. We derive the relation between the optical pr .er and the photocurrents. Some important noises in these photodetectors are also discussed. More extensive treatment of photodetectors can be found in .' Refs. 1-~. I

\

14.1

PHOTOCONDUCTORS

.'

14.1.1

Photoconductivity

Consider a uniform p-type semiconductor with a uniform optical illumination. The total electron and hole carrier concentrations, nand p, deviate . from their thermal equilibrium values, no and Po, by the excess carrier i concentrations, on and op, respectively, due to the optical excitation: n = no

+ on

p = Po

+

Bp

(14.1.1)

Since Po » no in an extrinsic p-type semiconductor, the net thermal recombination rate (2.3.14) can be reduced to

R':" n

=

on

(14.1.2) , 583

PHDTODETECIORS

------.~

E

-~--~---~---i ..-. D--?

Hole

I

.........

d

Electron!

+~ 111-_------------'

L..-

V Figure 14.1.

A simple photo conductor with an external bias voltage V.

where the low-level injection condition, on,op «Po, has been assumed. Thus the electron concentration will satisfy the rate equation a'6

-n=G at a

on

(14.1.3)

where Go is the net optical generation rate. For a uniform semiconductor with a de voltage bias V, as shown in Fig. 14.1, the electron and hole current densities are given by only the drift components, since there is no diffusion current due to the lack of spatial dependence (a/ax == 0): (14.1.4) The total current density is \ ( 14.1.5) where the conductivity is a

= q(ILnn + ILpP)

( 14.1.6)

The photoconductivity 6u is defined as the difference between the conductivity when there is an optical injection and the dark conductivity u o: 6u = a Uo

=

U

o = q(ILn

on

q(ILnn o + J-LpPI))

+

J.1 p 8p)

(14.1.7) ( 14.1.8)

The total current I is the current density J multiplied by the cross-sectional

14.1

PHOTOCONDUCTORS

585

area A = wd. I

fA

=

=

aEA

oAV =

(14.1.9)

t

where the electric field E = V/ t ,with t the length of the sample, has been used. The photocurrcnt 6.1 is defined as the difference between the total current in the presence of optical excitation and the dark current 10 = O"oAV/t. (14.1.10)

14.1.2

Photocurrent Responses in the Time Domain

Case 1: A Constant Light Intensity. If Go is independent of time, a/at at steady state, we obtain

=

0

(14.1.11) and op = on, since each broken bond creates one electron-hole pair. Thus, the photocurrent is given by

6.1

=

q(ILrr

+ IL p ) G o7

AV

ne

=:::

QJLn G o7

AV

ne

(14.1.12)

since usually J-L n » J-L p ' The above expression can also be expressed in , . ..rrns of the transit time of the electrons: , \

7/ =';

such that

.-

t

v;

-

t J-LnE

-

t

2

(14.1.13)

ILnV

!

(14.1.14) The above expression has a good physical interpretation. The term Go t A is the total number of electron-hole pairs created per second in the sample with a volume tA., The ratio 7 n/7/ gives the photoconductive gain, which is determined by how fast the electrons can transit across the electrodes and contribute to the photocurrent in the circuit before they can recombine with holes. As shown in Fig. 14.1, when an electron-hole pair is created, the photocurrent will be small if the electron and the hole immediately recombine before they can be collected by the electrodes (i.e.; the recombination

. . ...~,. .""":",.. ~ ""T"",.,....'7"" ....-. .,...~- ~ ................ - ~

~."fOo' ... ~ - ( t" _ ..•.,.- . - .... ...-. ' .- ... . . . _ -- .....

.. _ '~ -- -. • : . ... .

-.o:"........... ~ '"'::""••••

.... .

..,...

~~tJ:f:"!

t'" '~~•• ,~; .....,' ~

...~ -,':~•• \\":! .. ~ ~~t:"-:!'F'~~

PI-lOTOD ETEcrOR~;

586

lifetime T lI is much s ho rte r than the transit time). On the other hand, when the transit time T( is short, a significant number of photogenerated electrons will be able to reach the left electrode before they recombine with holes in the semiconductor. To preserve ch arge neutrality in the semiconductor, the left electrode will provide the same number of holes at the same rate as the electrons reaching that electrode per second , resulting in th e photoconductor current measured by the external circuit. For example, Tn I T( = 10 is equiva lent to having ten round-trips taken by the electron before it disappears by recombination in the photoconductor. The optical generation rate is equal to the number of injected photons per second, or photon flux (Po p t I hv') per unit volume (fwd) multiplied by the quantum efficiency T} : Go =

Po P t I h v T}

------'t w-d-

...... .:' ' .,..

:- j.....I'1'. ~._.-_

(14.1.15)

where w is the width, d is the depth of the sample ( wd = A), P o P t is the optical power (W) of the injected light, and 7] is the quantum efficiency or the fraction of photons creating electron-hole pairs. If the surface reflections and the finite thickness of the detector are considered, the quantum efficiency T} is just the intrinsic quantum efficiency 7] j multiplied by the absorbance derived in (5.3.51):

(14.1.16) where R is the optical reflectivity between the air and semiconductor, an d a is the absorption coefficient of the optical intensity. The injected primary photocurrent I ph is defined as

(14.1.17) The photocurrent is

(14.1.18) Again th e photoconductive gain is given by

(14 .1.19) If there are more electrons traveling across the electrodes before recombining with the holes, there will be more photocurrent appearing in the external circuit. The current responsivity R ). (AI ltV) is the photocurrent response per

.~.

14.1

PHOTOCONDUCTORS

587

unit optical incident power (14.1.20) and depends on the operation wavelength A. Case 2: Transient Response. If the constant light illumination is switched off at t = 0, that is, Go(l) = Go, t < 0 and 0 for t > 0, the response for t > 0 will be

a

-on(t)

on( t)

at

(14.1.21)

or

(14.1.22a) and the photocurrent response obtained from (14.1.10) or (14.1.12) is (14.1.22b) The result is shown in Fig. 14.2a. The decay time constant is the minority carrier recombination lifetime 'Tn. If the light injection is an impulse function, (14.1.23) Then,

a

-

at

on(t)

=

go o(t)

one t)

(14.1.24)

Integrating the above equation from t = 0 _ to 0 + will give

Assuming that initially on(O_) = 0, that is, the semiconductor is at thermal equilibrium before t = 0, and the excess carrier concentration is zero, we obtain for t > 0 This result is plotted in Fig. 14.2b.

( 14.1.25)

PHOTODETF CTORS

(a)

8n(t)

o

o (b)

(c)

Ga(t)::= Go (1 + m co s cot)

----.----i---.---r-l~wt

-T[

o

- - - - - i f - L . . - - - - -__ Wt

rt

Figure 14.2. The generation rate G oCt} and the excess carrier con centration Sn(t} for (a ) swi tch-off, ( b) impulse respon se , and (c) sinu soidal s teady-sta te response of a photo conductor.

Case 3: Sinusoidal Steady-State Response. If the optical intensity is modulated by a sinusoidal signal such that

.-

C o( t)

=

Cocos

( 14 .1.2 6)

co t

the sinusoidal ste ad y-s ta te re sponse can be found by letting I

(14.1.27)

on(t) = Re(on e- i W I ) wh ere Re me ans the real part of the follow ing quantity. W e have

a one t) -on(t) = Cocos wt-

at

Using G o cos cot

Re (G n e -

(14.1.28)

Tn

iw

{) ,

Bn

we o bt a in =

( 14.1.29)

Id .l

PhOTOCONDUCTORS

589

Thus

(14.1.30)

where ¢ = tan -l(WTn ) is the phase delay in the ac response. On the other hand, if the optical intensity or the optical generation rate consists of a de and an ac component,

+ m cos

Go (t) = Go (1

(14.1.31)

wt)

where m is the modulation index, the response of the excess carrier concentration will be (14.1.32) where ¢ is the same as in (14.1.30), The above results are shown Fig. 14.2c. Using (14.1.15)-(14.1.17), we find that for an optical input power P ( t)

=

+ m cos

Popt (1

In

(14.1.33)

wt )

the photocurrent response is

let) = lp[l +..; 1 +m

.2

W

2 Tn

cos(wt -

POP! Tn

I p

!

= qTJ--

hv

T

l]

\I I,

(14.1.34a)

(14.1.34b)

/

In other words, for a root-mean-square (rrns) optical power of which we replace the cosine function by 1/ Ii , (14.1.35)

P rm s =

the rms photocurrent signal is 1.

P

=

rm s

P - ( -t: qTJ hv Tt

J -,=.====-

(14.1.36)

.

I

,

590

14.1.3

PHOTODETECTORS

Noises in Photoconductors [6]

In this subsection, we present some fundamental concepts including the spectral density function for noises in photodetectors. Important noises such as generation-recombination or shot and thermal noise are discussed. Spectral Density Function SCI). The signal such as the photocurrent in the time domain i(t) of the photoeonductor is related to its Fourier transform i( f) in the frequency domain by

/'IJ i(f)e-

i(t) =

i27T / t d f

(14.1.37a)

-00

i(f)

Joo i(t)e

=

i 2 rr t /

dt

(14.1.37b)

-00

The average power P over a time duration T is

1 T / 22' 1 P=-! i (t ) dt = - J T

T

-T/2

li(f)1 df -00

2 -T 1 Ii (o f ) df 1 S(f) df o 00

=

2

00

00

2

1

=

(14 .1.38)

where we have used

lim

JT/2 e

T-HO

-T/2

i2Tr(f-f')r

dt

= 8 (f - f')

and defined the spectral density function St f

(14.1.39 )

) as

2 2 S(f) = T1i(f)1

( 14.1.40)

Shot Noise. For a sequence of events such as collection of charges within a time interval T, say ~

L

i( t) = i

=

h( t -

tJ

cs i « r

(14.1.41)

1

where N, = total number of events within time T, we obtain the -Fourier transform ief): NI

i( f)

2: h( f)exp(i27T ft i=l

j )

( 14.1.42)

i4.1

PHOrOCONDUCTORS

591

The ensemble average of Ji( f )J2 is 2

(Ii(/) 1)

~ (lh(/) 1{N,-/- i~ ~ exp[i27rf(t t,)]}) 2

j -

=

Ih(f) 12( ~ )

(N)Tlh(f)

=

2

(14.1.43)

1

where (N) = (Nt)/T is the average number of events occurring per second. The random distributions of t , and t j give the cancellation in the second term in (14.1.43) when i =1= j. The spectral density function is (14.1.44) Now we consider an inj ected electron between two capacitor plates with a separation f . The current is q

h(t)

=

- vet)

(14.1.45)

f

where uCt) is the instantaneous velocity of the ele ctro n and T t is the transit time. The average current ( I) is related to the average number of electron injections per second ( N) by (I )

q (N )

=

(14.1.46~

The Fourier transform of h(t) reduces to .'

h(f) =

jT,q u ( t ) e i2-rr f t d t ::::; fq o f

assuming the frequency density funct ion is

S(f)

=

IS

f

dx(t) dt dt

=

q

( 14.1.47)

low enough such that fT t « 1. The spectral I

2

2(N ) lh (f) 1 = 2(N )q 2 = 2q (/ )

( 14 .1.48 )

The power of this shot noise within a frequency interval between f + 6.f associated with the current is denoted by

(iHf»)

=S(f)6./

=

2q(/ )6.f

f

and

( 14.1.49)

Generation-Recombination Noise. In a photoconductor, the photogenerated carriers have a finite lifetime T , which is a random variable with a 'mean value ( T ) denoted as Tn for, . _the .. • electrons. The photocurrent due to 'On e injected

PHOTODETECTORS

592

electron is O:s;t:S;T

( 14.1.50)

otherwise Here we use u as the mean drift velocity and time. The Fourier transform of hCt) is h( [) =

foT

q

ei 27Tfl_

Tl

t/

=

lJ

as the mean transit

q (ei2 rrjT - 1 dt = - - - . - -

Tl

1

I21T[

Tl

(14.1.51) If we assume that the probability function of the random variable

T

obeys

Poisson statistics ( 14.1.52) with an average


=

JOT p( T) d-r =

Tn'

we find

( 14.1.53) Since the average photocurrent with the photoconductive gain
1p == (I) =
Tn

IT

l

due to

(14.1.54)

1

the spectral density function due to
( 14 .1.55)

The generation-recombination noise current is, therefore,

(ibR )=S([)Llj=

4q/(T /T)Ll[ p

1

n

(

+ (21T[1",;)

2

(14.1.56)

14.1

PHOTOCONDUCTORS

593

Thermal Noise (also Johnson Noise or Nyquist Noise) [6-8]. In a photodetector, the random thermal motion of charge carriers contribute to a thermal noise current. In other words, the thermal noise of a resistor R results in a random electric current i(t), which is characterized by a power spectral density at temperature T: 4 5(/) - R

hf eUI!/kaT) -

(14.1.57)

1

This thermal noise current adds to the photocurrent signal i /t) and affects the clarity of the detected signals. At low frequency hf « k BT, we have

S(f) -

(14.1.58)

The thermal noise is described by

(i})

=

1a si r, df = 4k RTts ] ~!

_ B_ _

( 14.1.59)

where the bandwidth /).f of the circuit is assumed to be much smaller than kBTlh. Signal-to-Noise Ratio. The power signal-to-noise ratio is therefore [8] Y]m 2 ( Po p t I hv) 8/).f[1 + (k BTlqRJp

)(Tt I Tn )(l + W 2 T: )] ( 14.1.60)

Noise-Equivalent Power (NEP). The noise-equivalent power is defined as the power that corresponds to the incident rms optical power (P r m s = m PoPt Iii) required such that the signal-to-noise ratio is one in a bandwidth of 1 Hz. Detectivity (D *). The detectivity is defined as

D*

=

{Ab:7 em (HZ)1/2 IW NEP

(14.1.61)

where A is the detector cross-sectional area in crrr'. More discussions on photoconductive detectors such as Hg .Cd I-xTe can be found in Ref. 9.

~

•. . . . ••. • •._', ~

~

_._,y . _

~

-

~"

"

"

"" ""'- " - ---'--- ' .-

•

. - - - _.

,

594

14.1.4

. -- ~ ~

·· ~ ~ ·/"'..-.-r' '''

' - ,-••

_ . , .~

• '_.-

PHOTODETECTORS Il-l-P-l

Superlattice Photoconductor [10, 11]

Recently, modulation doping in semiconductors has been introduced for novel device applications. An interesting example is a GaAs doping superlattice used as a photoconductor. For an extensive review of the compositional and doping sup e rlat t ice s, see Refs . 10 and ] 1. Here we consider a GaAs doped periodically n-type and p-type separated by intrinsic regions as shown in Fig. 14.3a. The electric field profile Et.z') can be obtained by noting that dE(z)/d z = p(z)/e , where p(z) = +qND in the n-doped regions, -qNA in the p-doped regions, and zero in the intrinsic regions. Th erefore, E(z) is either a linear profile with a positive slope qND / e in n regions, a negative slope -qNA /e in p regions, or a zero slope in the intrinsic regions (Fig. 14.3b). The potential profile for the conduction band edge Ec(z) = -qcp(z) = +qf~ooE(z')dz' is the integral of the electric field profile and is shown in Fig. 14 .3c . For an incident light with energy above the band gap, the photogenerated carriers will fall to the band edge and will drift or diffuse to the valleys of (a) Charge density p(z) +qN O

.........

+qN D

E

~

n

-E

i

p

i

-

-qN A

+qN n

E

n

i

E

p

i

E

- f-.n

z

-qN A

(b) Electric field E(z)

---r---~:-----:f----+---I---~z

(c) Energy band diagram

Figure 14.3. (3) The charge dens ity profile due to ionized donors and acceptors in an n-i-p-i doping superlattice using the depletion approximation . (b) The e lectric field profile due to the ch arge distribution in (a). Notice that the electric field E = 2E(z) is alternating between positive and negative di rections. (c) The energy band diagram of the n-i-p-i superlattice. The photogenerated electron -hole pairs are separated in real space because of the band profiles.

14.2

p-n

JUNCThJN PHOTODIODES

595

each band, as shown in Fig. 14.3c. The electrons arc separated from the holes in real space, resulting in a very long recombination lifetime Tn' This enhancement of a long lifetime has been found to be many orders of magnitude larger than the bulk carrier lifetime. Since the photocurrent response is proportional to CaT,l' an extremely large responsivity using the n-i-p-i superlattice as a photoconductor can be designed.

14.2

p-n

JUNCTION PHOTODIODES [12-15]

Consider a p-n junction photodiode as shown in Fig. 14.4a. The charge distribution p(x), the electric field Ei;x ) and the potential energy profile under the depletion approximation have been discussed in Chapter 2. Here we investigate the photocurrent response if the diode is illuminated by a uniform light intensity, described by a generation rate C(x, t ), which is the number of electron-hole pairs created per unit time per unit volume. Let us focus on the n stele of the diode. The charge continuity equation is given by (2.4.2)

aPn at

oPn

=

C(x,t) - -

.

Tp

1

a ax

- - - J (x) q

(14.2.1)

p

(a) A p-n junction photodiode

f I

II r-

p

n

r--

'I'l (b) Charge density

p(x) tqN D

+ +

+

----....,.;.ti-l.----+- x

(c) Electric field x

Figure 14.4. (a) A p-n junction diode under the "illumination of a uniform light. (b) The charge distribution p(x) under depletion approximation. (c) The electric field Et x) obtained from Gauss's law.

590

PHOTODETECTOlZS

where P" = Pno + op" is the total hole concentration in the n region, PnO is the hole concentration in the absence of any electric or optical injection, and oP n is the excess hole concentration due to the external injections. The minority (hole) current density in the quasi-neutral region (x ;;::: xn) is dominated by the diffusion component [12-14], as discussed in Chapter 2:

(14 .2.2) Since PlIO is independent of x and t, we have at steady state, if G( x , t ) = Go is independent of x and t,

(14.2.3) The above equation can be solved by summing the homogeneous and particular solutions:

(14.2.4 ) homogeneous solution

particular solution

where L p = jDpTp is the diffusion length for holes. The particular solution is due to the optical generation. If the n region is very long, we can set C z = 0; otherwise , oPn(x ----7 00) ----) +00, which is unphysical. We expect as x ----) +00 that oPn(x) will approach GOTp = total photogenerated holes. At x = xn' the hole concentration is pinned by the voltage bias V with the exponential dependence

( 14.2.5)

if the Boltzmann statistics are assumed. Therefore, we obtain ( 14.2.6) and

The current density Jp(x) is

= q ~p [Pno( eqv/kBT p

-

1) _. GOTp ] e -' (x-x n ) / L

p

( 14.2.8)

H.2

P-Il JUNCTION

1 HOTOnIODES

59~·

We obtain J/x) at the boundary of the depletion region

D

Jp ( x n ) -- q ~ L Pl10 (qV/kBT e

-

Xn

as

1) - q G 0 L p

(14.2.9)

p

where the first term is due only to the voltage bias, and the last term is due to optical generation. We see that only that portion of the photogenerated holes within a diffusion length L p away from the depletion boundary can diffuse (and survive) to the depletion region and be swept across the depletion region by the electric field and collected as the photocurrent by the external circuits. This means that the majority of the carriers on the p side have to supply this current immediately. A parallel (or dual) approach for the electron current density at x = -x p gives

(14.2.10) The total current I is the sum of J/x n) and JnC - x p) multiplied by the cross-sectional area of the diode A:

I=A[Jp(x n ) + In(-x p ) ] =

I o ( eqV/kBT

-

1) -

qAG o ( L p + L n )

(14.2.11)

where

(14.2.12) I

is the diode reverse current. The last term, -iqAGo(L p + Ln>, is the photocurrent of the diode, which is proportional to the generation rate of the electron-hole pairs, G. The I-V curves of ?l photodiode with and without the illumination of light are plotted in Fig. 14.5. When Go = 0, the diode reverse

-------::r---/L........,;.,.------.v f-

Io(eqVIkBT-l)-qAGo(Lp+ L n)

With light Figure 14.5.

The I-V curves of a photodiode with and without illumination.

PHOTODETECTORS

598

current - l ois the dark current , whi ch is usually very small compared with the photocurrent I pl1 = - qA Go( L p + L,) under a reverse bias condition. Therefore, the photocurrcnt is proportional to the generation rat e G o, wh ich is proportional to the incidcn t optical power P o p t ' The problems with the p-n junction photodiodes are as folJows: 1. Optical absorption within the diffusion lengths L p and L" is very sm all , that is, over narrow regions of L p and L; near the depletion region. Since L p and L" are very sm all , the contributions of the photocurrents are not effective. 2. The diffusion process is slow, which results in a slow photoresponse if the optical intensity varies with time . 3. The junction capacitance C , is sim ply cA /x w ' where X w is the total depletion wid th derived in (2.5.24) sim plified for a hornojunction diode with c p = CN = e :

( 14 .2 .13) whi ch can slow down the response by the RC~ time delay. For example , if A = (l mm)2, e = l 1.7co , N D = 10 15 cm : «NA fo r a p+-n photodiode , and - V = 10V » Vo, we obtain Cj = 30 pF and the 3-dB cutoff frequency f 3d B = 1/(27TRC j ) = 100 MHz for R = 50 n . RoA Product [15]. A useful figure of merit for the p-n junction photodiodes is the RoA product. Since the photodiode is operated at zero-bias voltage in many direct detection applications, the differential resistance at zero-bias voltage R o multiplied by the junction are a A is commonly used: (RA) o

-1

d/l

dJI

1 =-- . -A dV v =o - dV

.

V=Q

iI

\

(14.2.14)

So far, we have derived the dark cu rre n t 10 contributed by the diffusion processes in thi s se ctio n. Us ing (14.2.11 ) and (14.2.12), we obtain

(14. 2.15 )

.. '

14.7.

!J-Il

JUNCTION PHOTODIODES

599

where we have used the relations Pno = NDln;, npo = NA In;, and the Einstein relations Dpl/-Lp = Dili/-Ln = kBTlq. The first term in 04.2.15) is the contribution to 1/(R oA) from the diffusion current on the n side of the photodiode,. and the second term is from the diffusion current on the p side. There can also be other contributions to RoA products, such as the generation-recombination current in the space-charge region, the surface leakage current, and the interband tunneling current, which depend on the material properties, device geometry, and surface conditions. For a root-mean-square photon flux density

(14.2.16) where A is the photodetector illumination area,

1.\

I~I

Ii

R

'» PI.

q

=-=7]A

hv

r~ r: i' ",

( 14.2.17)

The signal-to-noise ratio SIN (current) is

S

( 14.2.18)

N

The detectivity for the above SIN is defined as.'

(14.2.19)

For a photodiode at thermal equilibrium (i.e., no externally applied voltage and no illumination of light), the thermal noise depends on the zero bias resistance R o using (14.1.59): (i~) -

• ( 14.2.20)

PHOTODETEC1'ORS

600

When not in thermal equilibrium, the I-V curve is I( V) = I o ( e ( q VjkaT)

Iph =

-

1) - 1 ph

(14.2 .21)

Q7]cfJ n A

(14.2.22)

where ep n is the photon flux density due to the background radiation. The mean-squared shot noise current has contributions from three additive terms [15]: (1) a forward current, which depends on voltage, 10 exp(qVB/knT), (2) a reverse diode saturation current, and (3) the background radiation induced photocurrent. Since these shot noise currents fluctuate independently, the total mean-squared shot noise curr e n t is

(14.2.23 ) At an operation voltage V = 0, R 0 1 = (dI/dV) v=o can be wri tt en as

=

qlo /kBT and 04.2.23)

(14 .2.24) The detectivit y at zero bias voltage is then obtained from 04.2.19)

(14.2.25)

For a thermally limited case , i.e., when the thermal noise is dominant over the background radiation induced signal and other noises, we have

~ 14.2.26)

whi ch re lates th e R oA product to the th ermally limited detectivity. If the photodiode is ba ckground radiation lim ite d, which means that the background radiation-induced photocurrent is dominant, we have ( D A* ) BUP -_ _h1v

!

!

7J_ 2B

(14.2.27)

which is the detectivity of the background limited infrared photodetector (BLIP).

14 .3

14.3

601

p-i-l' Pl-IOTODIODES

p-i-n

PHOTODIODES [8]

To enhance the responsivity of the photodiode, an intrinsic region used as the major absorption layer is added (Fig. 14.6a). For a light injected from the p + side with an optical power intensity Jo p t (W/ ern"), the generation rate is

(14.3.1 )

G(x)

The optical power intensity Jo p t is the incident optical power PoPt divided by the area A (Jo p t = Popt / A). Note that the total injected number of electrons

(a)

x=Q

x=W n

Optical power (b)

L

t I

p(x)

+<jNo

0-----9

'-------+ )

x

-qN A E(x)

(c)

i

= JX p(x) dx'/E o

.t

", 'I

-00

(d)

(e)

(f)

~(x) =-~~(X') dx' -______:~;

~

-;.1_I;

X

~----.L----__.)

o

G(x)

w

= Optical generation rate = G o e-ax G 0 -- Tli I opt ( l-R)/hv x

Figure 14.6. (a) A p-i-n photodiode under optical illumination from the p + side, (b) the charge density p(x) under depletion approximation, (c) the static electric field profile Ei.x), (d) the electrostatic potential cp(x), (e) the conduction and valence band edge profiles, and (f) the optical generation rate G(x). .

.,,

PHOTODEfECfO!<..S

602

per unit area per second is

So

oo

=

fo

G(x) dx

lOPl

=

(1 - R)"7;hv

(14.3.2)

where lOPl / h v is the number of photons injected per unit area per second, and "7i is the internal quantum efficiency for the probability of creating an electron-hole pair for each incident photon. The energy band profiles for the p-i-n diode can be obtained graphically based on the well-known depletion approximation, as shown in Figs. 14.6b-e for the charge density p(x), the electric field Et x), the potential cjJ(x), and the band diagram. At steady state, the total photocurrent consists of both a drift and a diffusion current: (14.3 .3) Considering the p + region to be of negligible thickness, we look at the contribution in the intrinsic region-O < x < W:

(14.3.4) where the minus sign accounts for the fact that the draft current flows in the -x direction. The above expression also shows that an increase of W» 1/ a enhances the photocurrent because of the increasing amount of absorption. For x > W, the analysis is similar to that of the p-n junction diode, where the hole (minority) current density on the n side is due only to diffusion:

a

er; at

0=

J diff = -qDp-Pn(x) ax 8Pn 1 a =G(x)- "» q-axJp(x)

(14.3.5) (14.3.6)

Therefore, we solve

1 - --G(x) Dp

(14.3.7)

The solution for 8Pn(x) consists of the homogeneous solution and the particular solution 8P

rt

(

x) = Ae -(x- W) /L l' + Ce -{xX '--- - ~ - - -

... ~

homogeneous solution

particular solution

( 14.3.8)

lid

p-i-n PHOTODIODES

603

and we have discarded the term e +(x- W)/L p in the homogeneous part since oP,/x ~ co) should be finite. C is contributed by G(x) and is obtained by substituting Ce -ax into (14.3.7): (14.3.9)

The coefficient A is then determined by the boundary condition

oPn (W)

=

PnO (eqV/kBT

-

1) =

-PnO

(14.3.10)

which is pinned by the reverse bias voltage. Therefore,

A

-Pn O

=

-

Ce- a W

(14.3.11 )

and (14.3.12)

The hole current density on the n side is

J diff

d

= -

qDp - Pn ( x) dx

~

qD a(1 -

=

-

_1_)C

p

q

a

(

x=W

Lp

1

e- aw _ qDp P Lp

SoaL p e- aw + P

1 + aLp

Dp

nOL

nO

'J

(14.3.13)

p

The total current density is

(atx=W)

J=Jdr+Jdiff

~ -qSo( 1 -

1

:-::J -

qPno

/

~:

(14.3.14)

The quantum efficiency is aW

77

=

J/q Iopt/hv

=

77i(1-R)(1 -

e1 + aLp

1

(14.3.15)

neglecting the contribution of the diffusion term m (14.3.14). Note that if W ~ co, the current density is dominated by .l

=

-qSo

(14.3.16)

, .•

' ", ~

• •'- '

..,

,..,...

";.". . _

.

~

••• ,

0".••, ' " • ._

•. ,.

•• ., -

, ' , ""

' _.

- • ..'

PHOTODETECTOR~

604

and 17 = 77;C1 - R) as; expected. Long-wavelength p-i-n photodiodes for high-speed receiver applications are of gre at interest for optical communication systems. For high-sensitivity optical receivers, photodiodes with a small junction area around a few tens of microns in diameter and a low doping depletion region of a few microns are desired [16]. At low doping, the center absorption region can be depleted at a small bias voltage and reduce the tunneling leakage current. Most of the long-wavelength photodetectors use In0.47Gao.53As grown on an InP substrate as the absorption region. The leakage current in InGaAs p-i-n diodes is dominated by the interband tunneling at high reverse-bias voltages and generation-recombination processes at low voltages [17, 18]. Hybrid p-i-n / FET receivers have been assembled for high-speed photoreceivers, which offer better sensitivity than other p-i-n photodiode receivers [16, 19]. Ultrawideband p-i-n photodetectors have also been shown to have a great potential for high-speed applications [20] with an impulse response in the picosecond scale.

14.4

AVALANCHE PHOTODIODES

To enhance the photocurrent response, some built-in multiplication processes may be utilized such that more photocurrents can be extracted in the external circuits at a given optical illumination. Ideally, we would have a single carrier-type photomultiplier; the carrier concentration will grow if impact ionization occurs in a region with a large electric field. In semiconductors, both electrons and holes can impact ionize more electron-hole pairs, as sh own in Fig. 14.7a. A schematic diagram for an avalanche diode is shown in Fig. 14.7b. A feedback process occurs since electrons and holes travel in

hv~ I

•I I I

I I

o

~

Jp(W)

w

hv

>x

Hole injection

Figure loot? (a) The energy band diagram for an avalanche photodiode with the electron and hole ionization coefficients an and (3p. The electron and hole i nje cti ons are given by 1)0) and fpC W) . (b) A schematic diagram for an aval anche photod iode .

14.4

AVALANCHE PHOTODIODES

605

opposite directions. Let us define the electron ionization coefficient (l / em) = the number of electron-hole pairs generated by one incident electron per unit distance f3 p = the hole ionization coefficient (I / em) = the number of electron-hole pairs generated by one incident hole per unit distance

an

=

In general, an =I=- f3 p for most semiconductors. They are functions of the applied electric field E with an exponential dependence: 'i

.r:

where the constants ao, {3o, Cn' and C; depend on the materials. For a general overview of the fundamentals of -avalanche photodiodes, see Refs. 21-23.

14.4.1 Ideal Avalanche Photodiode-Single Carrier Type Capable of Ionizing Collisions [22]

We now consider the special case in which only electrons can impact ionize. Suppose we have an incident current density J/x) at a plane located at position x (Fig. 14.8a). Over an incremental distance L1 x, the total generated electron-hole pairs is a nL1x multiplied by In(x), since a nL1x is the number of ionized electrons per incident electron in a distance Li x. Therefore, the current density In(x + L1 x ) is the sum of the incident (or primary) current density and the ionized (or secondary) current density: (14.4.1)

(a) Electron impact ionizations only In(X)

Jp(X)

I

~

..-7 ..-7 e-7

~

~

~

~

~

H

I

I I

Ifx Figure 14.8.

Jp(X -t1x)

I

e-7

•

In(x + t1x)

(b) Hole impact ionizations only

I

Ax~1

x+Ax

I

t--1;> I I

x-Ax

I

I I I I

X

Schematic diagrams for only (a) electron and (b) bole impact ionizations.

i,1

"

60,j

~ .

-, .'

-1,- .,. --. '"

PHOTODETECTOR S

or (14.4.2) Its solution is (14.4.3) if all == an(x) is not uniform, since the electric field Ei x) may not be uniform. The multiplication factor M n for the electrons is defined as (14.4.4) For a uniform an' we have ( 14.4.5) and (14.4 .6) Here the multiplication factor is finite since W is a finite width. A similar procedure for holes propagating in the -x direction, as shown in Fig. 14.8b, leads to ( 14.4.7) and its solu tion is ( 14.4.8) The hole multiplication ratio is defined as ( 14.4.9) For the special case in which f3 p is independent of the position x , we have ( 14.4.10) and (14.4.11)

" ',, " '"

' ~ .. ..

14.4

607

AVALANCHE PHOTOL-IODES

14.4.2

Both Electron and Hole Capable of Impact Ionization

Let us write the complete coupled equations when both electrons and holes cause impact ionizations in the presence of optical generation as well: d

- In(x) dx

=

a n ( x ) ln(x) + !3 p( x ) l p( x ) + qG(x)

=

an(x)ln(x) + f3 p( x ) l p( x ) + qG(x) (14.4.12b)

d

- dx1p(x)

(14.4.12a)

Note that each impact ionization process creates an electron-hole pair, as does the optical generation rate G(x) per unit volume. We see that the above two equations lead to

:'1

( 14.4.13) :i

or the total current density

( 14.4.14) which is independent of the position, as it should be since this is a onedimensional problem. Any current passing through a surface at x has to be the same at steady state. The two coupled first-order differential equations can be solved using one of the two variables, for example, I n ( x ) :

where 1 is independent of x and is determined later by the boundary conditions. Noting that the first-order differential equation .'

d

-y(x) + p(x)y(x) dx

=

Q(x)

( 14.4.16)

has a solution of the form

(14.4.17) we obtain by setting all initial conditions atx o = 0: 1ft dx'!3p(x')e-CP(X') + Qf6"'G(x')e-CP(X') dx'

e -cp(x)

+ I n(O)

(14.4.18)

6C

~

PHOTOD'C.TECTORS

where ( 14.4.19) We then match the boundary conditions at x = W. Suppose In(O) and Jp(W) are given; we then find lll( W) Using and

cp(W) - cp(x') = fW[an(xl/) - ,Bp(x")] dx" x'

(14.4.21) we find J immediately:

.

Therefore, we have

In(x) = Je
o

0

( 14.4.23) where I is given by 04.4.22). The hole current density is then i

I p ( x ) =1~Jn(x)

(14.4.24 )

Alternatively, we can start with 04.4.12b) using J p as the variable and find _ f(w dx'[a n ( x ' ) 1 I p( x) -

+

qG(x')]eJ~r[(t,,(X") -f3p(x")JdX"

+

Jp(W)

e).tla,,(X')-Pp(X'»)d X'

(14.4.25)

and obtain another expression for I using the boundary condition at x I p(O) = J - .In(O): .

I I =

n

(

0) + I

p

W) e -
=

0,

(

(

(14.4.26)

ll

Let 1.I,S consider three special cases with only electron injection at x = 0, or hole injection at x = W, or optical injection G (x) at a position x.

-. . ..

14.4

AVALANCHE PHOTODIODES

609

Case 1: Only Electron Injection at x = 0 (Jp(W),= 0 and G(x) electron multiplication ratio is obtained from (14.4.22): I

M

n

=

=

0). The

e~(W)

=

InC 0)

Jaw dx'l3p(x')efxft'[an(X")-f3p(X")]dX"

1 -

(14.4.27a)

An alternative form, if we start with I p ( x ) from the- beginning and find I following the above procedures, is

I

Mn

=

I

.i

1 =

n(O)

Jaw dx'al1(x')e-fJ'[an(X")-f3p(X")]dXlf

1 _

Case 2: Only Hole Injection at x = W (J/O) multiplication ratio is, using (14.4.26),

=

(14.4.27b)

0 and G(x) = 0). The hole ~

I

Mp

I

=

i'~

e -!peW)

p(W)

1 _

=

Jaw dx' an(x')e-fu'[an(X")-f3p(X")]dxIf

Ii 'I

(14.4.28a)

'.I II

which is also a dual form of (l4.4.27a). A dual form of 04.4.27b) is

I M p - I p ( W) - 1 For example, if we consider

=

Jaw dx'f3 p( x')efx'f[an(XIl)-f3p(X")]dx"

an = f3 p

lAx) Using I n(W)

1

=

j :1 it

(14.4.28b)

-!

= constant, and G( x ) = 0, we find

If3p x

+ I n(O)

( 14.4.29)

I - I p(W), we find

I n(O) + I p ( W)

1=-~----

1 -

( 14.4.30)

f3pW

The results of I n ( x ) and I p ( .r ) are plotted in Fig. 14.9.

Current densities In(O)

I n(O)

o

w

Figure 14.9. The electron and hole current densities, InCx) and jp(x), and the total current density I as a function of position x for a special case Q n = f3 p = constant and the only injections are determined by the values 1)0) and IpWl).

.'

PH0TODETECTORS

610

Case 3: Only Optical Injection G(x') = GoMx' - x) (If/(O) = 0, Jp(W) 0). We find that the total current density is

=

(14.4 .31) and the multiplication rate depends on the position of the optical injection x:

We can also check two special optical injection positions x = 0 and x = W. We find

M(x = 0) = M;

(14.4.33)

-"vfp

(14.4.34)

and

M(x

=

W) =

These are the same expressions as in (14.4.27a) and (l4.4.28b), as expected. By controlling the electron injection in the p region or the hole injection in the n region, the multiplication factors Mn(V) and M/V) as a function of the reverse biased voltage V can be determined [24-31]. They usually increase very slowly at a low .reverse bias and show an exponentially increasing behavior above a certain large bias voltage, as shown in Fig. 14.10. These reverse bias voltages can be of the order of 30 or 40 V. Once M; and M p are measured, an and {3p are supposed to be found from (14.4.27) and (14.4.28). If both an and {3p are independent of the position x, the integrations can be carried out analytically. We can express an and f3 p in terms of M; and M p : a

n

=

( 14.4.35)

(14.4.36)

Figure 4.10. Multiplication factors for the electron dud hole injections, M'; and M p , respectively, are plotted as a function of the reverse bias voltage.

Reverse bias voltage V

14.4 AVALANcHE PHOTOD10DES

611

From the above discussions, we see that in order to determine the impact ionization coefficients from the multiplication of photocurrerit, proper experimental conditions [24-26] have to be met: (1) .Mn(V) and M/V) for pure electron injection and pure hole injection, respectively, have to be measured in the same diode, but not in complementary p + nand n +p devices, (2) the primary injected photocurrent (without multiplication) must be determined accurately as a function of bias voltage, and (3) the electric field should .be slowly varying in space and uniform in the active region. The dependence of the electric field on the position and bias voltage must be accurately known. It is not easy to meet all of these conditions without approximations. Experimental data for Si [27], GaAs [24, 25, 28), InP [26, 29, 30], InGaAs, and InGaAsP have been reported, for example, GaAs [25] InP [26]

a

=

f3:

=

(1)

. -,

(2)

(3)

InP [30]

1.899 X 10 5e - (5.75x 105 / E)UI2 0 j ern) 2.215 X 10 5e -(6.57 x 105/E)1.75 0 j ern) 240 kY jcm < E < 380 kY /cm, N = 1.2 X 10 15 cm " :' an = 1.12 X 107e -3.JI XIO hiE (Ly crn) (3p = 4.76 X 106e -2.55 X106/E (Ly cm) 360 kY jcm < E < 560 kY jcm, N = 3.0 X 10 16 cm "? an = 2.93 X 10 6e-2.64XIO hiE (Ly cm) (3p = 1.62 X 106e-Z.lJXI06/E Ojcm) 530 kY jcm < E < 770 kY jcm, N = 1.2 X 10 17 cm ":' an = 2.32 X 105e-7.16X 10 II IE! (l y'cm) f3 p = 2.48 X 10 5 e -6.Z3 x 10 11/ £ 2 (l j ern) 6 an = 5.55 X 10 6e - 3.10x 10 IE OJ em) {3p = 1.98 X 106e-Z.Z9X106/E (Ly cm)

InGaAsP (E g = 0.92 eY) [30] an = 3.37 X 106 e-2.Z9 X10 6 /E (ljcm) f3 p = 2.94 X 106 e-2.40 Xl0 6/E Cljcm) Ino .53Ga0,47As [30]

an

=

f3 p

=

2.27 X 106e-1.l3 Xlo6/E (ljcm) 3.95 X 106 e - 1.45 x 10 6 I E (I j em)

where the electric field E is in V jcm in the above expressions. For strained In o.2Ga o.sAs and Ino.15Gao.63Alo.2zAs channels embedded in Al o.3Ga o.7As material, the an and f3 p values have been found to be higher in In o.2Ga o.8As channels and lower in the Ino.J5Gao.63Alo.22As channels compared with the unstrained GaAs channels using hole injection in a lateral p-i-n diode configuration [31]. For an rms optical power 04.1.35)

(14.4.37)

6]2

PI-I0TdDETECTOR'.;

where m is the microwave modulation depth in 04.1.34), and P OP 1 is the optical input power. The multiplied rms photocurrcnt response is .

p rms

_

q7J-- M

lp -

( 14.4.38)

hv

where M = M n for the electron multiplication and M m ultiplica tion.

=

M p for the hole

Excess Noise. Since the multiplication processes are random, an excess-noise factor can be defined for the multiplication factor M, which is treated as a random variable, (14.4.39) wh ere < >means ensemble average. It has been shown that these excess-noise factors [32, 33] can bewritten in terms of the ratio of the impact ionization coefficients an / f3 » :

<M > +

[1

Fp = ~(M > +

(1

Fn = f3

p

an

n

a

f3 p

P

::)[2- <~,» a, )(

f3 p

2 -

electron inj ection (14 .4.40a)

11

<M,,>

hole injection

(14.4.40b)

>,

Note that if no avalanche multiplications exist, (Mn (lvfp ) , Fn , and Fp .are all equal to unity. In Fig. 14.11, we plot F; vs. ( M n ) for various f3 n / a n = k ratios. We see the increase of the excess-noise factor F; with increasing f3p/an for electron injection. The physical reason for this is that , in the case 1000

u..o, ~

o ~

...ou

.-~~...............---y-~~.......--~

k or 11k =

. . . . . . ""7""""""

100

~

c,

~

(I)

'0

10

Z

Figure 14.11. Th e excess-n oi se fact o r F vs. th e m ultip lica tion facto r ( M) fo r different valu es of the rati o of the e lec tro n and ho le ionization coeffi cients. For ele ctron inje ction, k = /3p /a" and for hol e injection, an1/3" = 11k should be used in the ratio.

o 10

100

Multiplication Factor Mn or Mp

1000

14.4

AVALANCHE

PHOTODIODE~

613

of electron injection from the p + region in Fig. 14.7, the secondary electron-hole pairs also cause impact ionizations. The holes propagate in the opposite direction to that of the electrons. Therefore, if (3pla is increased, the backpropagating holes will create more impact ionization currents. The measured amplified current at the end electrodes will have more fluctuating signals since these secondary or higher-order impact ionization processes will contain more random characteristics. For hole injection, 1 If!, should be used in the same diagram . The multiplication or gain noise is given by ll

(14.4.41) The mean-squared shot-noise current after multiplication is generalized from the shot noise in (14.1.49) by adding the factor (M 2 ) «~) = 2q( t; =

+ Is + I D)<M 2 ) B 2q( I p + Is + ID)(MiFB

( 14.4'.42)

where I p is the average steady-state photocurrent (l4.1.34b), IBIS the background current, I D is the dark current [34-36], and B = tif IS the bandwidth. The thermal noise is (ij) -

(14.4.43)

where 1 I R e q = 1 I R j + 1 I R L + 1I R j , accounting for the junction resistance R j' the external load resistance R L and the input resistance R j of the following amplifier of the photodiode. For the modulation depth m = 1, the signal-to-noise ratio (power) for the avalanche diode is [8, 21]

N

t( q'YJPopt I

·2

S

lp

(i~)

+

(i})

hV)2 <M)2

2q( i, + Is + ID)(M/FB + 4k B TB I R e q

( 14.4.44)

Since F(M) > 1 and is a monotonically increasing function of the average multiplication ratio <M), the above signal-to-noise ratio can be optimized at a particular value of <M) . High-speed detections using avalanche photodiodes and their time dependence or frequency response have been investigated [37-40J. For M o > ani f3 p » the frequency-dependent multiplication factor is

Mo

M(w) - - - - - - 2] 1/ 2 [ 1 + (MOWT t )

( 14.4.45)

where T t is an effective transit time through the avalanche region. The effects of the avalanche buildup time have also been reported [39].

. .. .

I " "

("

,.r' .. .,.,

j,. .. .,...

~~

~ ,

,

_~.

"

' '," .,

- ,,.. .....

~~

-." -

- ~,..

I" ·

.. •.•

.• •

PHOTO DETECT':)R:>

614

Separate Absorption and Multiplication (SAM) APD [29, 30, 41, 42]. Various contributions to the dark current of an APD have been investigated , which include the generation-recombination via midgap traps in the depletion region, tunneling of the carriers across the band gap and a surface leakage current across the p-n junction in InGaAs APD, for example. When the reverse bias is above a certain value before the breakdown voltage VB' it has been found that the tunneling current is dominant, unless the doping density N D in the absorption region can be reduced below a certain value in which the tunneling current can be reduced to be smaller than the generation-recombination current. A separate absorption and multiplication structure has also been proposed (41] to reduce the tunneling current. The geometry is shown in Fig. 14.12a, where a low-field InGaAs region is used as

(a) SAM APD

Multiplication region

n-InP

Absorption re ion n·lnGaAs

V
p(z)

z -qN 0 2 -qN 0 1

(c) Electric field profile E

'--

= i. E(z)

-qN 0 3

E(z) I-----.-~-----..-+----------+__l......---~z

~

Small bias Large bias

Figure 14.12. (a) A schematic d iagram for a separate absorption and multiplication avalanche photodiode (SAM APD), where the absorption occurs at the narrow bandgap InGaAs region and the photogene rated carriers are swept into the InP multi plica t.ion region where the electric field is larger. (b) Charge density profile pC z ) under a large reverse bias. (c) The electric field profile (solid lines) under a large reverse bias. Dashed lines show the electric field profile for a small bias voltage.

•.

~

."

.~

.

14.4 AVALANU-IE PHOTODIODES

615

the absorption region and the photogenerated carriers are swept into the high-field InP binary region where avalanche multiplications occur [16]. Since the InP layer has a larger band gap than that of the InGaAs absorption region, the tunneling current can be reduced. The electric field profile E(z) = iE(z) can be obtained using the charge density profile p(z) based on the depletion approximation at a large reverse bias voltage since JsE(z) /Jz = p(z). Since 6 is slightly different in InGaAs and InP layers, a slight discontinuity in Ei.z ) occurs at the InGaAs/InP interface. Multiple-Quantum-Well APD. Heterostructure avalanche photodiodes have been fabricated for high-speed low dark current operations [41-43] since the late 1970s. Research on quantum-well photodiodes was started in the early 1980s. A plot of the excess factor F shows that the excess noise factor is minimized if f3 p / an « 1 using electron injection or an / f3 p « 1 for hole injection. In those limits, an ideal single-carrier-type multiplication process will dominate, and the excess noise caused by the feedback process of the impact ionization caused by the secondary electrons or holes in the opposite direction can be minimized. Ideas using multiple-quantum-well (MQW) structures for APD applications have been proposed and explored both theoretically and experimentally [44-52]. For example, consider GaAs / AlxGal_xAs MQW structures as the impact ionization region for electron injection (Fig. 14.13). Since s e.r s «, = 2:1 s, = 0.67 fl£g' fl E; = 0.33 fl £g), the electrons coming from the left barrier region will gain a larger kinetic energy fl E; when entering the barrier region than that of the holes traveling in the opposite direction. Therefore, the electron impact ionization coefficient an will be enhanced compared with f3 p • We expect a n / f3 p » 1. The excess noise factor is expected to be minimized. Measurements of the effective ionization coefficients an and f3 p show an enhancement of a n / f3 p from 2 in a bulk GaAs to about 8 in a GaAs/AlxGa1_xAs

cs

p

N

Figure 14.13. A multiple-Quantum-well avalanche photodiode using' GaAs/AlxGa1_xAs with the property that the ratio of the impact ionization coefficients an /f3p is much larger than one since t1 E; is larger than t1 E,..

I

I.

I' I

i

II .

'I

616

PHOTODFI'ECiORS

multiple-Quantum-well structure [45, 46] and M n , = 10 at an electric field E = 250 kY/cm. It has also been reported [47] that this ratio cx n / f3 p varies (not monotonically) with the aluminum mole fraction x. At higher values of x (~ 0.45) above the onset of indirect electron transitions, the noise is increased. Intersubband Avalanche Photomultiplier. Avalanche photomultipliers using an intersubband type (bound-to-continuum state transition) in an n-type doped or p-type doped multiple-quantum-well structure have also been proposed and investigated both theoretically [50, 51] and experimentally [52-54]. The idea is to introduce electrons in the quantum-well regions by doping the wells n type. Incident photogenerated carriers will "impact ionize" those carriers confined in the wells and kick them out of the wells via Coulomb interaction contributing to the avalanche multiplication current. Since this is a single-carrier type of photomultiplication, the excess noise is expected to be minimized. Experimental results on this intersubband avalanche multiplication have been reported [52-54] .

14.5

INTERSUBBAND QUANTUM-WELL PHOTODETECTORS

In Section 9.6, we discussed intersubband absorption in a quantum-well structure. To provide carriers for the intersubband transitions, donors for n-type electronic transitions have to be introduced in the quantum wells (or barriers) to provide free electrons which will be confined in the well regions at steady state without a bias voltage. When an incident infrared radiation illuminates the QW detector, electrons may absorb the photon energy and jump to a higher energy subband and be collected by the electrodes with an applied voltage. Theory and experiments on intersubband absorption and quantum-well intersubband photodetectors (QWIP) [55..:.87] have been investigated for long wavelength applications, which may be competitive with HgCdTe detectors. The advantages include the mature GaAs growth and processing technologies for high uniformity and reproducibility. For an extensive review of the subject, see Refs. 86 and 87 and the references therein. For n-type multiple quantum-well photodetectors, the optical matrix selection rule shows that the optical polarization must have a component along the growth (z ) axis, i.e., it must be TM polarized, as discussed in Section 9.6. For TE polarized light, the absorption is expected to be very small. However, . for p-type doped quantum-well photodiodes, the valence-band mixing effects due to the heavy-hole and light-hole states show that the x- and y-polarized light can have as large an absorption coefficient as the z-polarized light [87-90]. Therefore a normal incidence geometry is possible for p-type QWIP. In this section, we discuss mostly n-type QWIPs because of their potential for 3 to ~-,um and 8 to 12-,um photodetector applications. In Fig. 14.14, we

14.5

INTi:RSUBBAND QUANTUM-WELL PHOTODETECTORS

(a) 45°-Edge-coupled QWIP

617

(b) Grating-coupled QWIP

V

Multiple quantum-well k'"" absorption

y,

region

~11~~~~~b ~

~

j---±---t---~---I

Incident radiation '------tF-------l

hv

n" contact

AlAs reflector Thick GaAs substrate

Incident radiation Figure 14.14. Schematic diagrams of (a) a 45°-edge-coupled quantum-well infrared photodetector (QWIP) and (b) a two-dimensional grating-coupled QWIP.

show two examples using (a) a-4SO-coupled QWIP and (b) a two-dimensional grating-coupled QWIP. These designs provide the necessary polarization selection rule such that the infrared radiation will have a component along the growth direction of the multiple quantum-well absorption region. Our theory in Section 9.6 shows that the absorption spectrum is a Lorentzian function: a(hw)

r/(27T) =

ao

(E 2 1

-

hw)

2

+ (r/2)

2

(14.5.1)

where the intersubband energy E 2 l = £2 - E 1 is the subband spacing in a simple single-particle model as presented in Section 9.6. If the Coulomb interactions and screening effects are included, E 21 will have a slight shift due to the many-body effects [91-94]. The measured absorption spectrum [87] for a bound-to-bound transition is shown in Fig. 14.15 as a Lorentzian 2 shape. Note that the absorption is dependent on 1<1 ; therefore, a rotation of the polarization as a function of the polarization angle
PHOTODE rL CTOR.~

618

1.4r-------~----------------_,

1.2 1.0

WAVEGUIDE

w 0.8 u z
CD

~ 0.6 (f)

CD


0.2

oL-------2000

1800

1600

1400

PHOTON ENERGY

1200 11

1000

800

(em-I)

Figure 14.15. Measured QWIP abs orption spectrum for a multipass waveguide geometry. (Afte r R ef. 87.)

oscillator strength (i.e ., the intersubband dipole moment) and the escape probability can be optimized. For a bound-to-continuum state transition (Fig. 14.17b), the electrons have a greater probability to transport into the barrier region and be collected by the electrode and contribute to the photocurrent, although the intkrsubband dipole matrix between the ground state £1 wave function and the highly oscill atory continuum-state wave function may be smaller. For a simplified ana lysis [87 , 97, 98], we look at Fig. 14.17b, where P c is treated as an effective capture probability for an incident current I p for both the case (a) bound-tobound and the ca~e (b) bound-to-continuum transitions. Vie obtain Pc I p as th e fraction of incident current captured by the well and (l - P c)Ip as th e remaining curre nt transmitted to the next period. The incident infrared radiation cre ates a photocurrent i p: (1 4.5. 2) wh ere

l4.5

INTERSUBBAND QUANTUM WELL PHUTODETECTORS I .O@---~----------

6~9

---.

0 .6

I

0 .6

8-73-

.q:

e N

~

~

0 .4

0 .2

O'----'--......L---'---l---L-_--l..._-4I~--=~~.

30 40 50 60 POLARIZER ANGLE ¢ (DEG)

90

Figure 14.16. Experimental results for the polarization selection rule showing the peak absorption YS, the polarization angle eP where eP = 0° is TM polarization and eP = 90° is TE polarization, all at an angle of incidence Os = 73°, the Brewster angle. (After Ref. 95.)

(a) Thermionic emission (b) I p -+_-f::::::j~A.C:~Tunneling ~

.-

+-L~

Figure 14.17. (a) Bound-to-bound state transition and (b) bound-to-continuum state transition in a biased quantum-well infrared photodetector. The well width L w in (b) is designed to be small enough such that only one bound state exists in the quantum well and the second level £2 is pushed into the continuum.

~~'§,

_..

- -- "_ . . - ----" .

-

620

PHOTODETECTORS

From current continuity, we have

(14.5.3) Therefore,

(14.5.4) The total net photocurrent is (14.5.5) where g is defined to be the overall photoconductive gain, and 7J is the overall quantum efficiency of the MQW photo detector consisting of N w quantum wells. We have 7J ~ Nw7Jw if 7J « 1 since 7J is proportional to the absorbance of the structure . From 04.5.2) to (14.5 .5), we find

g= -

1 7Jw

- -=

1

(14.5.6)

Pc 7J

which gives the value for gain . The capture probability Pc decrease almost exponentially with the applied voltage [87].

IS

found to

Dark Current [87, .99, 100]. A simple model for the bias-dependent dark current I D is to take the " effective" number of electrons n*(V), which tunnel out of the well or are thermally excited out of the well into the continuum states, multiplied by the average transport velocity v( V), the cross-sectional area of the detector A and the electron charge q:

(14.5.7) where m*

n*( V )

,e

7Tfz-L p

f ( E)

f

I [

00

f(E)T(£ ,V)dE

(14.5.8)

£1

1

-

.-I

--=-----:::-:-~

(14.5.9),

where E F is the Fermi level measured from the conduction band edge (same as the first subband level £1)' and L p is the length of a period. T(E, V) is the tunneling probability through the triangular barrier with a bias voltage V. The velocity is

(14.5.10)

I

14. 5

I

Nt cRSUB

.

. .

BAN

D QU ANTUM-WELL PHOTODETECTORS

62 1

10-~

---

-..... <:{

10- 6

-

"U

~-

:2: lu

0:: 0::

10- 8

~

u

~

0::

<:{ C)

1 2 3 4 5 Figure 14.18 BIAS VOLTAGE, Vb(V) functi on . bark CUr re of the bias 1 n ts from me asured (solid curves) and calculated (dashed) data as a 107 . /-trn . (A.fte r Ref. 9v9~)tage a t various temperatures for a QWIP with a cutoff wavel ength

Where . /J.. IS the 1 bi la s Volta ge V and e ectron mobTt Vel' I 1 y, F is the average field determined , by the . OCity. the overall MQW width and u is the saturation drift The abo v . s ' a ve e slrnplifi d .h ry gOod ag e model has been used to explain the dark current WIt even mOre sin-. rel~firnent, as .up 1 ed d shown in Fig . 14 . 18 for a 10.7 p..m QWIP (99). An rno el [loa, 101] assumes that

T( E ) where E . II I B IS the ban i vv e obtain

er

hei

~ {~

( 14. 5 .11)

.

eIght on the right-hand sid e of the qu antum well.

(14.5.12)

where E

c :::: E B

-

E . 1 is the Spectral cutoff energy. The dark current becomes

(14 .5.13)

620

Pi-JOTODETECTORS

From

r

is determined from

(14.5.14) plotting In(JDIT) vs. (E e - E F ) should give a the result of E e - E F can also be compared with the optically measurea spectral cutoff energy Ec- This simple model has been reported to agree with experimental observations [87, 99, 102]. F)k 8 T ,

PROBLEMS 14.1

Consider a photoconductor (Fig. 14.0 that is an extrinsic semiconductor bar with a thickness d = 0.1 mm, a width w = 1 mm, a length t = 4 mm, and an acceptor doping concentration 10 15 em -3. Assume that the electron mobility f..L n = 3000 em" V-I S-1 (» J.L) and the applied voltage V -= 4 V. (a) If the photoconductor is illuminated by a uniform steady light such that the optical generation rate of electrons is G n , find an expression for the photocurrent 1p h = 1 - 10 , where 10 is the dark current, and 1 is the current when there is illumination of light. Find a numerical value for 1p h if G n = 10 16 em -3 s -) and 3 Tn = 10 s. (b) If the photoconductor is illuminated by a uniform light with a sinusoidal time variation, that is, GnU) = g cos co t, show that the photocurren t is given by the form

What are I p and ¢ in terms of g and Tn' etc.? What determines the 3-dB cutoff frequency in the frequency response of the photocurrent? (c) If the light has a dc (steady) and an ac component as may be used in optical communication, Gn(t) = GoO + m cos wt), where the constant m is usually caned the modulation index, find an expression for the photocurrent using the results from parts (a) and (b).

IT,

14.2

Expla in why the photoconductive gain than 1.

14.3

Derive 04.1.34a) and (l4.1.34b).

14.4

Derive the junction capacitance C, for a heterojunction using the depletion approximation in Section 2.5 for a p-N junction.

Tn

can be much larger

PROBLEMS

623

14.5

Derive the RoA product in 04.2.15).

14.6

Replot Figs. 14.6a-e for a p t-n "-n + photodiode, where a plus superscript means heavy doping concentration and a minus superscript means light doping concentration.

14.7

Derive (14.4.26) and (14.4.27a).

14:8

An avalanche photodiode with the electron and hole ionization coefficients an and {3 p is assumed to have a uniform field in the impact ionization region such that an and f3 p are independent of the position x. The electron and hole injections are given by In(O) and I/W), and the generation rate due to optical injection is G(x). (a) Write the two equations for the electron and hole current densities and solve for the hole current density as a function of x in terms of the injection conditions In(O) and I/W). Find the total current density I. (b) We assume that the electric field in the avalanche region is . - ... uniform such that an and (3p are independent of the position x. (i) If G(x) = 0 for all x, and In(O) = 0, find the multiplication factor for holes, M p , defined by

(ii) On the other hand, if G(x) = 0 for all x, I/W) multiplication factor for electrons

1M \

(c) Using the results in (b) for

n

=

0, find the

I =--

InCO)

M; and M p show that an and {3p can

be determined from (14.4.35) and (14.4.36) once M n and M p are measured:

!

14.9

an

=

(3p

=

Discuss the physics for the excess noise factor F( M) How can this excess noise be minimized?

=

<M 2) /

(M)2. !i

u

ir \!

I~ ~

;,

674

Pi ;OTODETECTORS

14.10

Derive 04.5.12) and (14.5.13).

14.11

Discuss the polarization selection rule for an n-type doped quantumwell infrared detector using intersubband transitions. Why are the configurations such as a 45°-edge-collpled structure or a grating-coupled structure used in the designs of these intersubband photodetectors?

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REFERENCES

62~

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i:

.,I

626

PHDTODETECTORS

34. H. Ando, H. Kanbe, M. Ito, and T. Kaneda, "Tunneling current in InGaAs and optimum design for Ln Ga.As y' In P avalanche photodiodc," Ipn, l. Appl. Phys, 19, L277-L280 (J 980). 35. S. R. Forrest, O. K. Kim, and R. G. Smith, "Analysis of the dark current and photoresponse of InO.53Ga0.47As/lnP avalanche photodiodes," Solid State Electron. 26, 951-968 (1983). 36. F. Capasso, A. Y. Cho, and P. W. Fay, "Low-dark-current Jaw-voltage 1.3-1.6 ,urn avalanche photodiode with high-low electric field profile and separate absorption and multiplication regions by molecular beam epitaxy," Electron. Lett. 20, 635-637 (1984). 37.

c. A.

Lee, R. L. Batdorf, W. Wiegmann, and G. Kaminsky, "Time dependence of avalanche processes in silicon," J. Appl, Phys, 38, 2787-2796 (1967).

38. R. B. Emmons, "Avalanche-photodiode frequency response," l. Appl: Phys . 38, 3705-3714 (1967). 39. H. Ando and H. Kanbe, "Effect of avalanche build-up time on avalanche photodiode sensitivity," IEEE J. Quantum Electron . QE-21, 251-255 (1985). 40. G. Kahraman, B. E. A. Saleh, W. L. Sargeant, M. C. Teich, "Time and frequency response of-avalanche photodiodes with arbitrary structure," IEEE Trans. Electron. Devices 39, 553-560 (1992). 41. K. Nishida, K. Taguchi, and Y. Matsumoto, "In GaAsP heterostructure avalanche photodiodes with high avalanche gain," Appl. Phys. Lett. 35, 251-252 (1979). 42. N. Susa, H. Nakagome, O. Mikami, H. Ando, and H. Kanbe, "New InGaAsjlnP avalanche photodiode structure for the 1-1.6 ,urn wavelength region," IEEE 1. Quantum Electron. QE-16, 864-870 (1980). ~L.;.

G. E. Stillman, V. M. Robbins, and N. Tabatabaie, "111-V compound semiconductor devices: optical detectors," IEEE Trans. Electron. Devices ED-31, 1643-1655 (1984). -

44. R. Chin, N. Holonyak, Jr., G. E. Stillman, J . Y. Tang, and K. Hess, "Impact ionization in multilayered heterojunction structures," Electron. Lett. 16,467-469 (1980). 45. F. Capasso, W. T. Tsang, A. L. Hutchinson, and G. F. Williams, "Enhancement of electron impact ionization in a superlattice: A new avalanche photodiode with a large ionization rate ratio," Appl. Phys. Lett. 40, 38-40 (1982). 46. F. Capasso, W. T. Tsang, and G. F. Williams, "Staircase solid-state photomultipliers and avalanche photodiodes with enhanced ionization rates ratio," IEEE Trans. Electron. Devices ED-3D, 381-390 (1983). 47. T. Kagawa, H. lwamura, and O. Mikami, "Dependence of the GaAsjAlGaAs supcrlattice ionization rate on AI content," Appl. Phys, Lett. 54, 33-35 (1989). 48. T. Tanoue and H. Sakaki, "A new method to control impact ionization rate ratio by spatial separation of avalanching carriers in multilayered heterostructures," Appl. Phys . Lett. 41, 67-70 (982). ' 49. K. Brennan, "Theory of electron and hole impact ionization in quantum well and staircase superlattice avalanche photodiode structures," IEEE Trans. Elecron. Devices ED-32, 2197-2205 (1985).

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628

PHOTODETECTORS

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99. B . F. Levine, C. G. Bethea, G. Hasnain, V. O. Shen, E. Pelve, R. R. Abbott, and S. J . Hsieh, "High sensitivity low dark current vl O J.Lm GaAs quantum well infrared photode tectors," Appl. Phys. Lett. 56, 851-853 (1990). 100. M. A. Kinch and A. Yariv, "Performance limitations of GaAs/AIGaAs infrared superlattices," Appl. Phys. Lett. 55, 2093-2095 (1989). 101. S. D. Gunapala , B. F . Levine, L. Pfeiffer, and K. West, "Dependence of the performance of GaAs/AlGaAs quantum well infrared photodetectors on doping and bias," Appl, Phys . Lett. 69, 6517-6520 (1991). 102. B. F. Levine , A. Zussrnan, J. M. Kuo, and J. de Jorig, "19 J.Lm cutoff long-wavelength GaAs/Al ...Ga1_xAs quantum-well infrared photodetectors," Appl. Phys. Lett. 71,5130-5135 (1992).

Appendix A The Hydrogen Atom (3D and 2D Exciton . Bound and Continuum States) [1-5] A.I

THREE-DIMENSIONAL (3D) CASE

The hydrogen atom is a two-particle system described by

H

=

-

tz2 --V2 2m 1 I

-

tz2 - - V 2 + VCr - r ) 2m 2 1 2

(A.I)

2

A general method to solve this problem is to change the variables to the center-of-mass coordinates R and the difference coordinates r:

.;

J J

I

.,

(A.2) (A.3) We define the total mass M

(AA) and the reduced mass m ; 111

-=-+m,

ml

m2

(A.S)

The first two operators in H can be rewritten as

(A.6)

where VR and Vr are the gradients with respect to Rand r. Equation (A.6) 631

THE

I-fYC:;:~ OGEJ~

A1\)M (:;D AND 2D EXCITON STATES)

can be derived using the fact that

a ax a ax a m1 a a -=-- + --=--+ax! aX l ax aX l ax M ax ax a ax a ax a m2 a a -= --+--=---aX 2 aX 2 ax aX 2 ax M ax ax

etc .

We can show the relation in (A.6) for all terms involving a2/ax~, a2/axi, a2/ iJ X 2 and a2/ ax 2 , then use an analogy for other components. The original Schrodinger equation

(A.7) can be separated into two parts if we let

where

(A.8) (A.9) and the total energy E T is

(A.I0) The eigenenergy and eigenfunction of the first equation (A.8) in the centerof-mass coordinates are simply those of a free "particle" with a mass M: EK

1i 2 K =

-

2

(A.ll)

-

2M

and eiK

f(R) =

·R

(A.12)

IV

Once !/J(r) is solved from (A.9), the complete solution is given by

e iK rJ; ( r 1> r J =

'R

Vv

Iff (r)

(A.13)

A.I

THREE-DIMENSIONAL (JD) CASE

6,~3

Solutions of \fler) for the Coulomb Potential

Using the definition of Vr 2

(A.14) the solution to Equation eA.9) with the Coulomb potential V ( r)

=

-

e2 -4-rr-c-r

(A.15)

can be obtained by the separation of variables:

ljJ(r) " "

=

R(r)e(e)ep(~) = R(r)Y(e,~)

(A.16)

and d2

--ep + m 2 ep

=

0

(A.17)

m 0 + A0 sin' e

=

0

(A.IS)

+ rVer) - E]R(r)

=

0

(A.19)

d~2

1 d ( ~- -- sin sin e de

11 2

1 d2

- - - - e r2R ) + Zm , r dr

de e-

de

11 2 A --R Zm , r 2

1-

2

Spherical Harmonics Solutions yfmeO, '1')

It is easy to see that the solution for

1

.

= - - e 1m cp

(A.20)

J2rr

where m is an integer, since
~

with a period 2rr.

g = cos e, leads to

(A.21) One notes that the above equation is even in g, that is, if g ---) - g, or equivalently, e ---) -tr - e, the same equation is obtained. Thus the solution is symmetric or antisymmetric with respect to the x-y plane.

'~Hi:

(j34

LYDROCJE;,r ATOi;

em A'..m 2D EXCJTON STATES)

If m = 0, the above equation becomes

-d [ (1 - t 2 )

dg

dP]

-

dg

+ AP = 0

(A.22)

which can be easily solved using the series expansion: co

(A .23)

A recursive formula is then obtained by substituting CA.23) into CA.22): k(k+l)-A

(A.24)

(k+l)(k+2)a k .

--

If the series does not terminate at some finite value of k,ak+2Iak ~ kl + 2), and the series will not converge at PCg = 1). Thus A must be an integer A = t Ct + 1) for some finite value t, that is, k = t, where t is an integer. The solution is

(k

P (C) _ f

S

-

1

[t /2] ~

( -

1) k (2

t-

2k ) !

~f-2k 2 t k'-::O (t-k)!k!(t-2k)r s

(A.25)

where [ t 12] means the largest in teger < t 12. Ip general, the solution to CA.2l) is given by the associated Legendre functions satisfying

where

1 - 2ft!

__

_~

(1

-

2) m /2

e

dt+m

de f + (2 e - 1) r m

(A.27) for positive m .::;

t.

For negative m with

1m!

~

t, 1m! should be used in

CA.27)

A.l

us

THREE-DIMENSIONAL (3D) CASE

since the differential equation (A.26) should give the ' same results for The first few polynomials are

t=o,

Pg(t)=l

t= 1,

P10(t)

=

t

pl(t)

=

11 - t 2

t=2,

+ m.

pf(t)=~(3t2-1)

Pi(t)

=

3tl1 - t

p}(t)

=

3(1 - t

2

2

(A.28)

)

.!

Noting that

;.

:. .'

f with 0 <- m < c,' -taine d :

t,

[PF(t)]2dt=

l

2t

-1

(t+m)! + 1 (t - m)!

2

the normalized solutions for the angular dependence can be

2t

"+

1

(t - Iml)! ( t + Iml)!

- - - ---~- (_1)(m+lml) /2 p).ml(cos

4'1T

and

(A.29)

e) e i m tp (A.30)

"

f

2 7r

(7T

J,

2

IYf ( e, cp) I sin ede d cp

=

1

(A.31)

cb =O 8=0

The first few spherical harmonics are

t

=

0

(s orbit)

Yoo

1 =--

J4rr

(A.32a)

r·

636

THE

1= 1

;r ( I ) i ~O G E N

}\ T O M OL AND 1 D EXCITON STf'. TES)

(p orbits)

{;3 Of; -

477

Y , ±lO,'P) =

cos f3 =

{;3

Z

_ . - == 4'77" r

3 5in Oe± ;. 8",

~

Of;

IZ) 3 x ± iy 8", r

1

- + - jX±iY )

(A .32b )

,fi

1= 2

(d orbits)

Y , oCO,'P)

~ ; 16rr 5

Y z ± 1( (3, 'P) = + Y

((3 Z± 2

m) ' T'

=

/

~; 16rr 5 [3Z r

2

(3C05 'O - 1)

{!ls -

-8 sin f3 cos '"

(3 e

::!:

15 . - sin? (3 e ± 2 1"., = 32",

-

2

.

1'1'

=

fi'S

+ -

{ff

1)

-

(x + iy ) z

-2

8",

. -,-

r

15 (x + iy Z ) 32", rZ

-

(A.32c)

Radial Functions for the Bound and Continuum States The r adial function Rt r ) satisfies Il Z

h2

1 d2

- ---z(rR) + Zm , r dr

t (t +

Zm , ;

.'

r

2

1)

R + [VCr) - E]R = 0 ( A.33)

\,

which may be rewritten in the form r

d Z _ t( I+1 ) Zm , 'I { _ + ~ z [ E - V( r )] f~u (r) z

dr

r

2

rt

=

0

(A.34)

where

u(r ) = rR(r)

(A .35)

Let us define the Bohr radius

(A.36)

A.I

THREE-DIMENSIOl'{AL (3D) CASE

and 1

ko

=

The Rydberg energy R y is defined as

(A.37)

In the following, we consider the bound and continuum states. Case 1: 3D Bound State Solutions (E < 0). For bound states, we define the variables

(A.38) 2r

p=v«;

(A.39)

Equation (A.34) reduces to

(AAO)

If we look at the asymptotic behavior of the function u(p) at p ~ co from (AAO), we find (d 2/d p 2 - t)u(p) = 0 and the solution should behave as exp( p /2) or exp( - p /2). The latter term exp( - p /2) should be chosen, since the former blows up as p approaches infinity. If we look at p ~ 0, the differential equation (AAO) is

d 2

[

.

-d 2 p

t( t + p2

1)] u(p)=O

.'

(AAl)

for t =/= 0, and its solution is either p t+ 1 or p -to The former should be chose~ since ui p) should be a regular function at the origin. Thus, in general, we .may assume u( o) of the form u( p) = e -p /2 p i' + l/( p)

.'

(AA2)

!

'~

~ -.

'- '

.,... \ ...... .'"t , -I' . ,; ,• •• •..,......... "' ••••

THE HYDE.O (;[:

638

,~

_~ ,, --

~ . _ .

}\T O M U P AND

_ .

~

...

~

.~ D

•• •• .

~

. . .... . ~ •. ~ - . ,- -' _ .

- ~

"

. ,

EXCITON STATE.S)

Equation (A.40) reduces to

d 2f ( p ) df(p) P .d 2 +(2t+2-p) dp +(y-t-1)f(p)=O p

(A.43)

By checking a series solution to the above equation, it is found that (y 1) must be an integer N, N = 0,1 ,2, .... Or

=N+t+ 1

y

t-

(A.44)

=1l

Using n as the principal quantum number, we have

t=

0, 1,2, ... ,n - 1

(A.45)

The solutions of the differential equation of the form d2

df

p - f + ({3 - p)- - af= d p2 dp

°

(A.46)

are the confluent hypergeometric functions,

F(a,f3 ;p)

=

a p a( a + 1) p2 a( a + 1) (a + 2) p3 1 + f31! + f3(f3 + 1) 2! + (3({3 + 1)(f3 + 2) 3T + (A.47)

Note that Ft;a, f3 ; 0) = l. We can identify that solution for ui p}:

a =

-

n

+ t + 1 and f3

=

2t

+ 2 and obtain the (A.48)

We can check from the definition in (A.47) that the polynomial F in (A.48) terminates after a finite number of terms since t < n - 1. Alternatively, the associated Laguerre polynomials L';:(p) are used. It is related to F by m

L';:(p) =( -1)

(n !)2

I( _ ),F(-(n-m),m+ l;ZY ' m. n m.

(A.49) Note that Lr;:(O) = (_l)m(n!) 2/[m!(n - m)!]. Therefore, ui r ) can also be

A.I

THREE-DIMENSIONAL (3D) CASE 63~

written in the form

(A.50) where C is a constant to be determined by th e normalization Condition . Using the in tegral identity

lo

co

e

-p

P

2!+2{L 2f'+ I ( p ) J 2 d n+f'

2n[(n +1)1)3 (n -1-1)!

p= ~

(A.51)

or

f

jOOe - p U +2[F ( _ n +f+ 1,21'+ 2;p)]2 dp = p

.1

2n[(2f+ 1)!]2(n '-f-l)'

(n+I)!

o

f

I

.

f I

(A.52) we find that the radial wave function satisfying the nor . . mallzatlOn Condition

I

f I' f

J

l°OR2(r)r2dr = 1 o

i

(A.53)

IS

R

n ,

3/2(

2 ) ( ( r) = na o

(n - 1 - 1)! ) 2n[(n

+ f)

11'

)3/2[

(2/na o (2/+1)!

--

x

(n + 1 )! 2n(n-I-1)!

1/2

}l/2

I'

e-

' / ( OOO

)(!f:) L~~i' (!f:)

. 0;;;;;

e-r/na (

t

2r)

F(- + 1+ 1+ 2;~) 1, 2

n

nao

As r ~ 0, Rn !(r) ~ (2r /ncz.o)t,. which is nonzero only when The complete wave Iunctions are 1 = O.

.

_

1

\1;;

.'

(A.55)

The first few wave functions are .1, _ 'f/ 100 -

.'

(A.54)

(1)3/2 _

ao

e -r/a o " __ •

. ( 15 state). I

I.

i

! ;

{ i

-. ...-

1i~~'-~.

.._....-~._.- ....... '\._ ._ .. -' .... __..,.,.,.

. .. ',. . - "':.-0

riu.

,

~_

•.-

~-

·

C_..'f'7"'~_·.~.· _

"~-:'~~ ' ''1"' .- " ., :-r-; "

-"f" '. ,. '

._

LYI)[tJI,_',!::: .J ATOM (}D /\N9:D EXCITON

,..,

..

STATES)

(2s state)

.f, 'PZ1O

r/J21 ± 1

1

= 2v~ 27T

(J -

«:

r

= O,l/J

=/:-

r

--

)

e- r /

Z lI o

cos (]

(2p state)

z»;

1 (1 )3/Z(

= - - -4[;; «;

Note that at

)3/2(

r)

,

- - e -r/Za o sin (] e ±

z«,

t

0 only if

= m =,

iii'

(2p states)

(A.56)

0 (s states). We have

2 (

na o ) 3/

2

and

(A.57)

The wave function at r = 0 will be useful when we study the exciton effects on opt ical absorption in Chapter 13. Case 2: 3D Continuum-State Solutions (E > 0). When the energy is posi, tive, the solutions will not be quantized. Instead they become continuous. The solution to (A.34) for the radial wave function is -obtained following the change of variables as in (A.38) and (A.39),

yZ

,

=

(A.58)

2r p = - - = 2iKr vI a 0

We choose Ill' = ix, where number, since

K =

..jEIR y

E=

2m,

-

(A.59)

=klk o is the normalized wave

(A.60)

A.l

THREE-DIMENSIONAL OD) Cf\SE

641

and the Rydberg is

Ry =

(A.61)

and !: = r / a 0 is the normalized radial distance. The solution u(p) has the same form as in (A.48),

(A.62) We therefore look for the normalized radial wave function of the form -.

RKtC~) = N, ( 2Kr) e exp( - iK~) F( f

+1+

K

~, 2 f

+ 2; 2i«[J (A .63)

such that (A.64) in the ~ space. Later on, we change the normalized variable ~ to the real spacelistance r, and the 8(K - K') normalization rule to the 8(E - E') normalization rule. We note the asymptotic behavior of F,

(A.65)

as [z ]

~

00 ,

We thus have

_

f

- N K f ( 2 K r ) (2 t .

-

,

+ 1). [

exp ( 1. K !: )

(,..,.

d K~

r(G + iJ

1

) -t-l + i

+

,\

11K

)

/

/K

+

c.c.]

(A.66)

T H E rr ( i) R OG E f-.J' }\T !)1vi (3D AN D 2]) EXCITON STATES)

fA 2

where c.c. means complex conjugate of the first term in the large bracket. We use Tt z ") = [T(z I]", 2iK[ = exp]i 71"/2 + InC2K[)],' and

(2 'LKT- ) - t - ,l+i /K

=

(2'IKT ) - t-1(2 lKT ' ) i/ -

K

~

The radial function can be written as for

e-

(2 t'+1)! bKt=NKt

7T

00

(A.68a)

/ (2 K )

If(t'+l+i/K)1

K

T -')

(A .68b)

and the phase factor is

(A.68c) The normalization condition is d etermined by the asymptotic behavior of R Kt as!. -') :t:'. Since K [ » Cl/K)ln(2K[) and 0K '( , we find

(A.69 )

Therefore, we choose bK ( = V2/7T to .satisfy the O(K - K') normalization rule (A.64). In deriving (A.69) , we have expressed COS(KT) as the sum of two exponentials and use !

(A.70) noting that

K, K'

are both positive..' F rom b

_ f2 N r - V-;; K

K

K {

,

we find

+ 1 + i/K)I (2 t' + I)! ,

e'TT/ (2K )lf( t'

(A.?! )

A.,

THREE-DIMENSIONAL (3D) CASE

643

and RKrC!.) is given by (A.66). The quantity [I'( ated noting that for

t

t+1+

i/K)1 can be evalu-

= 0,

(A.72) For t> O,fC(+ 1 + ilK) ilK), and

~)

r(t+l+

=

2

Ct+ i/K)(t- 1 + ilK) ... (l + i/K)fO +

~)r(t+l- ~]

=r(t+l+ =

[n (S2 + ~)] 5 =

1

K

2

TrIK

sinh( tr I

(A.73) K)

Therefore, we obtain

»:

=

['2:

V;

Ke /(2K) [n" (S2 + 1T

(2t+1)! 5=1

K_ ]

I _1 )] 1/2[_.7r_ K

2

1/2

smh(7rIK)

(A.74)

If we change the normalization rule to the physical quantities in terms of R E f(r) . '1 the real space r and use the energy normalization rule,

(A.75) we obtain _i_, 2 ka o

t+

2; 2ikr)

(A.76)

(A.77) Again, as r ~ 0,

J!.EeCr)

~ C2kr)!, which vanishes except for

t

=

O.The

!'HL

644

}-!YDPOGE~' ATC,l':l

(38 AND 2D EXCITON STATES)

complete wave function is given by

(A.78) where the spherical harmonics Yt m(e, 4» are given in (A.30). At r = 0, we find !/JEtm(r) =1= 0 only if t = m = O. Therefore, using CEO = [e 1T / K j sin h( 7T j K)JI/2 and Y oo = Ij,J4'IT, we obtain 1T

K

1 [ e 1 I!/FEoo(r = 0)1 = - ya R 3 . h( / K ) -4 SIn 'IT 'IT o 2

A.2

/

]

(A.79)

TWO-DIMENSIONAL (2D) CASE

In this case, the position vectors of the two particles r l , r 2 , the center-of-mass coordinates R aJ;1~L the difference vector r are all in the x-y plane. All the expressions still follow the equations (A.l)-(A.ll), except that only the .r-y dependence exists, i.e.,

VCr) = where r = xX + YY =

(A.80)

47TSf

x 2 ) i + (YI - Y2)Y' The solution is

(Xl -

e iK · R

!/F(r 1 , r 2 ) = -

a

!/F(r)

(A.8l)

where A is the cross-section area, and !/F(r) satisfies

(A.82)

The solution is of the form

(A.83) and the radial function satisfies 1 d d -; dr r dr [

m

7

2

2m r

+

~

l (

2

) 1 e E + 4;';; R( f) = 0

(A.84)

A.2

TWO-DIMENSIONAL (2D) CASE

Case 1: 2D Bound State Solutions (E < 0). Using a change of variables, the same as those in the three-dimensional case (A.38) and (A.39),

Ry

-E

(A.85)

2r p=

(A.86)

we obtain d2 ( d p2

+

1 d

-~

P dp

m

2

- -

p2

1]

Y P

+ - - - R(p)

=

4

0

(A.87)

As p --) 00, we find the dominant terms above are (d 2jd p2 - i)R(p) Therefore, we set Ri.o) = e- p / 2h ( p ), where h(p) satisfies . 2

d [ d p2

+

=

O.

-......

(1- p) d P dp

1( Y p

1 2

+ -

m2 p

-

-

) ]

h(p)

=

0

(A.88)

As p --) 0, we find that the dominant terms are 2

1 d m d2 + - - ( dp2 P dp p2

Therefore, R(p) behaves like find that f( p ) sa tisfies 2

• [ P-2 d + dp

p1m l.

)

h(p) = 0

(A.89)

We then assume that h(p)

=

d + ( Y - -1 - Iml J] f(p) (2Im/ + 1 - p)-

dp

2

p,m1f(p), and

=

.

0 (A.90)

If we compare the above with the definition of the confluence hypergeornetric function Fi cc, {3; p ) in (A.46) and (A.47)

d2 . d ] P d 2 + ({3 - p) dp - ex F(~,{3;p) [ p

=

0

(A.91)

we obtain

f ( p)

=

F ( Im I +

t -- y, 21 m I + 1; p )

(A.92)

The above polynomial should not approach infinity faster than any finite power of polynomials, and the acceptable solution is only when jml + -i - Y

Tli F I-!'(DRO ::JEl': ATOM

646

( 3~)

AND 20 EXCTON STATES)

t

:L n

is a negative integer, i.e. , y ~ Iml + or use 'Y = n n ~ Iml + 1. The radial function is given by . R( p) = e-p/2pltniP(lm l

'Since 'Y = n -

!, we

+

1 - n , 21ml

+

= 1,2,3, ... and

1; p)

(A .93)

n=1,2,3, ...

(A.94)

find the energy levels

E ri =

Using the relations

(n!)2

m

Lm(z)

=

(-1)

m!(n-m)!

n

F[

-en - m),m + l;z]

(A.95)

and 2 (n+m)! e-pp2m[L~:m(p)] pdp = (2n + 1 ) - - o (n-m)! 00

1

(A.96)

we obtain the normalized radial wave function Rnm(r) satisfying

(A.97)

.-

4!

(n - Iml - I)! a o V (2n - 1)3(n + Iml- I)! .'

x e- p/ 2p lmIL 2Iml (p) 11+1171 1- 1

4

(n + Iml - I)!

ao

(2n-1)\n-lml-l)! e- p /

X

2

(2Iml)! p1m1p( -n + Iml + 1, 21ml + L; p) (A.98) 2r

p=

(n - t)a o

n =1,2,3, ...

and

Imls:;n-] (A.99)

A.2

TV,'O-DIMENSIONAL (2D) CASE

647

The complete wave function is (A.IOO) As r ~ 0, Rnm(r) ~

p1ml,

which does not vanish only if m

It/Jno(r

=

0)1

=

2 1 IRnO(O)1 27T

-

- --

2

=

O. We obtain

1

(A.lOl)

------::-

7Tao2 ( n - '21)3

The above result is l/(n - i)3 per area of the circle determined by the Bohr radius Qo' .. Case 2: 2D Continuum State Solutions (E > 0). The procedure similar to that in the 3D case:

IS

very

(A.I02) p

= 2iKr

=

(A.103)

We chce 1 = IK ]I

and define ~

=

.-

\ " ,

rI Q o again. The radial function Ri p) behaves like

R(p)

=

Canst.

e-p /2p lmIF

.' l '

( Iml + 2 + ~, 21ml + 1; p

)

(A.I04)

i

If we assume that

and follow the same steps as that in the 3D case, we obtain

~ (2Iml)! err/(2K)

N

= « m.

-

7T

(

1

i )

2

K

r Iml + - + -

(A.106)

··· n

.,~ .

· . ' r, ~

. ...-1'"O"'~ .. - - "O;" ~.

- :... ~ ,

-r,,,,,• •': _ ~

" ".',~ - "i ~'"

648

.~

.._. """'. ' _ _. •_ .

THE HYlJROGEN ATOM C3D A ND 20 EXCITON STA"j'ES)

and R'(I/!) satisfies the (j(K - K') normalization rule

(A.107)

The only minor difference between the 2D and 3D radial wave functions is that in 2D, as r ---+ co, the prefactor in front of the cosine function is given by

l l{i.

-

(A.I08a)

where

fT e N Km{2Iml) ! V-;; If(lm l + i

-'TT j{ 2 K)

b,m l

=

«: ~

(Iml +

~):

.

+ ilK) I

+ arg [ r ( ml +

~

+

(A.108b)

~)]

(A.108c)

and bKm = -/2 1Tr is determined by the 8(K - K') normalization rule. If we change back to the physical quantities in the real space r instead of the normalized dist ance v = rlao, and use the 8(E - E') normalization rule, noting that

.(A .109)

.'

we find for

(A .lJO )

RKm(r ) /2KR y «;

(A.lIl)

A.2 TWO-D iM ENSIONAL (2D) CASE

Using

r( 1+ K)i ~ r(1

i)

2

2

+ K

2

r(1

~o J

2 -

cosh( 7T /K)

(A.112)

and

1

r[lml + 2 +

i K

)

2_

n

[(s _ ~)2

+ 1]

s = 1 2 K2

(A.113)

7T"

cosh ( 7T /

K)

we obtain

CE'( ) (2kr)lml ., e -ihF ( 1· J f2R; Iml + - + -'- , 21ml + 1; 2ikr 2R a 21ml ! 2 y

K

o

(A.114a)

(Ao114b)

Note that E = h 2k 2/2mr> and tion is given by

K

= k/ k ; =

ka o . The complete wave func-

(}\.115) The wave function approaches (2kr ) Iml as r except for m = 0:

0, and it vanishes a

---+

r

=

°

.'

2 /

1 -

- - - ; ; - - - - ; : - --

-

-

-

-

R }' a ~ 2 7T" [1 + exp( - 2 7T" /

-

"7"

K) ]

Important results of this appendix are summarized in Table 3.1.

(A.116)

650

THE HYDROGEN ATOM (3D AND 2D EXClTON STATES)

REFERENCES 1. For both bound and continuum state solutions in the three-dimensional case see L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3d ed., Pergamon, Oxford, UK 1977, p. 117, and H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, Berlin, 1957 . 2. For an n-dimensional space, n :2: 2, see M. Bander and C. Itzykson, "Group theory and the hydrogen atom (0 and (IO," Rev. Mod. Phys. 38, 330':..-345 (1966), and 38, 346-358 (1966). 3. C. Y. P. Chao and S. L. Chuang, "Analytical and numerical solutions for a two-dimensional exciton in momentum space," Phys . Rev. B 43, 6530-6543 (1991). 4. E. Menzbacher, Quantum Mechanics, 2d ed., Wiley, New York, 1970. 5. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, World Scientific, Singapore, 1990.

.'

Appendix B Proof of the Effective Mass Theory B.!

SINGLE BAND

To prove the effective mass theory (4.4.5) in Section 4.4, let us write the Bloch function

(B.l) satisfying (404.3). Since {l/Jnk(r)} is a complete set of basis functions, we may expand the solution l/J(r) in terms of these functions:

l/J(r)

=

Lf n

=

B.Z.

Lf n

B.Z.

d 3k

(27T)

3anCk)l/Jnk(r)

d 3k

(27T)

. 3

(B.2)

a n(k) Ink)

where the integration is over the first Brillouin zone (BZ) in k space. Multiplying (4.4.4) by l/J:k(r) and integrating over the Volume, we find \,

where

(Bo4) and the orthonormal relation (nkln'k') = us consider the Fourier expansion of Utr)

s; n,8(k -

k') has been used. Let

,

(B.5) 651

PROUF OF T HE

652

EFFI~CTIVE

MASS THEORY

We find

(B.6) Since the product u~k(r)un'k,(r) is periodic in r, we may expand this product in terms of a Fourier series u~k(r)Un'k,(r)

=

E C(nk, n'k' , G)e iG '

r

(B.7)

G

where G sums over all reciprocal lattice vectors. Therefore, ( n k IV In' k') =

E C ( n k , n' k' , G) [;

k - k' - G

(B.8)

G

Here k and k' are in the first Brillouin zone. We assume the following: 1. The perturbation !V(r)1 is small enough that there is no mixing between the bands (single-band case, n = n''). 2. VCr) is slowly varying in r. Thus O, is falling off rapidly for large k,

(B.9) or equivalently, Ll~kLlnk' = 1 is assumed. 3. The integration over k is mainly contributed from k = k o (an extremum or zone center) since we are interested in the behavior of energy near ko' For convenience , we take k o = O.

.-

The resultant equation for a)k) becomes .' (B.lO)

Define an envelope function F(r)

(B .ll) We obtain [E n ( -iV) + U(r)]F(r) = EF(r)

(B.12)

from the inverse Fourier transform of (B.lO). The above equation is the effective mass equation in real space.

B.2

DEGENERATE B/\.NDS,

653

The total wave function is

fa

=

(k')eik'· r

d 3k'

u .o( r(217)3 )-nk

ri

(B.13)

F(r)unko(r)

=

using the leading order approximation for unk(r). Using the k . p theory in Section 4.1, we have

EnC k )

En(O) +

= .-

I: Da{3kak{3

a,

p

(B.14)

Thus, the effective mass equation can also be written as

[L

a,p

d_)

D {3 ( _i_d ) (_i_ CX

dX cx

+ U(r)]F(r)

=

[E - En(O)]F(r) (B.I5)

dX{3

where Da{3 is given by (4.1.13). Alternatively, Da.{3 = (h 2/ 2)( 1/ m * )a{3 ' and we can write (B. 15) as the effective mass equation (4.4.5).

B.2

DEGE!'lERATE BANDS .~

To prove the effective mass theory (4.4.11) for degenerate bands [1], we choose a complete set of basis functions: (B.16) Expand the wave function !/fer) for (4.4.9) in terms of these basis functions:

(B.17) We proceed as before and obtain

654

PROOF OF T1 [E EFFECTIVE

MASSTHE01~Y ,

For the choice of the basis functions as in Section 4.3 for the Luttinger-Kohn Hamiltonian, Pji = 0 for all i, j = 1, ... ,6. That is why we have to go to a second-order perturbation theory. Define,

(B.19) We obtain in k space

[E - Ej(O)JaJk) (B.20) Therefore, we obtain the effective mass equation in real space:

t [E D/;P( -i~ I(_i_a

j'=1

a,f3

\

aX a I

J + v(r)OJJ']Fj,(r)

=

[E -

Ej(O)JFj(r)

aXf3

(B.21) where 2

D':'~

))

h [ = -- 0

2m o

0

0

,(5

11

af3

+"

p?' pf3.. J'Y

'YJ

+ p~rt pa" 'Y1

]

c: m 0 (E 0 - E y ) 'Y

(B.22)

and the wave function in the effective-mass theory is given by 6

ljJ(r) =

E Fj(r)ujo(r) . .' j= 1

(B.23)

\

for the leading-order approximation for ujk(r) =:: ujo(r). Notice that if VCr) == 0, the solution to (B.2}) is indeed the plane.' wave function, that is, Fj(r) = a /k)e ik· r, j = 1, 2, ... ,6, where {a j(k), j = 1, ... ,6} is an eigenvector of (4.3.16).

REFERENCE 1. J. M. Luttinger and W. Kahn, "Motion of electrons and holes in perturbed periodic fields," Phys. Rev. 97, 869-883 (955).

Append.ix C

Derivations of the Pikus-Bir Hamilton-ian for a Strained Semiconductor In this appendix, we derive the Pikus-Bir Hamiltonian [1, 2] discussed in Section 4.5. Using the coordinate transformation between the uniformly deformed coordinates and the undeformed coordinates (4.5.2) and (4.5.4), we obtain 3

r;

=

ri

+ ~

C i j rj

(C.la)

j = l

noting that both (x', y', z') and (x, y, z ) are components using the same basis h " y, '" an d'" vee t ors x, z, were r 1 = x, r 2 = y, T 3 = z, an dr'1 = x," r 2 = Y, , T; = z'. For example, x' = x + cxxx + cxyY + ExzZ. In a one-dimensional case with strain along the x direction only, we have E xx = (x' - z ) / x and Cr y = C x z = 0 . If a shear-type strain is introduced, in the x-y plane, we have E x y = (x' - _\ .' y, when E x x = E .r z = O. In vector form, we have r'

=

r

+£.r

(1 + £) . r

=

(C.lb)

The inverses of (Ci l a) and (C.lb) are

r.

=

r; - ~ cijrj

(C.2a)

j

r =

(1 -

~)

. r'

(C.2b)

In the uniformly deformed crystal, the potential is still periodic except that the function VCr') is a different potential from Vo(r). We write the Schrodinger equation for the Bloch function in the deformed crystal coordinate system r':

(C.3) 655

•

656

PIKUS-BTR EAMILTONI AN

pan A STRt-\fNED <)EMICONDU,CTOR

Using the chain rule and (C.2b), we have

orj a

or' I

a

a

L - - = - - L£jj -a . ar~ ar. Br,I . r), J J I

(CAa)

)

or p'

= p . (1 - e)

(CAb)

Similarly,

(C.S) The periodic potential can be expanded as

V[(l + s) . r] = Vo(r) + L f/;jCjj

(C.6)

i, j

-

. .. .

where

Therefore, Eq. (C.3) becomes (C.7)

where (C .8)

(C.9) Noting that we intend to treat He as a perturbation due to strain, and H o contains Vo(r) with a period equal to that of the undeformed crystal, we write the Bloch function as

l/!nk,[(1 + s) . r)

=

eik"r'unk,(r')

= eik"(I +E')'runk,[(l + s) . r] =

eik'ru~k(r )

(C.IO)

where k = (l + s) . k' ha s been used, and we rename undO + e) . r] as u~k(r), for the strained Bloch periodic part, which is to be determined .

C.lSINGLE-BAND CASE

657

Substituting (C.IO) into (C.7), we obtain Ho[eik'ru:;k(r)]

(C.Il) (C.12)

Therefore, we have the equation for the wave function u~/r):

(C.13a) (C.13b) (C.13c) (C.13d) (C.13e) The above result is similar to that in the original paper of Pikus and Bir, if we note that k' = (1 - £) . k. Our conventions are that primes k' and r' are used for the deformed crystal and k and r are used for the undeformed crystal. Since the perturbatior term H k vanishes in the first order (Hk\m = 0 because of parity considerati .1, we seek perturbation terms second order in k and first order in CaW

C.l

SINGLE-BAND CASE

For example, a single conduction band is probably the simplest case. The perturbation theory for the single subband leads to U~k( r)

= u n O( r)

Houno(r) = En(O)unO(r) h h2 (k . p) (k' p ) . En(O) + - - + (HJnn + - 2 L nn' . n'n 2rn o rna n'*n En - En' fz " (k : p) nn' ( H fJ n'n + (He) nn' (k . p) n'n 2k 2

E

=

+-

LJ

rna n'*n

=

En(O) + (Hc)nn +

( C.14a)

En - En'

LD

Ct{3k

ak{3

(C.14b)

a,/3

where

Da{3

is the same as (4.1.13). We have used the fact that the term in the

658

PIKlJ~ ·_PIR

HAMlf TC.':'iIAN FOR A STP..AINED

Sf~MICONDtJ: ~TOR

perturbation theory

where H' = (fl/mo)k . p + He + H ek, has nonzero contribution only from the first term (Ii /mo)k . p because of parity consideration, and we keep terms up to the second order in k and the first order in e. The diagonal term (nIH"ln) leads to ( C.15) because of the isotropic nature of the conduction band edge. Therefore, the single-band (conduction band) strained dispersion relation is given by

(C.16 ) and ( C.17)

C.2

DEGENERATE BANDS

We assume in general that A .

.-

B

Laj(k)u;o(r) + Lay(k)uyo(r) j

(C.18)

y

as in Section 4.4, where class A consists of those bands of interest, such as the degenerate valence bands: two heavy-hole, two light-hole and two spinsplit-off bands. Class B consists of those bands outside of class A. We follow the same procedures in Lowdin's perturbation method presented in Section 3.6 and obtain H' H' iv vi' E E 0 y

(C.19a)

E Oy ) aj' ( k) = 0

(C.19b)

A U} ·,=H.+ j}j

LB v rii'

A

L (Lj; /

from 0.6.12) and (3.6.5), respectively. The result turns out to be almost the

C.2

DEGENERATE BA1'\DS .

6"::9

same as that in the Luttinger-Kohn Hamiltonian: u~, Jj

=.

0.·,£-(0) + ~ '" DCi~k jJ J 11 a k{3 a,{3

(C.20)

or

H

=

H

LK

+ He

(C.21)

=

The first two terms in (C.20), E/O)ojj' + La,{3Dj~~kak{3 H j7!'" are exactly the elements of the Luttinger-Kohn Hamiltonian, and have been expressed in matrix form in (4.3.14). The third term in (C.20)

( H ) 11.., = E

jjCi~E: {3 J1 a

'~ "

(C.22)

a,{3

is the linear strain Hamiltonian. The last term in (C.20) vanishes for a lattice of diamond type, in which there is a center of inversion, and either Ph or (H)'Yj vanishes. For crystals that do not li:e a center of inversion, this term may not be zero. It means that an extremum in k space can be shifted due to deformation. However, this term is considered negligible for most III- V .compounds, We can also express the third term due to strain by identifying \,

(C.23a) Therefore [3],

h2 Y l -2m o

~Dd = u

h2 y z

-2m o

h2 Y 3

-2m o

Du ~

3

D'u ~-

3

-al-'

- -

(C.23b)

b -

2

(C.23c)

-d -

213

(C.23d)

PJKUS-BIR j-1,\MJ: ,,01\;1 AN FOR A STRAINFD SEMICONUJCjOP

660

where the constants D td , D t l , and D:/ come from different components of D/;r In matrix form, we have [3, 4] 1

P+Q

-S

R

0

--s fi

fiR

If, f>

-s+

P-Q

0

R

-fiQ

Ifs

R+

0

P-Q

S

{fs+

fiQ

If, ~) I~ _ 1.)

1

R+

s+

P+Q

-fiR +

--s+ fi

--S+

- fi Q

ffS

-fiR

P+~

0

fiR -+-

{fS+

fiQ

--S

0

P+~

H=-

0 1

Ii

1

Ii

2'

11

2'

2

_1) 2

It, t> 11.2' - 1.) 2 (C.24)

where P = P,

+P

E

R = R; + R B

Q = Qk + Q S = Sk

+

SE

B

(C.25a)

(C.25b) .'

PE = -

a

L' (

E: x x

+

E: y y

+

E: z·z )

( C.25c) where the wave vector k is interpreted as a differential operator - iV'; cij is the symmetric strain tensor; Yl, Y2' and Y3 are the Luttinger inverse mass parameters; au' b, and d are the Pikus-Bir deformation potentials; and A is the spin-orbit split-off energy. The basis function I i, rn) denotes the Bloch

REFERENCES

6Gl

wave function at the zone center and is listed in Eq. (4.3.3). Here the energy zero is taken to be at the top of the unstrained valence band. The Hamiltonian (C.24) has been used extensively to study the strain effects on the band structures of semiconductors.

REFERENCES 1. G. E. Pikus and G. L. Bir, "Effects of deformation on the hole energy spectrum of germanium and silicon," Sov. Phys i-Solid State 1, 1502-1517 (1960). 2. G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors, Wiley, New York, 1974. 3. S. L. Chuang, "Efficient band-structure calculations of strained quantum wells using a two-by-two Hamiltonian," Phys. Rev. B 43, 9649-9661 (1991). (Note that au in this paper should be taken as -au to compare with data in the literature.) 4. C. Y. P. Chao and S. L. Chuang, "Spin-orbit-coupling effects on the valence-band structure of strained semiconductor quantum wells," Phys. Rev. B. 46, 4110-4122 (1992).

!

Appendix D Semiconductor Heterojunction Band Lineups in the Model-Solid Theory [1] Semiconductor heterojunctions and superlattices have been under intensive investigation both theoretically and experiment ally for th e past three decades. The potential device applications using heterojunctions are tremendous. In this appendix, we discuss a simplified model to determine the energy band lineups of semiconductor heterojunctions based on the model-solid theory . Jl -4]. The goal is to develop a reliable model to predict band offsets for a wide variety of heterojunctions without the need for difficult calculations such as in the local-density-functional theory or ab initio pseudopotential method. The relation of the model-solid theory to the fully self-consistent first-principles calculations can be found in Refs. 2 and 3. The major idea is to set up an absolute reference energy level. All calculated energies can then be put on an absolute energy scale, allowing us to derive band lineups. In the model-solid theory, an average energy over the three uppermost valence bands (the heavy-hole , the light-hole, and the spin-orbit split-off bands) E u • av is obtained from theory and is referred to as the absolute energy level. The values of E; av for different semiconductors are usually tabulated [1] (Table K.2 in Appendix K) so that no calculations for these values are necessary. These results should be compared with those of the first-principle calculations whenever possible to justify the model. An . estimate of the maximum possible error is about 0.1 eV. Band offsets should be checked with experimental data such as those in Refs . 5-19. The modelsolid theory provides a simple guideline for estimating the band offsets for materials, especially ternary compounds with varying compositions for which experimental data may not be always available.

D.I

UNSTRAINED SEMICONDUCTORS

If materials A and B have the same lattice constants, we may have an ideal heterojunction and there is no strain in the semiconductors. For this case, the heavy-hole and light-hole band edges (E H H and E L H ) are degenerate at the zone center, and their energy position is denoted as E u : ~

E L• 662

-

EV , 3V +

3"

(0 .1)

D.I

UNSTRAINED SEMICONDUCTORS

. 663

where ~ is the spin-orbit splitting energy, and the spin-orbit split-off band-edge energy E so is 26-

(D.2)

3

The conduction band edge is obtained by adding the band-gap energy E g to

e; (D.3) Note that in the model-solid theory, the spin-orbit splitting energy 6- and the band-gap energy E g are taken from experimental results. The only input provided by the model-solid theory is the tabulated Eu,av value. This Eu,av value is essentially the same as the p-state energy E p in Fig. 4.3a. With the above results, the band lineups between materials A and Bare shown in Fig. D.l. We have

and the band-edge discontinuities are

6.E c

E cA

=

-

E cB

6.E v

tx E; + 6.E u

=

E yB

-

E vA

6.E g

=

(D.4) (D.5)

The partition ratios of the band-edge discontinuities, Q; = 6.E e/ilE g and Qu = 6. E u / il E g' are obtained from this theory and can also be compared with experimental data.

. \\

.'

-

Material A

-

-

-

-

-

E~

-

Material B

Absolute - - - zero energy

- -

t L\E c

E~

I

B

Eg

A

Eg

___ ~ 3_

L\E y

- -

tJ.A

E~,av -

"3

-

- -

-

Figure D.l. Band lineups in the model-solid theory, E" ,",v in each material region is obtained from the model-solid theory and is tabulated in Appendix K. The bandgap energ-y E g and the spin-orbit splitting A ofeach material are taken from experimental 'results.

., I

"

! :1 !

664

D.2

13'\ND UN'El7?:: IN THE MODEi SOLiD THEORY

STRAINED SEMICONDUCTORS

If a material A with a lattice constant a is grown on a substrate with a lattice

constant a o along the z direction, we have Ex x

=

Ey y

=

ao - a

(D.6a)

a

and

Cl2

- 2C- £ xx

(D .6b)

11

The band-edge shifts are (D.7a) (D.7b) The position of the average energy of the valence bands £0 av under strain is shifted from its unstrained position £~, av in (D.n by - Pe: ' .

(D.S) We thus have the center of the valence-band-edge energy

(D.9) The heavy-hole, light-hole, and spin-orbit split-off band edges are

E HH = E ~ - P, E LH = £~ - PE: E

so

d. I

lL\

2

(D.10) Qe 1 + 2 + 2"[L\2 + 2L1Qe + 9Q;f / 2

=£0 _ P _ L\ + Q e U E: 2 2

_

!-[L\"2 + 2L1Q + 9Q"2] 2 e e

1/

2

(D.l1) (D.12)

The conduction band edge is shifted by P, given by (D.7b):

(D.l3) Note that in the limit of a large spin-orbit split-off energy L\ » IQel, we can ignore the coupling of the spin-orbit split-off band and £Ll-l

=='

£,0 - P,

+ Qe

E so """ £,0 - P, - L\

( D.14a) . (D.14b)

D.2

STRAINED SEMICONDUCTORS

665

For a ternary alloy such as A x B I-xC with a lattice constant a(x), a(x)

=

xa(AC) + (1 - x)a(BC)

(D.lS)

which is a linear interpolation of the lattice constants a(AC) and a(BC) of the binary compound semiconductors, we use the following formula to calculate an energy level E (= EL~, av for example): E ( A x B 1 -x C) = xE ( A C) + (1 - x) E ( B C) b.a

+ 3x(1 - x)[ -av(AC) + auCBC)} -

ao

(D.16)

where the last term accounts -for a strain contribution to the ternary alloy, 'I'. and S:« = a(AC) - a(BC) is the different between the lattice constants of two compounds AC and BC. Once E~, av is determined the- b a n d - e d g e r energies for the strained ternary compound can be calculated following I (D.6)-(D.13). Many theoretical parameters for the electronic and optical properties such as those listed in Table K.2 in Appendix K can be found in the data books compiled by various groups (such as Refs. 20-23), review papers (such as .t Refs. 24-27), and research papers (Refs. 28-37). Example Ga As/ 'Al As Heterojunction GaAs and AlAs have almost the same lattice constants. Therefore, the heterojunction has a negligible strain. We see from Table K.2 in Appendix K that

Et.,avCGaAs)

=

-6.~2

Ev,av(AlAs)

=

-7.49 eV, b.(AlAs)

eV, b.(GaAs)

= =

0.34 eV, Eg(GaAs)

=

1.52 eV

0.2S e V, Etr(AlAs)

=

3.13 eV

.'

Therefore,

EuCGaAs) -

0.34 -6.92 + - 3

=

-

E[:(AlAs)

=

0.28 -7.49 + - 3

=

-7.40 eV

6.E v

=

-6.81 + 7.40

Also, the band-gap discontinuity is b.Eg discontinuity ratio is b.Et./b.Eg = 0.37.

=

6.81 eV

0.59 eV

1.61 eV, and the valence-band

=

•

BAND LlNFUPS H' THE l\nnEL-SOLID THEORY

666

Example

InO.53GaO.47As /InP Heterojunction

a(GaAs)

o

=

a

a

5.6533 A, a(InAs) = 6.0584 A, a(InP) = 5.8688 A

E u,3v(In 1 _ xGa xAs) = xEu.av(GaAs) + (1 - x)Eu,av(InAs) Aa

+ 3x(1 - x)[ -av(GaAs) + aJlnAs)]a

Aa = 5.6533 - 6.0584 = -0.4051

A

a(In1_xGaxAs) = xa(GaAs) + (1 - x)a(InAs) A(In1 _xGaxAs)

= xA(GaAs)

+ (1 - x) 6.(InAs)

For x = 0.47, In0.53Ga0,47As is lattice matched to InP. Therefore, we do not have the strain terms (Pe = 0, Q e = 0). We obtain Eu,av(Ino.53Ga0.47As) = -6.77CJ eV, 6. = 0 .361 eV

Using Eu,av(InP) = -7.04 eV and EvOnP) = -7.003 eV, we find AEu = 0.344 eV. From room temperature data for the band gap, EgClno.53Gao.47As) = 0.73 eV, E/lnP) = 1.35 eV, and AEg = 0.62 eV, we obtain the ratio AEu/AEg = 0.55 = 55%. •

D.3 SOME EXPERIMENTAL REPORTS ON BAND-EDGE DISCONTINUITIES There has been a considerable amount of experimental data on band offsets, mostly on unstrained systems. For strained semiconductors, the band offsets are complicated by the deformation potentials, which also shift the conduction and valence-band edges. Therefore, fewer data are available for strained heterojunctions. .'

Eg(GaAs) = 1.424 eV

(300 K)

r

Eg(AlxGa1_xAs) = 1.424' + 1.247x eV

(300 K)

6. E g ( x) = 1.247x eV AEc = 0.67 AEg

A E L = 0.33 AEg

(AE c =0.69 AEg, AE v

== 0.31AE", Ref. 11)

D.3

EXPERIMF~TAL

REPORTS ON BAND-EDGE DISCONTINUITIES

667

2. Ino.53Ga0.47As/lnP ('" 0 K) [18]

E g ( lnP)

=

1.423 eV

Eg(InO .53Ga0.47As)

=

0.811 eV

. .t1Eg

=

0.612 eV

6. E c

=

0.252 e V

=

0.41 6. E g

se;

=

0.360 eV

=

0.59 t1E g

,

3. In o.52Al O.4S As/ InP( '" 0 K) [18] Eg(lnP)

=

1.423 eV

Eg(lno.52Alo.4SAs)

=

1.511 eV

t1E g

=

0.088 eV

t1 E;

= 0.252 eV = 2.86 t1Eg ,

t1E v

=

-

(Type II)

0.164 eV

The above results for Ino.53Ga0.47As/ InP/ In o.52 AlO.4SAs band offsets and their transitivity relations are illustrated [18] in Fig. D.2. The transitivity relations give ts E; = 0.504 eV = 0.72 t1E g , and ts E; = 0.196 eV = 0.28 6.E g for an Ino.53Ga0.47As/ln0.52Al0.4SAs heterojunction. .~

t

0.252eV

0.504 eV 0.252eV 0.811eV

1.423 eV

1.51leV

/

O.SUeV

O.3dn j---J.64~V rlO.196~V Band offsets and transitivity of Ino.53GA0,47As/lnP (!1E c = 0.41 !1E g , !1E L. = = 2.86 ti.E g > !1E g = 0.088 eV) at low temperatures (0 K) [18].

Figure D.2.

0-59 tJ.E g ) and Ino.52AI0,48As/lnP (tJ.E c = 0.252 eV

~, ~

..

~""""",_",_."""~ ._,,,~_,

_~

..

~_.

__ •."",

...., .•'

-.1

,,' ...•

_~

•. ":"., .. , .. _

,',.

~r

' .. 'l .·.~~'7'

\.~"

"J"'l-~t'.o..j:""""~.":f'_.Y"· ·

,'

--""~,"""_r;'.-

,~-

~~,

,w~_.~

·~.'

iJ/.ND :,TNEliPS IN 'fEE MODEL-SOLlD

668

_.

··· ·.,···' - : '"!""' :r,..· ·-...·,.~ ·•__••··..~···'!~

THEO~Y

4. In 1 _x G a xAs v PI-V/ InP lattice-matched system [22] For Inl_XGAxAsyPI_Y quaternary semiconductor lattice matched to InP substrate, 0.1896y X= - - - - - - - -

0.4176 - O.0125y E 8 = (Inl_xGaxAsYPI_Y) = 1.35 - O.775y.+ O.149 y 2 eV ~Eg( y)

= O.775y - O.149 y 2 eV

~Eu( y) = O.502y - O.152 y 2 e V

~Ec(Y) = ~Eg(Y)

where

~Eu( y)

- tlEv(Y) = O.273y + O.003 y 2 eV

was determined experimentally.

•

Some reports on strained InxGa1_xAs/lnP, InXGal_xAs/lno.52Alo.4sAs, InGaAs/InGaAsP, and In .Ga l -xAs /GaAs can be found in Refs. 7-9, 14, 15, and 19.

REFERENCES 1. C. G. Van de Walle, "Band lineups and deformation potentials in the model-solid theory," Phys. ReL'. B 39, 1871 -1883 (1989). 2. c. G. Van de Walle and R. M. Martin, "Theoretical calculations 0: semiconductor heterojunction discontinuities," 1. Vac. Sci . Techno!. B 4, 1055-1059 (1986). 3. c. G. Van de Walle and R. M. Martin, "Theoretical study of band offsets at semiconductor interfaces," Phys. Rev. B 35,8154-8165 (1987). 4. c. G. Van de Walle, K. Shahzad, and D. J. Olego, "Strained-layer interfaces between II-VI compound semiconductors," 1. Vac. Sci. Technol. B 6, 1350-1353 (1988). 5. R. People, K. W. Wecht, K. Alavi, and A. Y. Cho, "Measurement of the. conduction-band discontinuity of molecular beam epitaxial grown Ino.52Al0.4SAsj Ino.5JGao.47As, N-n heterojunction by C-V profiling," Appl. Phys, Lett. 43, 118-120 (1983). 6. R. C. Miller, A. C. Gossard, D. A. Kleinman, and O. Munteanu, "Parabolic quantum wells with the Ga As-Al Ga.Lj As system," Phys. Rev . B 29, 3740-3743 (1984). j

7. R. People, "Effects of coherency strain on the band gap of pseudomorphic Irr.Ga I -x As on (OOl)InP," Appl, Phys. Lett . 50, 1604-1606 (1987). 8. R. People, "Band alignments for pseudomorphic InP/In
T::"~

REFERENCES

669

10. C. D. Lee and S. R. Forrest, "Effects of lattice mismatch on In xGa l-xAs / InP heterojunctions," Appl. Phys. Lett. 57, 469-471 (1990). 11. L. Hrivnak, "Determination of r electron and light hole effective masses in Al xGa i . ... As on the basis of energy gaps, band-gap offsets, and energy levels in AlxGa1_xAs/GaAs quantum wells," Appl, Phys. Lett. 56, 2425-2427 (1990). 12. B. R. Nag and S. Mukhopadhyay, "Band offset in InP/Gao.47Ino.53As het-erostructures," Appl, Phys, Lett. 58, 1056-1058 (1991). 13. M. S. Hybertsen, "Band offset transitivity at the InGaAs / InAlAs / InP(001) heterointerfaces," Appl . Phys. Lett. 58, 1759-1761 (1991). 14. B. Jogai, "Valence-band offset in strained GaAs-InxGa1_xAs superlattices," Appl. Phys, Lett. 59, 1329-1331 (1991). 15. J. H. Huang, T. Y. Chang, and B. Lalevie, "Measurement of the conduction-band discontinuity in pseudomorphic InxGal-xAs/lno.52Al0.4SAs heterostructures," Appl. Phys. Lett. 60, 733-735 (1992). 16. S. Tiwari and D. J. Frank, . "Empirical fit to band discontinuities and barrier heights in III-V alloy systems," Appl. Phys, Lett. 60,630-632 (1992). 17. R. F. Kopf, M. H. Herman, M. L. Schnoes, A. P. Perley, G. Livescu, and M. Ohring, "Band offset determination in analog graded parabolic and triangular quantum wells of GaAs/AlGaAs and GalnAs/AllnAs," J. Appl. Phys. 71, 5004-5011 (1992). 18. J. Bohrer, A. Krost, T. Wolf, and D. Bimberg, "Band offsets and transitivity of InxGal_xAs/lnl_yAlyAs/lnP heterostructures," Phys. Rev. B 47, 6439-6443 (1993). 19. M. Nido, K. Naniwae, T. Terakado, and A. Suzuki, "Band-gap discontinuity control for InGaAs / InGaAsP multiquanturn-well structures by tensile-strained barriers," Appl. Phys. Lett. 62, 2716-2718 (1993). 20. K. H. Hellwege, Ed., Landolt-Bornstein Numerical Data and Functional Relationships in Science and Technology, New-Series, Group III 17a, Springer, Berlin, 1982; Groups III-V 223, Springer, B~rIin, 1986. ;

21. For a brief version of the data book in Ref. 20, see: O. Madelung, Ed., Semiconductors, Group IV Elements and I/I-V Compounds, in R. Poerschke, Ed., Da t a in Science and Technology" Springer, Berlin, 1991.' 22. S. Adachi, Physical Properties of III-V Semiconductor Compounds, Wiley, New York, 1992. 23. S. Adachi, Properties of Indium Phosphide, nTSPEC, The Institute of Electrical Engineers, London, 1991. 24. J. S. Blakemore, "Semiconducting and other major properties of gallium arsenide," 1. Appl. Phys. 53, R123-R181 (982). 25. S. Adachi, "Material parameters of In Ga l_xAs yP 1- y and related binaries," 1. Appl. Phys. 53, 8775-8792 (1982). j

26. S. Adachi and K. Oe, "Internal strain and photoelastic effects in Ga1_xAlxAs / GaAs and InxGal_XAsyPl_y/lnP crystals," 1. Appl. Phys. 54, 6620-6627 (1983). 27.S. Adachi, "GaAs, AlAs, and Ga1_xAlxAs: Material parameters for use m research and device applications," 1. Appl. Phys . 58, R1-R28 (1985).

[, i' r r

67U

BAND IJNEUPS If\: THE MODEL-SOLID THEORY

28. P . Lawaetz, "Valence-band parameters in cubic semiconductors," Phys, Rev. B 4, 3460 -3467 (1971). 29. R. E. Nahory, M. A. Pollack, and W. D. Johnston, Jr., "Band gap versus composition and demonstration of Vegard's law for In1_xGa xAs yP1 - y lattice matched to InP," Appl, Phys . Lett. 33, 659-661 (1978). 30. K. Alavi and R. L. Aggarwal, "Interband magneto absorption of Ino.53Gao.47As," Phys. Rev. B 21, 1311-1315 (1980). 31. A. Raymond, J. L Robert, "and C. Bernard, "The electron effective mass in heavily doped GaAs," J . Phys. C: Solid State Phys. 12, 2289-2293 (1979). 32. L. G. Shantharama, A . R. Adams, C. N. Ahmad, and R. J. Nicholas, "The k . P interaction in InP and GaAs from the band-gap dependence of the effective mass," J. Phys. C: Solid State Phys . 17,4429-4442 (1984). 33. W. Stolz, J. C. Maan, M. AJtarelli, L Tapfer, and K. Ploog, "Absorption spectroscopy On Ga0.47Ino.53As/AJo.4slno.52As multi-quantum-well hetero-structures . 1. Excitonic transitions," Phys . Rev. B 36, 4301-4309 (1987). 34. W. Stolz, J. C. Maan, M. Altarelli, L. Tapfer, and K. Ploeg, "Absorption spectroscopy on Ga0.47Ino.53As/Alo.4sIno.5zAs multi-quantum-well hetero-structures; II: Subband structure," Phys, Rev. B 36, 4310-4315 (1987). 35. L. W . Molenkamp, R. Eppenga, G . W. 't Hooft, P. Dawson, C. T. Faxon, and K. J. Moore, "Determination of valence-band effective-mass anisotropy in GaAs quantum wells by optical spectroscopy," Phvs, Rev. B 38, 4314-4317 (1988). 36. D. Gershoni and H. Temkin, "Optical properties of III-V strained-layer quantum wells," J. Luminescence 44, 381-398 (1989). 37. R. Sauer, S. Nilsson, P. Roentgen, W. Heuberger, V. Graf, A. Hangleiter, and R. Spycher, "Optical study of extended-molecular fiat islands in . lattice-matched In o.53Ga 0,47/\ <.;/lnP and InO.53GaO.47As/lnl_xGaxAsyPl_y quantum .ve lls grown by low-pressure metal-organic vapor-phase epitaxy with different interruption cycles," Phys, Rev. B 46, 9525-9537 (1992).

Appendix E Krarners-Kronig Relations The induced electric polarization P in a material is due to the response of the medium to an electric field E (ignoring the spatial dependence on r):

pet) =_cof~oox(t - T)E(T) d-r cofoox( T)E(t - T) d r o

=

(E.l)

The integration over T is from - 00 to t in the first expression, since the response p(t) at time t comes from the excitation field before t. In other words, the function X( T ) has the property that

<0

(E.2)

coE(t) + Pt z)

(E.3)

c(w)E(w)

(EA)

for

T

i.e., the system is causal. Since

D(t)

=

we find in the frequency domain

D(w)

=

by taking the Fourier transform of (E.3), where

(E.5) Since X( T) is a real function (it is in the time domain), we see that

c( -w)

=

c*(w)

(E.6)

If we write c(w) in terms of its real and imaginary parts,

c(w)

=

C'(w) + ic"(w)

(E.7) 671

KH/tvlFRS-KRONI(, HE] ATIONS

672

we find that e' ( -

w)

£/1 ( _

w) = -

e' ( W)

=

£" (

(E.8)

w)

(E.9)

that is, the real part of c(w} is an even function, and the imaginary part is an odd function of w. Another physical property of £(w) is that, in the highfrequency limit, £(w) tends to eo, because the polarization processes PC!), which are responsible for X, cannot occur when the field changes sufficiently rapidly [1]:

(E.lO) Define an integral I in the complex

Wi

plane,

I = _I_A:.. e( Wi)

27Ti':Yc

Wi -

£0

-

w

dw

'

(E.ll)

where the closed contour C is shown in Fig. E.1. It is in the upper Wi plane. The function e( w) is analytic in the upper-half plane, since when Wi = w R + iw j , the integrand in (E.5) includes an exponentially decreasing factor ewhen W j > O. Since the function X( T) is finite (for a physical process) throughout 0 < T < 00, the integral in (E.5) converges. Since the function e(w') - £0 is analytic in the upper Wi plane, the closed contour does not have any poles; therefore, the integral I vanishes using Cauchy's theorem. By breaking the contour into three parts-CO Oll the real axis, - R < Wi < w - 8, w + 8 < Wi < R, (ii) along the infinitesimally small semicircle near w with a radius 8, and (iii) along the big semicircle with a large radius R-we find that the contribution due to part (iii) is zero as R ~ 00. Therefore, as 8 ~ 0, the integration along part (i) is the principle W

{ 7"

-R Figure K!. The integration contour on the upper-half w' plane for the derivation of the Kramers-Kronig relation.

KRAMERS-KRONIG RELATIONS

. 673

value (denoted by P) of the integration along the real axis from and we obtain . 1.

- - p 27Ti

f oo -

00

+ --

dw'

w

Wi -

1

£0

c; ( Wi) -

f

Eo' ( Wi)

27Ti 0

£0

-

WI -

dw '

=

0

00

to

+ 00,

(E.12)

(tJ

The second integral equals - i[e(w) - eo], which can be evaluated simply by a change of variable from Wi to 8, Wi - W = 8e i 9 at a constant radius 8. We find 1 7Tl

E(W)-£O=-'p

foo

e(w') - eo I 0)

-00

w

-

dw '

(E.13)

. We separate (E.13) into real and imaginary parts and find 1

e' ( w) -

= -

£0

P

"( ) -

pf

7T

d WI

I

(E.14)

-ooW-W

7T

1 £w---

£"( Wi)

f cc

e

£' ( WI) -

00

0

I

W

-cc

-

d to I :.

(E.15)

W

The above results are the Kramers-Kronig relations, which relate the real and the imaginary parts of £(w) to each other. If we make use of the even property of the real part c'ew) and the odd property of the imaginary part £If(W), we obtain alternatively 2

e' ( w) - e 0

=

-

f oo

P

2w =

-

7T

pf

I

dw

Ow-w

7T

. e" (w)

Wi Elf ( W' ) ,2 2

oc

0

e' ( Wi) 2 W'

-

-

(E.16)

E

w2

0

dw'

(E.17)

In a homogeneous, isotropic medium with a complex permittivity function £(w) = £'(w) + iE"(w) and a permeability }-La, the propagation constant of an electromagnetic wave at an angular frequency co is

k(w)

=

wVf-Lo£(w)

=

wJf-Lo£o [£~(w) + iE~(w)r/2

=

-n

w c

(E.18)

\,

t,

~ .

674

KRI\ ~v1ERS ···KRONIG

". 'l. _W .. ;: . _ .

RELAT10NS

where we have used the real and the imaginary parts of the relative permittivity

e( w) erC w) e~(

w)

= -

e' ( w ) =

-

eo s~ (

So

w) -

s" (

w)

So

(E.19)

and the complex refractive index

n=n

+ ix

(E.20)

We obtain the relations

(E.21a) .

2nK = s~ (w)

-

... ,

(E.21b)

Here the real and the imaginary parts of the refractive index new) and K(W) can be expressed as

n 2 = %[ e~(w) + K2 =

t[ -e~(w)

Vs~\w) + 8~2(W) ]

(E.22a)

+ Ve~\w) + 8~\W) ]

(E.22b)

Experimental data for nand K of GaAs and InP semiconductors as a function of optical energy and wavelength are tabulated and plotted in Appendix J.

REFERENCE 1. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski i, Electrodynamics of Continuous Media, 2d ed., Pergamon, New York, 1984.

~"

........

_

.... . . . ~

Appendix F

Poynting's Theorem and Reciprocity Theorem F.l

POYNTING'S THEOREM

The power conservation is a very useful law. From Maxwell 's equations in the time domain .

-. ...

a at

vx

E

v x

H = J

=

--B

(F.l)

a at

(F.2)

+

-D

Dot-multiplying (F.r) by Hand (F.2) by E, and taking the difference , we obtain

v . (E x where 'V • (E x H) Poynting vector as

=

H)

=

a at

aD at

-H . -B - E . -

H . 'V x E - E . V

- E .J

(F .3)

X

H has been used. Define the

S=E xH

(FA)

which gives the instantaneous energy flux density (W/m 2 ) . For an isotropic medium, D = E:E and B = ,uH, the electric and magnetic energy densities are E:

' w =-E·E e

2

,u wIn = -H· H 2

(F.5)

Therefore, Poynting's theorem in the time domain is simply

a

V'S= --(w at e +wm )-E'J

(F.6) 675

676

POYNTING'S THEOREM AND RECIPROCFfY THEOREM

If integrating over a volume V enclosed by a surface S, we obtain Poynting's theorem in the form

(F.7) i.e., the power flow out of the surface S equals the decreasing rate of the stored electric and magnetic energies plus the power supplied by the source, - fffE' JdV. A complex Poynting's theorem can also be derived from Maxwell's equations in frequency domain:

v . (tE

X

H*) = -iwOE . D* - ~B . H*) - iE . J*

(F.8)

If J = J d + Jf , where J d accounts for dissipation, e.g., J d = O"E is the conduction current density in a conductor, we then have

1 ) w__ 1-1 V· ( -ExH* +i-(E'D*-B'H*)+-E'J*=---E'J* (F.9) 2 2 2 d 2 I where the right-hand side is the time-averaged power supplied by the source JI , and the terms on the left-hand side are the time-averaged power flux, the difference in the electric and magnetic stored energy density, and the time-average dissipated power, respectively.

F.2

RECIPROCITY THEOREM [1, 2]

Consider two sources J(1) and J O ) producing two sets of fields in the same medium described by D = e E and B = ,uH: E(I )

= -

V'

X

H(l )

=

V'

X

E(2)

a = - -B(2)

V' If we take E(2) •

H (2 ).

(Fi l Oa) -

(F.10b), we find

a at

vX

X H( 2)

E (I ) .

( F .10a)

-B(I)

J (l)

+

a

(F .10b)

- D (I )

at

(F.lIa)

at

=

a

J (2 )

+ - - D (2)

(F.lIb)

at

(F. l lb) and subtract by

H(l) •

(Frl La) -

F.2

RECIPROCITY Tl-jEOREM

677

Integrating over an infinite volume and using the divergence theorem, we obtain

rt:. (E(l) ~

X H(2) -

E(2) X

H(!)) .

dS

=

f

E(2) •

J (1) dv -

v

f

dv

E(l) . H(2)

v

(F.13) Using the property in which the surface integral on the left-hand side goes to zero as the radius of the surface goes to infinity, we obtain

f

E(2) •

J(1) du

=

VI

f

E(l) • J(2)

du

(F.14)

Vz

Since the two sources J I and J 2 are distributed over finite regions, such as those of two dipoles, the volume integrals are only over the regions of the. two sources VI and V 2 , respectively. E(l) and E(2) would then be the electric fields due to J(1) and J(2\ evaluated at the positions of the other sources· occupying volumes V2 and VI' respectively. This is the reciprocity relation. We may generalize the above relation to field solutions in two different media, described by t;C!)(r) and e (2)(r), and the same permeability j..L. We have, using the frequency domain representation,

(F.15) .'

If we integrate the above equation; over an infinite volume, we obtain similar relations to (F.14) except for the extra term due to the difference between two dielectric functions. However, if we apply the above relation to dielectric waveguide structures which are translationally invariant in the z direction, and integrate over only a volume of a disk shape with a thickness 6. z ~ 0 and a radius R ~ 00 (see Fig. Fvl ), we find I

111/' AdXdYd~'~ (If" + If" + n}· dS ffA' alz+Cl.z

i dx dy +

ff A . ( - i) d x d y atz

(F .16) where A = E(1)

X H(2) -

E(2) X H(l)

and the fields at SOX) vanish. Since

J(I)

670

POYNTING'S Tf fECRE i\,; I,ND REGPR OCiT \ TH LORLM

t---...I

f----""'\

Dielectric waveguides

v f\

f\

-z -4--+-

Z

Figure F.l. The volume V for the space be twe e n z and z + /).Z enclosed by the surface 5 = 51 + 52 + 5 00 for the derivation of the reciprocity relation for dielectric waveguide structures.

and J(2 ) are zero in the dielectric waveguides, we obtain a reciprocity relation for the waveguide: J

- ff(E (l)

X H( 2 )

-

Jz

E( 2 ) X H(l») . i dx dy f

= iw ff(

S(2)(X,

y) -

S(l)( X,

y)]E(l) .

E(\)"dx dy

(F.17)

I

REFERENCES 1. L. D. Landau, E . M . Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuotls Media , 2d ed. , Pergamon , New York, 1984. 2 . S. L. Chuang, "A coupled-mode formulation by reciprocity and a variational pr inciple," J. Lightwa oe Technol. LT-5, 5-15 (l98~), and "A coupled-mode theory for multiwave guide systems satisfying the reciprocity theorem and power conservalion, " 1. Lightwave Techno!. LT-S, 174-183 (987).

Appendix G Light Propagation in Gyrotropic Media-Magnetooptic Effects [1-3] In this appendix, we present a general formulation to find the electromagnetic fields with the characteristic polarizations of gyrotropic media. The magnetooptic effects are then investigated.

G.!

MAXWELL'S EQUATIONS AND CHARACTERISTIC EQUATION

The constitutive relations for gyrotropic media are given by [1] D

E·

=

E

o

(G.la)

~]

(G.lb)

B = jLH

.

(G.2) .'

One example is a plasma with an externally applied de magnetic field in the i direction: "

(G.3a)

£g

£,

~ ~

£0 [

w( :;~w~;) ]

(G.3b)

:~ 1

(G.3c)

£0 ( 1

-

where w

qB o m

=C

(GA) 679

680

PROPAGATION IN

GYROTRO[)I':~

IviEDj,\--MAG;'\TETOOPTIC EFFECTS

is the cyclotron frequency, and

(G.S) is the plasma frequency for a carrier density n. Here m is the electron mass. For the free carrier effects in semiconductors, the effective mass of the carriers should be used. Note that the geometry of the medium considered is rotation-invariant around the z axis. Thus, we may consider the propagation wave vector k to be in the x-z plane without loss of generality: k = xk ,

+ zk z

(G.6)

We repeat Maxwell's equations for plane-wave solutions here: k X E

=

wB

(G.7a)

k X H = -wD

(G.7b)

k· B = 0

(G.7c)

k' D = 0

(G.7d)

Using (G.7a), (G.7b), and (G.2), we obtain k

X

(k x E) = w,uk x H =

-W

2

J.t E . E

Or, equivalently, the above vector equation can be written in a matrix representation following Table 6.1 in Section 6.3 of the text:

\ -k z

o o

E~J[~:]

(G.S) {

We carry out the square of the matrix on the left-hand side, and move to the right to obtain , I

k; .

lW

W

-iW 2J.t E!!

2,u E

2

J.tE g

-kxk"

k,~

-k",k z

+ k; "- (J./p.E 0

[EX

0 k2 'x

-

W

2

J.tE z

l.

~: ~O

(G.9)

The dispersion relation is obtained by setting the determinant of the matrix

G.l

1'.1AXV/ECl'S EQUATIOJ'iS A:t
681

to zero for nontrivial solutions for the electric field, We obtain the characteristic equation after some algebra (k.~

+ k; -

W

4

W

2

fJ.- E) (

-W

J-L2E: ( k ; -

W

2

fJ.- Ek ; -

2J-L C ) z

W

2J-L E k ; z

+

4

J-L2EEz )

(G.lO)

0

=

W

Rewrite the wave vector k in terms of the angle () with respect to the z axis, k

=

ik x + ik z = ik sin

f)

+ ik cos

f)

and define the constants K

=

w"j J-LE

(G.lla)

(G.llb) (G.llc) Equation (G.lO) reduces to

(G.12) which has the form

The solution k 2 is easily obtained from

k

where

2 =

B + VB 2 -

-

-

-

-

2A

4AC -

-

(G.13)

6~ 2

PROPAG ATION IN G YR O'I R()P IC MED i A-MAUN ETOOPT1C EI'PE ::TS

Usin g (G .9) again , we have k 2 cos " e

-

2

K

'K g2 1 - k

2

sin

(G.14)

e cos e

o

where k ? is given by the two possible roots in (G.13). Since the above determinant of the matrix is zero, Eq. (G.14) contains three algebraic equations, which are linearly dependent. Using the second and the third equations in (G.14), we find

( G .15a) k 2 sin 2

=

k 2 sin

e -:-..K-;

(G .15b )

e COS.e

.

where two possible valu es of k 2 are given by the roots in (G.l3 ).

G.2

SPECIAL CASES

Case 1: e = 0 (the wave is prupagating parallel to the magnetic field). Equation (G .14) becomes

,

.-

k2

-

K

2

k2

2

"

- i K 152

1"K g

-

K

0 2

0

0

[EX] ~: .~ 0

0

(G.16)

- K z2

We have £2 = 0 , since

K; =1=

0

(G.17 )

and

(G ,18) We obtain two ro ots for k

2

,

(G.19)

.\

G.2 .::iP;2CIAL CA(;ES

683

Substituting the above roots back into (G.16), we find 2 +K g 2 1'K g

- +i

(G.20)

Combining (G.17) and (G.20), we see that if the wave is propagating parallel to the magnetic .field, the wave will be circularly polarized. Let us assume that the wave is propagating in the + i direction; we have either

( a)

( G.21)

the wave is left-hand circularly polarized (LHCP), or

(b)

(G.22)

the wave is right-hand circularly polarized (RHCP). Case 2: 8 = "'IT /2 (the wave is propagating perpendicularly to the magnetic field). Equation (G.14) becomes -K 2 'K g2 1 0

-iK g2 k? -

K

2

o o

( G.23)

0 .'

Possible nontrivial solutions for the electric field will be \,

( a)

(G.24)

If k 2 = K; is true, we find that Ex =0, E y = 0 from the first two equations in (G.23). Therefore, we obtain that the characteristic polarization is linear polarization in the z direction E = iEz and the propagation con:' stant is k = K, = WJJ.LE z . (b) - K 2(k 2 - K 2 ) - K; = 0, which leads to

(G.25)

If (G.25) is true, then k 2

"* K;,

E;

Ex

=

= 0, and K g2 - i -2 E K Y

(G.26)

684

PROPAC-iATION IN GYROTR()PIC MED :\ - --MAGNETOOl'T1C EFFEC,'S

The wave is generally elliptically polarized in the x-y plane with a propagation constant k = K 4 - K;) / K 2 . The wave vector is always on the x-y plane since (j = 7T /2. Furthermore, if E g = 0, the medium becomes uniaxial. Equations (G.24) to (G.26) show that either the electric field is polarized in the z direction and k 2 = K'1, or polarized in the y direction (since Ex = 0, E, = 0 from (G.26)) and k 2 = K 2. Both waves are linearly polarized propagating with different velocities. This birefringence is called the Cotton-Mouton effect.

J(

G.3

FARADAY ROTATION

Let us consider a slab of gyrotropic medium with a dc magnetic field applied in the +2 direction and the wave propagated parallel to the dc magnetic field. This is the special case 1, (j = 0, discussed before, and the two characteristic polarizations are left- and right-hand circularly polarized with corresponding propagation constants given by (G.21) and (G.22). Consider an incident plane wave as shown in Fig. G.1 with E = xEoe i h

(G.27)

Upon striking the interface at z = 0, the wave will break up into two circularly polarized waves: E =

E E (x - iy)_o eik,.,z + (x + iy)~eiLZ 2

2

(G.28)

These two circularly polarized waves propagate with two different wave numbers, k + and k _. As discussed b~fore, k ± = J-L( E ± E g ) . At z = d, we have

w-l

.' (G.29)

•

x E =x~Eoe ikz

k::: Figure G.!. A linearly polarized plane wave E incident on a gyrotropic medium experiences Faraday rotation after passing through the medium at z = d .

zk H y

REFI:RENCES

685

Thus, we have the ratio of the y component to. the

x component

of the

electric field E;

E

eik +d _

-

ei/cd

-leik+d+eik_d

·x

(G.30) which is a real number. Thus the electric field at z polarized making an angle

=

d is again linearly

(0.31) with the i axis. We conclude that the incident linearly polarized wave (in the i direction) is rotated by an angle eF at x = d, which is called the Faraday rotation. Since the Faraday angle eF depends on the difference between k + = wVJL(e + £g) and k_= WVJL(£ - £8)' the carrier density n and the effective mass of the electrons can be measured from the magnetooptic effects using the Faraday rotation as discussed above. Another setup is called the Voigt configuration for which the propagation direction is perpendicular to the direction of the applied de magnetic field zBo' The incident wave is chosen to be linearly polarized at an angle of 45° with respect to the static magnetic field. The transmitted wave does not experience a rotation; it becomes elliptically polarized, however. The phase angle or the amount of ellipticity is determined by the difference of the propagation constants of ·~he two characteristic polarizations, which is related to the plasma frequency W P and the cyclotron frequency we' For more discussions on the rnagnetooptic effects and their measurements in semiconductors, see Ref. 3.

REFERENCES 1. J. A. Kong, Electromagnetic Wave Theory, Wiley, New York, 1990.

2. K. C. Yeh and C. H. Liu, TheolY of Ionospheric Waves, Academic, !'jew York, 1972. 3. K. Seeger, Semiconductor Physics, Springer, Berlin, 1982.

;~.,

' ,....,.,"'

f':'~

.•._.~:-

~

_.h

_~....."

~~ .

•~ ~ "

_

_

Appendix H Formulation of the ImprovedCoupled-Mode Theory In Appendix F we derived a general reciprocity relation for waveguide systems. The relation is

a

az

ff

(E(l) X

= iw

H(2) -

ff [

E(2)

s(2)( x,

x

H(l») •

i dx d y

y) - e(1)( x, y)] E(l)

• E(2)

dx dy

(H .I)

where E(1)(x, y, z ) and H(l)(x, Y, z ) are a set of solutions in the medium described by s(l)(x, y) everywhere in space. Similarly, E(2)(x, y, z ) and H(2)(X, y, z ) are another set of solutions in another medium described by E(2)(X, y) everywhere in space. Let us consider three different media. Medium A:

Waveguide a Only

Suppose only waveguide a exists in the whole space (Fig. Hla). The medium is described by

The solution is

+ E~a)(x, y)] e i {3" z y) + H~a)(x, y)] e i /3 z

[E~{/)(x, y)

[H~a)(x,

(B.2) :

u

for the forward propagation mode and [E~a)(x, y) - E~{/)(x, y)] e- if3 " z

[ - H ~ a) (

X ,

y)

+ H ~a) (

x, y)] e -

(E.3)

i /3 a Z

for the same mode as (H.2) propagating in the - z direction. Note the 686

FORMULATION CF

~HE

IMPROVED COUPLED·MODE THEORY

(a) ___I

a

y_E(_a)c_x,y_)

687

_

------P:l~-----(b)

_Ia~,-(c)

_ "x

Figure H.1. Three different permittivity furictionsof interest for applications in deriving the improved coupled-mode theory using the reciprocity theorem. (a) only waveguide a exists in the whole space, (b) only waveguide b exists in the whole space, and (c) both waveguides a and b exist.

.'

relations between the field components of the forward propagation modes and those of the backward propagation modes. These relations can be obtained from Maxwell's equations and can also be checked from the z component of the Poynting vecto : for the power flow. We see that there is a sign change before H~a)(x, y) in CH.3). For example, the TE modes in a slab waveguide have a solution E = YE/x)e i {.1 Z and H = V X E/iwj.L = (-xi/3E y -+. zca/ax)Ey)/iwj.L for the forward propagation modes. T~e field expressions for the backward propagation modes are E = YE/x)e- ' {.1 Z and H = (xi/3E y + z(a/ax)E.)/iwf.L, with a sign change in the x component. Medium B:

Waveguide b Only

Suppose only waveguide b exists permittivity function is described by

III

the whole space (Fig. HcIb). The

The field sdlution is [E~b)(X, v)

[H~b)(x,

+ E~b)(x, y)] e i {.1 b z v) + H~b)(x, y)] e if3bz

(HA)

688

fORMULATION OF THE

jMP1~(iVEf) COUPl.ED-MODE

THE . )HY

for a mode guided in the z direction and

[E; b) (

X ,

y) - E ~b >< X , y)] e - i f3 b

[ - H~b)( .r ,

Z

y) + H~b)( x, y)] e -

(H.5)

if3b z

for the same mode as (H.4) propagating in the -z direction.

Medium C:

Both Waveguides a and b Exist-Coupled Waveguides

In this case, the medium is described by e(x,y)

which is shown in Fig. H.lc. The field solutions can be written as the superposition of two individual waveguide modes: E(x, y,

z) =

H(x, y, z)

=

a(z)E(a)(x, y) a(z)H(l1\x,

+ b(z)E(b)(x,

y)

y) + b(z)H(b)(x, y)

(H.6)

Let us consider three applications of the general reciprocity relation given by Eq. (H.n, which requires two sets of field expressions in two media, e(1)(x, y) and e(2)(x, y).

Application 1; ! Suppose we choose the first medium to be the coupled waveguide system described as medium C above, e(l)

=

e(x, v)

and the field solutions E(l) and H( l), as in (H.6), the coupled-mode solutions. Choose the second medium to be only waveguide a, e(2)(x, y) = e(a\x, y) and choose the solutions to be the corresponding guided mode propagating in .the - z direction (H.3). E (2) = [E ~ H(2)

(7 ) (

.r , y) - E ~') ( x, y)] e -

i /3,,:':

= [-H~a\x, y) + H~tl)(x, y)] e- iP,, 2

(R.7)

FORMULAT:ON OF THE IMPROVEr, COUJ>LEL'-!v'iODE THEORY

689

We obtain from the reciprocity relation (H.I)

dd Z

[f fac

z)e -

i~", ( - E~a, X H~'"

+ ffbCz)e-i~"'(-E~bl

X

-

E~a, X

ma' )

0

Z dx dy

H~a, - E~a' X H~b')

oZdXd Y ]

- iUJJf 6.c(G)(x,y)[E?a ). E~b) - E~a). E~b)] dxdyb(z)e- i ,l3 a z (H.8) Or d db(z) "d;Q(z) + C. dz = i(f3a + Kaa)a(z) + i(f3aC + Kba)b(z) (H.9)

where

(H.1D) 1 C p q = -ffE(q) x H {p) . i dx dy 2 I t

K pq =

w

4 Jf

.' (H .11)

~c(q)(x,Y)[E~P). E~q) -- E~P). E~q)] dxdy .'

(H.12) '

(H .13) I

Note that the fields have been normalized such that C aa 'an d Cbb

=

1.

Application 2

Choose e( l l( x , y), E(I) and H(l) to be the coupled waveguide system and the field solutions as in Application 1. Choose the second medium to be waveguide b only in the whole space

,. , ,. ,

~

~

_..-- --

.,

-

.

690

: • . , ..,,·t ....

F ORMU~ .ATION

,... ~_

·

p~

. ,; . . ..

"

" . • . .. . .

OF T H IC; ;:,1"1"'1<. ·. J'/2[' CC;UPLE D-M OO :-:

•

'i~HEO RY

with so lutions of t he guided mo d e propaga ting in the - z d ir e ct ion : E (2)

[E~b )(X , y) - E~b)(x, y) ]e- iJ3bZ

=

[ -H~b>Cx , y)

H (2) =

+

H ~h ) ( x , y) ]e-iJ3bZ

( H.14)

We obtain

Therefore, we fin d d dz

C-

[a(z)] = lQ . [a(z )] b(z)

(H. 16)

b(z)

where

c

=

[~ ~]

( H.1 7)

K aa

Q = [ ·r .1'" d b

(H.I S)

Application 3

Choose £(l)(x,

v)

E( 1) = [ E ~ a ) ( x I y) H (l)

£( G)(x,

=

+

= [H;U)(x, y) +

E~a)(

v)

x, y)] e il3 uz

H ~{f) ( x ,

y) ]e i13 uz

(H.19)

an d £ (Z)(x,

y) =

£(b )(X,

v)

E ( 2)

= [E ~ b ) ( X I v) - E ~b ) ( X , y ) ]e- iJ3bZ

H ( 2)

=

[ - H~b) (

x, Y)

+

H ~h l( x , v)

Je

- i13h2: .

(H.20)

•••- . ..- • •• , '"

'~

- .0::.'

••':':'" ~

FORMULATION OF THE LvIPROVED COUPLED-MODE THEORY

691

We obtain

(H.21)

(H.22) which is an exact relation. This relation also shows that for an asymmetric coupling system (f3 b =t f3 a), the coupling coefficients are not equal (K ba =t K ab). Only if the overlap integral C is small in the weak coupling case can K ba be approximately equal to K ab. Since

Qll =

e, + K aa

Q22 =

Q 12 = K ba + f3 aC Q21 = K ab

+

f3b C

e, + K bb

=

K ab + f3b C

=

K ba + f3a C

(B.23)

Q can also be written as (H.24) Therefore, d

[a(z)]

[G(Z)].

C ~ b( z)

= I[K + CB] b( z)

(H.2S)

0]

(H.26)

where B=[f3a

o

f3b

The coupled-mode equation can be written as d

[a(z)]

~ b ( z) M

=

. [a(z)] = 1M b ( z)

B + C-lK

(H.27) (H.28)

"i:jJa-;:IM

U

R.c"iSa·"-".-"

692

If

(H.29) th en 'Yo

=

'Y& =

Po

+

Ph +

K "" - CKba

1 _ C2 K bb - CK "b 1 _ C2

K ob - CK h b k a b = -------,-2

1- C

K b a - CK a a k b a = -------,,-2

1 - C

(H.30)

It is straightforward to show that (H.31) whi ch is an exact rel ation simil ar to (H .22).

Appendix I Density-Matrix Formulation of Optical Susceptibility The density-matrix theory plays an important role in applications to linear and nonlinear optical properties of materials in quantum electronics. The basic idea is that the density-matrix formulation provides a most convenient method to predict the expectation values of physical quantities when the exact wave function is unknown.

1.1

DENSITY-MATRIX THEORY [1, 2]

Assume that ljJ(r, t) is the wave function of the material system under a perturbation Hamiltonian, which can be due to an electromagnetic field or other excitations. The density-matrix operator p is defined as the ensemble average of the form

(1.1) where an overbar means the ensemble average. Explicitly, we can expand the wave function ljJ(T, t) using a complete set of wave functions ¢Jr): \, I

(1.2) n .'

Though not required, {¢n(r)} are usually chosen to be the solutions of the unperturbed Hamiltonian H 0' which describes the electronic states of the material system in the absence of any perturbation. The ensemble average of a physical quantity P is given by (P)

=

(ljJ(r, t)IPlljJ(r, t)

=.L: c:(t)cm(t) [m(r), P4>n(r)] In, n

=

L: Pnm Pm n m n i

=

Tr(pP)

(1.3 ) 693

DENSlTY-NiATfdX FORMULA.TIO:t' OF OPTICAJ SUSCEPTlBILITY

which is the trace of the matrix product Pnm and Pmn' where

Pmn

=

<4>m( r) IP! =

J¢~(r) P
Pnm = m> = (

(1.4 )

= c~(t)cn(t) 2

Note that Pnm = P':rln by definition, and Pnn = Ic n( t) has the physical meaning of the probability of finding the particle in the level n. The time-evolution of the density operator can be derived as follows. Use 1

a at

Hl/J(r, t) = ih-l/J(r, t)

(1.5)

or

(1.6) Multiplied by
. dCm(t)

dl--·= "" .i...J c n (t)Hrn n (ii

(1.7)

n

Thus

. a Ih-Pnm at

=

. acn(t) * _ ac;:'(t) cm(t) + Ihcn(t)--at at

Ifl

LHnkCk(t)C~(t) - LCn(t)ct(t)H::: k k

k

L (HnkPkm - PnkHkm) k

(1.8)

= [H'P]llm where H m* k -- H k m has been used. We conclude that

a

i fz - P = [H

at

'

pJ

= H P .- pH

(1.9)

Here H = flo + H' is the total Hamiltonian, which consists of an unperturbed part H 0 and a perturbation potential H', accounting for any external perturba tion.

1.2 DENSITY-MATRIX APPROACH TO THE OI'TICAL PROCESSES

1.2

695

DENSITY-MATRIX APPROACH TO THE OPTICAL PROCESSES

An advantage of the density-matrix approach is that it gives general expressions for the linear and nonlinear optical susceptibilities [2]. This approach has been shown to be very useful to study quantum electronic processes in different material systems. The density operator P satisfies the equation of motion as derived in (1.9):

(1.10) where the Hamiltonian operator consists of three parts: H = H o + H'

+

(1.11)

Hrandom

H o is the unperturbed Hamiltonian, H' accounts for the interaction such as the electron-photon interaction, -ror which 3

H'

=

-M' E(t)

L

=

MjEj(t)

(1.12)

}=1

where M is the dipole operator, • M

=

er

(1.13 )

and e = -I e] for electrons a, . .J + lei for holes. E is the electric field. Hrandom includes the relaxation effects due to incoherent scattering processes:

ata P

-i = ~l [H o + H',p] r,

(a p )

+ -

at

(1.14) relax

where ap )

( -a t

-1

- ----;;- [Hrandom ,p relax

(1.15)

- p(O)]

"

which can be written in terms of the T[ and T 2 time constants:

_ata (p uu _ p(O») relax -_

(0)

PUll - PUll

(1.16)

[Ill

( _ata Pili,:) relax: _

PUt'

for u

-=1=

u

(1.17)

for the initial distributions p~~2, = p~~~8ut;l which are diagonal.for e~ch state. It

09(;

DENSJT'. -MATRlX rORMU'J,TIO:'-J or Gl'T1CAL SUSCEPTIBILITY

is convenient to use

1 (LISa)

YUH

=

Yu u

= Y

1 U t'

for u

= (T) 2

=1=

u

(1.18b)

llU

Taking the uu component of the equation of motion (1.14) and using

(1.19) we obtain the density-matrix equation in the presence of an optical excitation:

a

-i

i

at PUll = ---,;- ( E ll

-

L; 2; (l'v1L'Pu'u -

Eu)Pll e + h

II

Puu,M!t'u)Ej(t)

}

(1.20) In general, we may define

Eu

!..-.'lIV

=

-

E;

(1.21)

--h-

and consider the interaction term H' = - M . E(l) as a small perturbation. The perturbation' series gives

( 1.22) I

One obtains a _p(n+l)

at

ltV

/

= ( -lW . 1w

-

I

(n+1) + Ylt e ) Puc

(Mi Ii " '-; " L...-

UlI '

LI

(11)

P/I'V

-

(n)MilI'U )EJi (t)

PUll'

j

(1.23 ) for n :2: O. Consider an optical field given by

E( t)

(1.24 ) <:<=1

1.2

DENSITY-MATRIX

APPR~)ACH

ornext, PROCESSES

TO THE

697

For example, E(l)

=

E( w) . E( w) e- l w t + e+ iw t 2 2

E(w)cos cot =

Thus, N = 2, WI = W, W2 = -W, and ~(w) = E(w)/2, ~(-w) Note that the zeroth-order solution is simply the initial condition p(O) = p(O)8

(1.25) =

E(w)/2 .

uu

(1.26)

p~or= f( E a )

(1.27a)

Ph°tJ

(1.27b)

uu

uu

or

=

f( E b )

as can be seen from (1.16). The general density-matrix equation (1.23) starts with the first-order p~}J(t) as the unknowns. The first-order solution IS obtained by substituting the zeroth-order solutions into (1.23). Let P (lu o)( t )

=

N ~ p(l )(W )e-iw",t ~ lit! a 0'=1

(1.28)

If only a single frequency co is considered, )(w)e- i w t P(ulc)( t ) = p d In'

-

+

p(I)(

-

Ul'

-w)e+ i w 1

(1.29)

By matching the e-iw",t dependence in (1.23), one obtains \

i

"

( 1.30) for u = a, b and v = a, b, using the two-level system shown in Fig. 9.1 with E; -;:; Ea' In terms of the components, we have (l ) _

Pa a

-

Ph~( t) = P~J( w)e -

p el) -

bb

0

F"

+ Ph:~(- w)e + icul

(1.31 )

-

( 1.32)

M ' E( ) ba _ W h(wbCl - W - lYba) .

(1.33 )

-

(0) _

Pao

iwl

( 0)

Pbb

.

DENSJTY-MXrRIX FCRMULATIO N OF ()PTI CAL S USCEPTIBILITY

E':J8

and p (O) _

Ii ( W ba

p (O)

aa

+w

bb

-

i v»; )

M

E(

s.:: ~

-

) W

(1.34)

Similarly (1.35) where

p~12(w)

is obtained by exchanging a with b from Pb1J(w) in (1.33), and P~16( - w) is obtained from Pb1J( - w). The polarization -per unit volume is calculated from the trace of the matrix product of the dipole moment matrix M and the density matrix:

pet)

=

1 V Tr[p(t)M] 1

= vL.Pu u(t)M v lI UU

1 =

V(PaaM aa + PabMba + PbaMab

+

PbbMbb)

The ith component of the polarization density, to the first order optical electric field , is given by

(1.36) In

the

1 ; i p,.(t) = -V (M p(l)(w 1l\ + M i p(L)(w»)e - iwt \ b

1

+ -V

a

_

a

b

a b

[M il p (l )( -w)

_

b

a

+ M'-ab_p (L)( _w»)e +iwt ba

b
The definition of the electric susceptibility

Xi j

(1.37)

is obtained using (1.38)

or in component form

PiCt) =

LC: oX ,-;( lIJ) £j ( (v )e "' j

il
+ LC: OXi J ( -(IJ ) E/ _w)e -t·i wt (1.39) j

1.2 DENSITY-MATRIX APPRO.\CH TO THE OPTICAL PROCESSES

699

Substituting (1.32)-0.35) into (1.37) and comparing with (1.39), we obtain

.t:( It W

M~b Mia ba -

W -

. ) 1'rba

](

(0) _

P aa

(0») Pbb

(lAO)

We find that the relation

(1.42) is observed. This can also be checked with the symmetry property of the complex permittivity function E'(W) + iE"(w), discussed in Eqs. (E.8) and (E.9) in Appendix E, in which the real part c'ew) is an even function and the imaginary part E"(W) is an odd function of w. When a distribution of states is considered, for example, the energy bands

(1.43a) and (1.439) for two sets of wave vectors k , and k b . We have to sum over the density of states in the k space. Assume that E b > E; and h co is close to E; - Ea' The second term in EOXi/W) will be the resonant contribution: "

The spins sa and Sb are considered. If the states \a) and Ib) db not include the mixing of spins, one finds that the spin summation gives simply a factor of 2 since the dipole operator M is independent of spin and . (1.45) that is, the spins s., and

Sb

must be the same. Thus; the spin selection rule

DENSITY-MATRIX FORMULATION \JF OPTICAL SUSCEPTlBILI'!"Y

700

gives a factor of 2 when summing over spins.

(1.46) For a plane wave propagating in an anisotropic medium with (1.47)

where (Eb)ij is the background dielectric tensor, the solutions are in the 'form of ordinary and extraordinary waves with corresponding propagation constants. Let us consider the simple case when (Eb)ij and Xu are diagonal:

(1.48) That is, the induced polarization has the same direction as the applied electric field direction, i = i: M bi a

e"· M b a

--

(1.49) (1.50)

Therefore E1 =

Eb

+ Re(EoX)

Ie' Mbal\E b

E a - hw) (E - E; - hw)2 + (f/2)2 (fa - fb) b

vEE 2

= E2

Cb

+

-

(1.51 )

= Im( coX) 2

=

VEE (E

b

-

Ie . M b a l2 ( f/2) E, - hW)2 + (f/2)2 (fa - f b )

(1.52)

where ti Yi« = f /2 is the half linewidth. We see that the expression for cz in (1.52) is the same as (9.1.38) if the conversion (9.1.31) is used. The expression for E I in (1.51) when I" = 0 reduces to (9.1.37) if we take the resonance approximation E b - E a = lu» in (9.1.37), in, - E a ) 2 - (hw? = (E b - E a hw)2hw. Note that the expression (9.1.40) for El(W) generally gives a better convergent result than 0.51) for semiconductors using a two-band model, for which the sum over the energy can be slowly convergent since the integrand in (1.51) is only inversely proportional to energy, and we are integrating over

~\EFE:\ENCES

701

the energy. The propagation constant is

= WVf.Lo E

k

= W"ff.LEb =

=

W/P:[E b + EOX(W)]

/ 1 + ~[XI(W) \ 2E b

+ iXfI(W)]}

a

+ k a 6.n + i -

(1.53 )

= the background dielectric constant

(1.54)

kanr

2

where nr =

I!i V EO

1

6.n = - X '

z»,

1 =

2n,E o

Re( EOX)

=

the change in refractive index (1.55)

and the " intensity" absorption coefficient a =.2 Im(k) is W

a

Im(EoX)

= flrCEO

REFERENCES 1. A. Yariv, Quantum Electronics, 3d ed., Wiley, New York, 1989. 2. Y. .'R . Shen, The Principles of Nonlinear Optics, Wiley, New York, 1984 . ,

\,

.'

(1.56)

Appendix J . - Optical Constants of GaAs and InP

70'2

703

OPTiCAL CONSTANTS OF GaAs AI'
Table J.1

Optical Constants ofGaAs

Optical Energy hw (eV)

Wavelength

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.35 1.4 1.42 1.425 1.43 1.435 1.44 1.45 1.47 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6

2.4797 2.0664 1.7712 1.5498 1.3776 1.2399 1.1271 1.0332 0.9537 0.9184 0.8856 0.8731 0.8701 0.8670 0.8640 0.8610 0.8551 0.8434 0.8266 0.7749 0.7293 0.6888 0.6526 0.6199 0.5904 0.5636 0.5391 0.5166 0.4959 0.4769

* References

A (u m)

Refractive Index n 3.3240 3.3378 3.3543 3.3737 3.3965 3.4232 3.4546 3.4920 3.5388 3.5690 3.6140

3.666 3.700 3.742 3.785 3.826 3.878 3.940 4.013 4.100 4.205 4.333 4.492

Extinction Coefficient K

Absorption Coefficient a

= 47TK/A

(10 4 ern -1)

References" a,b

0.0017 0.0271 0.0554 0.0572 0.0557 0.0568 0.0612 0.0664 0.080 0.091 0.112 0.151 0.179 0.211 0.240 0.276 0.320 0.371 0.441 0.539

0.0240 0.3900 0.8001 0.8290 0.8101 0.8290 0.8994 0.9893 1.216 1.476 1.930 2.755 3.447 4.277 5.108 6.154 7.460 9.025 11.174 14.204

a,c

a,d

,

are indicated on the first row for all items below. aE. D. Palik, "Gallium arsenide (GaAs)," pp. 429-443 in E. D. Palik, Ed., Handbook of Optical Constants of Solids, Academic, New York, 1985. bA . N. Pikhtin and A. D. Yas'kov, "Dispersion of the refractive index of semiconductors with diamond and zinc-blende structures," Sou. Phys. Semicond . 12, 622 (1978). cH. C. Casey, D. D. Sell, and K. W. Wecht, "Concentration dependence of the absorption coefficient for n- and p-type GaAs between 1.3 and 1.6 eV," J. Appl, Phys. 46, 250 (1975). d D. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and JnSb from 1.5 to 6.0 ev.: Phys. Rev. B 27, 985 (1983).

r

704

." -.,- •••

.•.~. ,,..••••

_.... '''''''

. ,~

'" ... ~"

, .. ...'

~ ...

'1-,. . ~ ..,

OPTICAL CONSTANTS OF G;,As !,ND Il'?

The refractive index n, the extinction coefficient K, and the absorption coefficient (Y of GaAs as a function of optical energy are plotted in Fig. J.1.

2.5

5.5

~ · . . ! cfb . ····················y······················f·········· ············1······················T" ··········l}···· ~ ; i ~ : : : : c9

GaAs

5.0

....................l.

4.5

•

3.5

o a

0

0: :

~

~

L.of?........•...

•

1.5

•

:

6 ~

•

0 O'

•

0~5"

...

0:

j

: :

~

K

h

h

:

••• •••••••

: . •

: - • • - :

1.0

1.5

..

u~ !1.~ ••

_h •

•

j;..

0.5

: :

j

2.0

1.0

0.0 3.0

2.5

Photon energy (eV)

20

GaAs

I

b

1.

: 6 0 ~.--C5.. 00Q~~ ~h

•• u_ •• m u .--

(a)

(j

.L

:

····················t····················t···········~···~··~··o-·~·······_····t·. ······~:······ ·

4.0

---e

l I

: j •

n

3.0

2.0

0.,

16

f-

:

f

........................-[

12

1

~

·

j

.

+

j : :

• •

.

c••••• ]•••• •• •••••••••••••-

: : ··············t··················.. · ·~ ·· · ·· · · ·· ·· · ··· · · -

---= .- 8 f-· · · · · · · · · · · t· · · · · · · · · · · l.· · · · · · · · · · · ·l· · · · · · .· ·~ · ·l· · · · · · · · · · ..ca....

0

0

~

.c

<

,. .

4

o 0.5

(b)

.

~.

i

1.0

i

;..................•...+ _ .~

-

:

j • -

.~

1.5

i 2~0

j:

-

:

.

2.5

3.0

i

Photon energy (eV)

Figure J.1. (a) Real and imaginary parts, nand K, of the complex refractive index and (b) the absorpt ion coefficient a of GaAs vs . photon energy.

OPTICAL CONSTANTS

Table J.2

Optical Energy trw (eV) 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.25 1.272 1.301 1.326 1.333 1.340 1.345 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

or GaAs

AND InP

705

Optical Constants of InP

Wavelength

Refractive Index

Extinction Coefficient

A (u m)

n

K

2.0664 1.7712 1.5498 1.3776 1.2399 1.1271 1.0332 0.9919 0.975 0.953 0.935 0.930 0.925 0.9218 0.8266 ·0.7749 0.7293 0.6888 0.6526 0.6199 0.5904 0.$636 0.5391 0.5166 0.4959

3.129 3.146 3.167 3.191 3.22 3,254 3.297 3.324 3.346 3.362 3.385 3.390 3.396 3.399 3.456 3.467 3.476 3.492 3.517 3.549 3.585 3.629 3.682 3.745 3.818

Absorption Coefficient a

= 47TK/A

00 4 ern -1)

. References

a, b

0.0000113 0.000281 0.0059 0.0109 0.0355 0.0571 0.203 0.218 0.242 0.270 0.293 0.317 0.347 0.380 0.416 0.457 0.511

0.0001456 0.003705 0.07930 0.1473 0.4822 0.7791 3.086 3.535 4.170 4.926 5.739 6.426 7.386 8.473 9.697 11.117 12.948

a,c,d

a, b, d a,c,d

a,e

"O. J. Glernbocki and H. Piller, "Indium Phosphide (Inf')," pp. 429-443 in E. D. Palik, Ed., Handbook of Optical Constants of Solids, Academic, New York, 1985. bA. N/Pikhtin and A. D. Yas'kov, "Dispersion of the refractive index of semiconductors with diamond and zinc-blonde structures," SOL'. Phys . Semicond. 12, 622 (1978). "G. D. Pettit and W. J. Turner, "Optical Constant," J. Appl. Phys . 36, 2081 (1965). "s. ,'0. Seraphin and H. E. Bennett, "Refractive index of InP," pp. 499-543 in R. K. Willardson and A. C. Beer, Eds., Semiconductors and Semimetals, Vol. 3, Academic, New York, 1967. eD. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, lnP, lnAs, and lnSb from 1.5 to 6.0 eV," Phys. Rev. B 27, 985 (1983).

..

,~.~ .

- "~"'.'''''''''-' .

'-~ ''

......

7U(i

OPTIC/\L CONSTANTS OF GaAs ANT:' IIIF

The refractive index 11, the extinction coefficient K, and the absorption coefficient a of InP as a function of optical energy are shown in Fig . J .2.

4.4

1.4

InP

.........-

4.2

_··

· ·· ..•.•...............

4.0

n

-

, ,,

... . .. :

-..-

-

..

.. . . -:- ..•................... : -

o

-

o

.

······· ··············r······· ···:···········l· ··· ······· ········ ···+····· ············~···f··········· ..·······

0.8

:

·l

:

:.

:

.: 0

:.

: :..: 0: : : : 0 :····················t..······· ·············j·· ····················6· ·'C·················j···.················

3.6

1.2 1.0

:

0-

.

.

~

3.8

--

~

:·

.-

0 . 6'. 0 0 0 :

.

3.4

: ····················-:-········7····

3.2

·~r···················+······~············t···········

3.0

0.5

-:-:

~

_

..,

: . .

: :

:

:

: dP

:

··

.

.

)..

1.0

1.5

2.0

2.5

-

-

.

0.6

•

:

.

0.4

.

0.2

0.0 3.0

Photon energy (eV)

(a)

20

InP

! f

;

!

.

.

•

16 I-····················-r······················i······················j······················t···.·············..-

12 ,...

.J ~

..1

i

1

~

1

.l

.1.

-

t - t.-1

i·

•

~

-

~

8 ,...····················t·················;···+·····················j···.···~············T····················-

~

4

f-

~

!

0.5

1.0

o (b)

;

J

.;

•

J

.

1.5

~ : ...t. ! :

2.0

.l :

-

: 2.5

3.0

Photon energy (e V)

Figure J.2. (a) Real and imaginary parts, nand K, of the complex refractive in dex and (b) the absorpt ion coefficient a of InP vs, photon energy.

k

Appendix K -Electronic Properties of Si, Ge, and a Few Binary, Ternary, and Quaternary Compounds

.

-.....

.'

.'

707

Table K.2 Important Band Structure Parameters' for GaAs, AlAs, lnAs, InP, and GaP

-t"

Materials GaAs Parameters a o (A) E g (eV)

5.6533

OK

1.519

300 K

1.424

AlAs

InAs

InP

GaP

5.6600

6.0584

5.8688

5.4505

0.42

1.424

0.354

1.344

3.13 2.229* 3.03 2.168* 0.28 -7.49 21.1

0.38 -6.67 22.2

0.11 -7.04 20.7 06.7)f -5.04 1.27 -6.31 -1.7 -5.6 10.11 5.61 4.56

2.90... 2.35* 2.78 2.27* 0.08 -7.40 22.2

Optical matrix parameter e, (eV)

0.34 -6.92 25.7 (25.0)f

Deformation potentials (e V) «, (e V) . ({L'. (eV) a = Q c - au (eV) b(eV) d (eV) C l1 (lOll dyne z'cm") C 1Z (lOll dync z'cm") C~4 (lOll dvne z'crn")

-7.17 1.16 -8.33 -1.7 -4.55 11.879 5.376 5.94

-5.64 2.47 -8.11 -1.5 -3.4 12.5 5.34 5.42

-5.08 1.00 -6.08 -1.8 -3.6 8.329 4.526 3.96

0.067 0.50 0.087

0.15 0.79 0.15

0.023 0.40 0.026

0.077 0.60 0.12

0.25 0.67 0.17

0.333

0.478

0.263

0.606

0.326

0.094

0.208

0.027

0.121

0.199

6.8 (6.85) 1.9 (2.1) 2.73 (2.9)

3.45 0.68 1.29

4.95 1.65 2.35

4.05 0.49 1.25

t1 (e V) E; av (eV)

Effective masses m;/m o mhh/mO mih/mO

1 mhh,z/m O = m1h.z/mO =

"YI "Y2 "Y3

"Yl - 2"yz 1 "YI

+ 2'Y2

20.4 8.3 9.1

-7.14 1.70 -8.83 -1.8 -4.5 14.05 6.203 7.033

* Indirect

band gap, E/X) value. aC. G. Van de Walle, "Band lineups and deformation potentials in the model-solid theory," Phys. Rev. B 39, 1871-1883 (1989). b p . Lawaetz, "Valence-band parameters in cubic semiconductors," Phys. Rev. B. 4, 3460-3467 (1971). C S. Adachi, "GaAs, AlAs, and Ga 1 -x AI,rAs: Material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R28 (1985). "o. Madelung, Ed., Semiconductors, Group TV Elements and llT-V Compounds, in R. Poerschke, Ed., Data in Science and Technology, Springer, Berlin, 1991. eK. H. Hellwege, Ed., Landolt-Bomstein Numerical Data and Functional Relationships in Science and Technology, New Series, Group III 173, springer, Berlin, 1982; Groups III-V 223, Springer, Berlin, 1986. . 'i, G. Shantharama, A. R. Adams, C. N. Ahmad and R. J. Nicholas, "The k . p interaction in InP and GaAs from the band-gap dependence of the effective mass," /. Phys, C: Solid State Phys. 17, 4429-4442 (I 984). 709

ELECI

RONlCPROPERTIE~C'F ~ [ ,

Ge, AND A FEW COMPOUNDS

Table K.3 Important Band Structure Parameters for AJ.~Ga 1- rAs, In , _ xGax As, Alj In , _ x A s , and GarIn 1 _ x A s ,P J _ y Compounds a -

General Interpolation Formula for Ternary Compound Parameters P: P(AxB1-xC) = xP(AC) + C1 - x )P(13C ) Al xGa 1-.~ As E/r) = 1.424 + 1.247x (eV) at 300 K for x < 0.4 1.519 + 1,447x - 0.15x 2 (eV) at a K for x < 0.4 m; /m o = 0.067 + O.083x mhh / m o = 0.50 + 0.29x (Density of states mass) mih Im o = 0.087 + 0.063x m;o/mo = 0.15 + 0.09x Y j(x) = xyJAIAs) + (l - x)YiCGaAs) (for calculating transport masses) In1_xGaxAs EgCn = 0.36 + 0 .505x + 0.555x 2 (e V) at 300 K 0.324 + 0.7x + 0.4x 2 (eV) at 300 K 0,422 + 0.7x + 0.4x 2 CeV) at 2 K m:lmo = 0.0250 -x) + 0.071x - 0.0163xCl-x) or l/m:Cx) = x/m;CGaAs) + C1 - x)/m;(InAs) Ino.53Ga0.47As EgCn = 0.813 (eV) at 2 K 0.75 (eV) at 300 K m:lm o = 0.041 mhh / m o = 0,465 / / [001] 0.5.6 11[110] mfh I mo = 0.0503 Al xlnl _xAs EgCn = 0.36 + 2.35x + 0.24x 2 CeV) at 300 K 0.357 + 2.29x (eV) at 300 K for 0.44 < x < 0.54 0.447 + 2.22x CeV) at 4 K for 0.44 < x < 0.54 Al OAS In o.5 2 As EgCn = 1.508 CeV) at 4K 1.450 CeV) at 300 K m:/mo = 0.075 mhh/m u = 0.41 mi" Imo = 0.096

e

Ref. a b

a interpol. f a interpol. c a

a

a

a

d

ELECTRONIC PROPERTIES OF Si, Ge, AND A FEW O)M20UNDS

711

Table K.3 (Continued) General Interpolation Formula for Quaternary Compound Parameters P: = xyP(AC) + (l - x)(1 - y)P(BD) + (l - x)yP(BC) + x(1 - y)P(AD)

P(AxB\._xCyDI_Y) GaXInl_xAsyPl_y

1.35 + 0.668x - 1.068y + 0.758x 2 + 0.078 y2 - 0.069xy - 0.322x 2y + 0.03xy2 (e V) at 300 K m:(x, y)/m o = 0.08 - 0.116x + 0.026y - 0.059xy + (0.064 - 0.02x)y2 +(0.06 + 0.032y)x2

Eg(x, y)

a(x, y)

=

=

5.8688 - 0.4176x + 0.1896y + 0.0125xy

a

(A..)

Lattice-Matched to InP: 0.1894y x

=

(0.4184 - O.013y) E/y) = 13.5 - 0.775y + 0.149 y 2 (e V ) at298K 1.425 - 0.7668y + 0.149 y 2 (eV) at 4.2 K m; / m o = 0.080 - 0.039 y m~h/mO = 0.46 . -_. 2 mih/mO = 0.12 - 0.099y + 0.030y m':o / m o = 0.21 - 0.01 y - 0.05 y 2 In I-x _yAJ-rGa y As E/x,y) = 0.36 + 2.093x + 0.629y + 0.577x 2 + 0.436 y 2 +l.013xy - 2.0xy(l - x - y) (e V) at 300 K Lattice-Matched to InP: (I n O.5 2 AJ O,4s)/In 0.53 Ga 0.47 )1-Z As x = 0.48z 0.983x + y = 0.468 E/z) = 0.76 + 0.49z + 0.20z 2 (e V) m:/m o = 0.0427 + 0.0328z

a

e

at 300 K

aK. H. Hellwege, Ed., Landolt-Bomstein Numerical Data and Functional Relationships in Science and

Technology, New Series, Group III, 17a, Springer, Berlin, 1982; Groups III-V 22a, Springer, Berlin, 1986. b M . EI Allai, C. B. Sorensen, E. Veje,and P. Tidemand-Petersson, "Experimental determination of the GaAs and Gat_..rAlxAs band-gap energy dependence on temperature and aluminum mole fraction in the direct band-gap region," Phvs. Rev. B 48, 4398-4404 (l993). "s. Adachi, "Material parameters in In..rGal_..rAs yP 1 - y and related binaries," J. Appl, Phys. 53, 8775-8792 (1982). . dp. Bhattacharya, Ed., Properties of Lattice-Matched and Strained Indium Gallium Arsenide, INSPEC, Institute of Electrical Engineers, London, U.K., 1993. eS. Adachi, Physical Properties of III-V Semiconductor Compounds, Wiley, New York, 1992. f H . C. Casey, Jr., and M. B. Panish, Heterostructure Lasers Part A: Fundamental Principles, Academic Press, Orlando, FL, 1978.

Index Absorbance, 213, 586 Absorption: coe fficient, 341. 354, 701 interband, 352, 358 intersubband, 373, 374 in quantum-well structure, 358 spectrum, 360 Acoustooptic: effects, 527 modulators, 508 wave couplers, 530 Airy function, 547 Ampere's law, 22, 25 Auger: generation-recombination processes, 4D, 79 recombination, 40-42,397,431 Avalanche photodiode (APD), 604 multiple-quantum-well, 615 separate absorption and multiplication (SAM),614 Axial approximation, 181 Band diagram, 3 Band-edge: basis fu nctions, 134-136 discontinuity, 6, 666 energy, 135, 176 profile, 186 Band lineups, 662 Band structu re. 129 of bulk semiconductors, 150-153, 156. 157 Kane's model, 129 Luttinger-Kohn's model. 137-141 of quantum wells, 175-185 of strained quantum wells. 185-190 Beat length, 301 Bernard-Duraffourg inversion condition, 357 Bloch function, 125,359 Bloch theorem. 124 Block diagorialization, 180 Bohr radius. 105,550 Boltzmann statistics, 27 Bonding diagram, 3,4 Bound-state solutions: for hydrogen atom, 102-106,637-640, 645-647

for square well. 91-97 Boundary conditions: for effective mass equations, 91 for Luttinger-Kohn Hamiltonian, 183 for Maxwell's equations. 23 for ohmic contacts, 34 Bragg acoustooptic wave coupler, 530 Bragg diffraction, 529 Chebyshev polynorn ial. 452 Coherence time, 501 Concentration: electron (n), 30. 398 hole (P). 30, 398 surface electron, 424 surface hole, 424 Contact potential: p-N junction, 52 n-P junction. 68 Continuity equation. 22 current, 25. 26, 44 r .bab ility density, 84 Continuum states, for hydrogen atom, 102-106,640-644,647-649 Coupled-mock: applications of. 303. 309 equationJ, 297,450 improved theory. 302, 686 theory, 283 in rime domain, 288 Coupler: 6.a. 312, 526 directional. 310. 526 I , grating, 285 prism. 284 transverse, 283 waveguide, 283, 309 Coupled optical waveguides. 294 Coupled resonators. 289, 291 Coupling: asynchronous. 293,302 codirectiorial. 297, 299 coefficient, 290, 296, 317, 689 contradirectional, 297, 299 synchronous, 293,302 Critical angle, 207, 261, 405 713

INDEX

71 "1 Current d ensity. 22. 63 elec tron . 25, 61-63 h ole. 25. 60-63 injection , 397. 488, 489 s ur fa ce.23 . C uto ff cond itio n. 246. 259 C u to ff fre quency. 246 Damping fact or. 495 Dark current. 585. 597. 598. 620 Deformation potential. 147.660 conduction band, 442 hydrostatic. 151 shear. 442 valence band. 44 2 Density-matrix. fo rm u la tio n o f optical su scep tib ility. 693 Densi ty o f states. 544 conduct ion band. 29 jo in t. 360 o n e-di me ns io na l. 90 fo r photons. 346 three-d imens ion al. 28 two-dimens ion al. 88 valence band, 29 Depletion : approx imation. 50. 67 width. 53.56.68. 69 D ete ct iviry. 593. 599. 600 background limited in fra red ph ot o dctector (BLIP), 600 Dielectric waveguides: rectangular. 263 slab. 242. 257 Diffusion : coefficient. 26 length. 48 Dipole approxim ation. 341 D ipole moment. 342 inters ubb an d .3 73 optica l. 374 Distributed Bragg reflector, 462 Di st ribu ted feed bac k: c ou pled-mode eq ua ti ons, 3 16 la sers. 4 57 stru ctu res. 315 Du al ity princ ipl e. 203-205.212.2 15.255 Dya di c Gre en 's fun c tion . 115 Effective in de x. 248 method, 270 Effective mass. 136 for degenerat e bands. 142. 6S3 parallel and perp e ndi cul ar. 153. 157

fo r a singl eba nd, 142. 651 theo ry. 141. 65 1 E ffect ive well width , 95, 122 E ins tei n 's A a nd B c oefficie n ts. 347 Eins te in rela tio n. 405 Elasti c s tiffness const a nts. 148. 149 El cctroab xorpfion m odul ators. 538 interb and .570 Electron-hole pair. 539 E lectron ic properties: o f binary. ternary a nd quaternary compounds, 707 of s., Ge. 707 Electrooptic: effects. 508. 509 m odulators. 508 Ex c iton: bind ing energy. 551. 552 b ound and cont inuu m s tat es. 81. 102.554. 631 e ffect. 550. 558. 563 three-dimensi onal. 102. 103. 550. 63 1 two- dimensional. 104. 105,552.644 Extra ord inary wav e. 233. 234 Fa b ry-Pe ro t m ode. 277.41 2 Fara d a y rotation. 684 Farad ay's law. 22 Far-field : a p p ro ximat io n. Z to pattern. 2 J8-210. 454 -456 Fermi-Dirac : distribution. 28 integral. 29.30 Fermi level. 31 in version formula for. 33 quasi-, 27. 58.362.423 Fermi's go ld e n rule . 119. 337. 340 Fil te r. 3 10 F ran z-Keldysh e ffect. 546 Frequenc y re sp on se fun cti on. 491 G a in. 356 d iffere n tia l. 489 interb und.35 8 line a r. 48 7 n on lin ea r. 493 pea k. 429. 43 8 in a quantum-well la ser. 421-437 sa tu ra tio n. 493 spe ctru m. 35 [, 363. 364,385. 42 5.427 with valence -b and -mixing effect s, 381 G a ug e: Cou lo m b , 203. 338

.'

\ I

715

INDEX Lore n tz, 202. 214 transformation. 202 Gauss's law. 22. 50 Generation: rate. 26 in semiconductors. 35-43 Goos-Hanchen phase shift. 208. '209. 220. 262 Guidance condition. 243. 261. 267 Hamiltonian : electron-photon interaction. 338 Luttinger-Kohn. 140 Pikus-Biro146.655 Harmonic oscillator. 82. 97 Heaviside step function. 88. 361 Hermite-Gaussian functions. 98.415 Hermitian adjoint. 340 Heterojunction : band lineups. 662 double. 3'93~ energy band diagram, 52. 57, 59. 66. 70-72. 76 n-N, 70-75 n-P.65-70 P-n-N. 76 semiconductor p-N. 49-65 Hydrogen atom. 82. 102.631 Impact ionization. 43. 605. 607. 611 coefficients. 43 Index ellipsoid. 235. 509. 511. 512. 520. 522 Interband: absorption, 352. 358 electroabsorption modulator. 570 gain. 358 Intersubband: absorption. 373.374 avalanche photomultiplier. 616 quantum-well photodetectors, 616 Intrinsic carrier concentration. 27. 38 Ioniza tion coefficien t: electron. 43. 605 hole. 43. 605 Isotropic media . 224 Junction heating. 469 Junctions: metal-semiconductor. 75 n-N. 70 n-P.65 P-n-N.76 semiconductor p-N, 49 K factor. 496

k-seiection rule. 352. 354 in quantum well. 359 k surfaces : extraordinary wave. 232-234 ordinary wave. 232-234 Kane's model. 129-137 Kane's parameter 132. 136.369.385 Kerr effect. 510 k· P method. 124 for degenerate bands. 137-141 for simple bands. 124-129 with the spin-orbit interaction. 129 Krarners-Kronig relations. 226. 344. 671 Kronig-Penney model. 166

r.

Laser arrays. coupled. 449 Laser oscillation conditions. 460 Leakage current. 401. 403 Lifetime: carrier. 37. 39. 488 carrier recombination. 398: 585-587 photon. 488. 489 Light-emitting diode. 405 Linewidth : broadening. 344 enhancement factor. 497.503 spectral. 497-504 Lorentz force equation. 388 Lorentzian function. 343. 344 Lowdin 's method. 114-116. 139 Luttinge. -Ko h n Hamiltonian, 140. 176-185. 659 Luttinger-Kohri's model. 137-141 Luttinger parameters (YI' Y2' Y)). 140. 183.709 Mach-Zehnder, interferometric waveguide modulator. 523, 524 Magnetooptic effects. 679 Marcatili's method , 263 Matrix elements , 117.341. 352, 368 Maxwell's equations. 21. 22, 200 , 223 in frequency domain, 204 general solutions to. 200-203 Mobility. 26 Model-solid theory. 662 Modes: EHpq . 263.267 HEp(/ ' 263. 264 transverse electric (TE). 243.257.261 transverse magnetic (TM). .254-257.261 Modulation: rate equations. 487 response. 491 of semiconductor lasers.A,S7

11': 1) EX

': 16

Mod ulatiou-doped q u r.n turu WGI I. l .:':O··)()6 Modulat ors. 508 aco us to o pt ic, 50 8 ampl itu de. 50S. 51 I. 515 directional coupler. 525.526 ele ctro absorp tion . SJS electroopric. 508 phase. 517 waveguide. 524 Momentum matrix elements. 366.3 83 of a bulk semicond uctor, 370 of quantum wells. 370 Noise: equivalent power (NEP). 593 excess. 612 gen erati on-recombination . 591 multipl ication (o r ga in). 6 13 in ph ot oconductors. 590 s ho t. 590 thermal (Johnson or Nyquist). 593

.'

Operator. 84. 85 annihilation. 99 creati on. 99 Optical confinement factor. 253-256. 27 5. 399.488 for index-guided la ser. 420 for quantum well's. 433 . 446 Optical constants of GaAs a n d In P , 702-706 Optical matrix element. 352 for excitonic transitions. 560 interband.359 for two-p article transition picture. 543 Optical processes . in semiconductors. 337 Optical transit ions. us ing Fermi's go ld e n rule. 337 Ordinary wave, 132-234

Periodic table. 2 Permea bili ty. ten sor. n Permitriviry: fu nctio n. 343 te nso r. 22 fo r uniaxia l media . 128 Perturbati on meth od. 106. 566 Lowdins. 114 Perturbation theo ry: ti m e-de pe nd e n t . 116 tim e-inde pe nd e nt. 106 Phasor.203 Photoconductive ga in. 46.585 in quantum wells. 617 Ph otcc on cluctivity, 45.583

Photoconductor. 583 n-i-p -i s upe rlat ticc. 594 noi se s in . 590 Photocurrent response. 585 Photodc tect ors. 583 in te rsub b a nd . 6 16 Photod iodes: avalanche. 604 p-i-n, 60) p-n junction. 595 Photon lifetime. 489 Pikus-Bir Ham ilt onian. 144-157.655 with out spin- orbit coupling, 147-154 with spin-orbi t coupling. 154-157 Plane wave: reflection from a layered me dium. 205 solutions to M axwell's equations. 123 Pockels effec t. 509 Poisson 's equat ion . 24. 25 Poisson's ra tio . 440 Polarization: tr ansverse electric (TE). 205. 243. 370-373. 385. 564, 565 transverse m agnetic (T M ). 208. 255. 371-374,3 85 .564.5 65,617 Polaroid. 238 Poynting's theorem, 675 Poynt ing vector. 339 Probability c u rr ent den sity. 83 P rop agation m atrix: b a ckwa rd. Yl O in elec tro rna g rie tics. 109

forward. 159. 167.211 for a one-dimensional potential. 157-160 for u period ic po tential. 166 Pr opagation: co ns ta n t. 226.248 in gyrotropic media. 679 Quantum c a sc a d e laser. 380 Quantum confined Stark effec ts (QCSE). 533 . 557 Quantum efficiency: differential. 400. 470 ex te rn a l. 401 internal. 400 Quantum mech anics . 82

Sc hrod inger e q u a tio n. 83 sq ua re we ll, 85 Q ua n tu m wel l. S5 lasers . 421-448 s trained , l S5-i90. 437-448 Qu arter-wave plute. 236. 238. 5l S QlI c\Si-c I ,.:'.'t ru~ tu t ic field s . 2.+