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
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I . Title'. I!. S e r ~ e s .
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i14-24'70 I
62 1.35 1'(i.15--~lc20
I.'rinteJ i n the IIniteJ Stares
ot't~.1!1*:riit:
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<i- !() (1(;.!-1[>.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<r] 4d"'
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.
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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<.'
SEhll<l'(.)~?>L!:,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=
<al Generation of electron-hole pair
(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!<h 1:Cor!3Lr(:TOi7, 1:LECiROT'ilCS
(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<O
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<o
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<I:)~C'CTOR
JTJNCTIONS
-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<l
c ~ c r g ylevels a n d wiive !functions.
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 <band.
~ n Thi? .first four basis
functions arc respectiveiy dl=ger:er;.~t~
with t h e I::L:;T, f o ~ l rbasis F~lnctions.The
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?,<UC"T'iJRES i N SEMICONDUCTORS
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 ((
<I
0).
,The heavy hole still keeps its
feature of a heavy effective mass along the ki direction, and it is above the
light-hole band at k = 0 for the conipression case, On the other hand, the
light-hole band is above the heavy hole band in the case of tension ( x > 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) <a,
a(x) = a,
/--
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<C3.;VIC~l3AND
STI'CUCIURES l N SEMICONDUCTORS
!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:</
1i1,C:i;i I
:b sy.-)teril.( l ~ j!Iii!>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> ;<it!! .r 0.32.5 in the harriri.~,
( 1 1 ) j! ,% cju;~~l~l:r-rl
ell.; \+ill1 .Y = 11.315 i n the l-:a:.r.ic~-s,
(c.) 73 ;
iL . ~ L L : ~ ~wells
I ~ L I with
~
.r = 0.32,
and ( d ) 9:.<
qi.lilnLum ~vel!!; ~ . i c l : .:.
il.38. The !~c.!~.i:r;i.lni:ll
; ~ s ~iss !tie w;ivcvcctor normalized by
,
/I , :
=j
;
;
4
f i
j.\fir;r.
q l ~ f''5..)
-
-
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<UC.I'I'UlIE:S OF SF:FiilCGY3!I:.:7'OR
<)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.;.<i., ut!!l tl-it>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:<Ii' I SEMICONT;!.r!_"rOR.;
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 <IF EI-EC i'KO?JIC B,>.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
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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<O :V C,C;1\T E'TICS
!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<t: k i n the
plane.
.:--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
'<Yh:~~I.~D
, ~. E1,EC
~ f ~ J C'5 ? C .sl4Ar: J v L- ,.V 1:'""
:?.3! . j ~ ~ j ~ ~ s
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<riicnt to choose rpr' = 0 for the eigenecluation.
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(<y,z)
satisfies the wave equation and should have solutions of the form exp(i k , x +
i k , z ) as 1 o n g . a ~ir;7 + li: = u 2 p & Therefore,
.
we see that a general solution
for E R ( x ,z ) will be
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.)
.
<I
c
i F PL.1 C.'kV171!lNS OF' OPI'I:2Al, %'A~.I'EGT
-1I):,,COIJFL!<R,S
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
.<urn ot' (11) an i~iipc'~.t~irbc'Cf
part t ( " ' ! i - ~ .which d ~ s c r r b e sa u n i f ( ~ r mslab waveguide, and Ic) a
perturbed pi1r.L A?
r--,- - :t-, or i i i t
F! - E.-, ill t . 1 ; ~corrugation regions.
-
-
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<i for three different v:~lurs of
?I;</.
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<z<?
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.
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and
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s.
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 =
=
.'
<l/Jblpil/t
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<t>,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 <p :T- + k sin 8 sin (p y + k cos 8 2
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>
+ 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 </JX + sin () sin </J Y + cos ez) 1'1')
[sin
1
- -I(
X'
/3
- iY')
I')
-
1
13
IZ' l ')
1
/3- I( cos e cos </J +
+ (cos
1
- IS
e sin 4>
i sin </J ) .X
- 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 <D ---. i cos lp ) S' - sin
(
. ..
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) +- <Ie' M c ~ 1', I:» xz- 2. AI/
----._-1
.
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, <P2 lerle;/
<P I)
<¢ 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
<OJ
<:>
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 , <!I1d F . (h) Ab sor p tion coeflic icn ts
fur t he five: sa m p le s in (a) . Noh: tn .u th e :lbs or p tic,n C\ ld iic ie n ts ,." . :-'; lI n p k, . /\ , 8 . a n d C ( left
ve r tic a l sca le) are s m a lle r th a n thn :::e: o f s um p le , E ;; oll F us in g tI 1 '~ rig h t sc ule . (Alt e r R d . 22. )
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).
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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.
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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
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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<Fc
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
<l:
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 <n r n, and Iv =
e -n/RIl,_ vs. the carrier concentration n for two values of R. We see that the
intersection point t. = t, determines the transparency density n I f ' at which
the peak gain is zero. This transparency density n tr is reduced when the mass
rate is reduced from R = 5 to R = 1. The peak gain is also plotted in
Fig. 10.2Sb for R = 1 and R = 5. We note that while the transparency
density nil is 'smaller at R = 1, the maximum achievable gain also saturates
at a smaller value when n increases, since gm is proportional to the joint
density of states, which is proportional to the reduced effective mass rn, =
1;1~~lJIj;/(m: -+- mj;) = m;R/(R + O. If FnO:; = m;. we have m , = m:/2. If
mot :..= 5m:', we have m , C~= 5nl'f,'/' 6.
1&
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 = <P, + Qt ' where
P, = - 2a l (l - C l 2 /C ll )e. Q ~ = -bU -I 2C l 21' C l! lt' , and th e in-plane s t ra in E = [a (..-)(lO J/afl ' a Cr) is the latt ic e con st ant of th e unstrained In 1- rG a .( As alloy arid au is that o f th e
s u bs trate. wh ic h is ulso t h e; in - p la ne : ; ~l t i ,x "'() n 'i t<~ l~ t of th e s t rain cd I n j . . G a •. As lattice .
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.)
<U
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. ( <I ) t\ sc he rn utic di.ig r.u» for a cO~I~)l.:d g :.liil -gUlckd se m ico nd uc to r las er array
[ l25 J. (b l Se nu co n d uc tor buried- rid,~l: lu:.er array ll-,(1 J.
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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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)
......
<l.l
~
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, <Inc! tht:' _. J dB rc's pl l,:SC :;i\::;: -; t he :' clB Ir c q ue n cy /',.113' (After Ref. 8 )
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,<p = ~¢'
«.
= -
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..l</>U)
(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<n
""
(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
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3. K. Lau and A. Yariv, "High-frequency current modulation of semiconductor
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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,
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1101-1112 (1982).
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7. W. St rcifer, D. R. Scifres, and R. D. Bu r ah.rm. "Spontaneous emission factor of
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506
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(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
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11. Y. Arakawa and A. Yariv, "Quantum well lasers: gain, spectra, dynamics," IEEE
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1.3 fLm InGaAsP high speed semiconductor lasers," IEEE 1. Quantum Electroll.
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13. R. Olshansky, P. Hill, V. Lanzisera, and W. Powazinik, "Universal relationship
between resonant frequency and damping rate of 1.3 fLm InGaAsP semiconductor
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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
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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
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(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
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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).
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Quantum Electron. QE-18, 259-264 (1982).
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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
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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 </J =
"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<U()
=
=
~ 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<lg~ dilfradlDn. In this case, the
incident k; vector has to come from a particular direction satisfying th:;:
:'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 <Inc! K. Thyagarajan, Optical Electronics, Cambridge University
Press. Cambridge, UK, 1(j80.
I O. M. Abramowitz and 1. A. St cgun, Eds., Handbook of Mathematical Functions with
Formulas, Graphs, and Mathenuuical Tables. Chapter (j, Dover, New York, 1972.
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 <P in (13 .1.2) an d 03.1.3) is the basis
function s used in their ex .ansions. In the envelope function <P for the
electron-hole pair states, 111e fast-varying Bloch periodic p arts ucCr) and
U ,"(r) , which behave like liS) and It , ± ~ > 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 <d'
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<t1i,)fh bd\\t:';i1 the W:1'.,: \";c:l,'r~. i\,,, k il , k. dilli 1(
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
<fie i k np" r
(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 <P1l (0) 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<V R- E ] ( 13.3. 10)
g
\
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<I>(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):
<f ie '
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>
<o _ ~
!"".
"...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<I11(p = 0) 12Ie ' pctlll"ml 2 o ( E x
-
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 !<V '( p = 0)/2 give s a Sommerfeld enh ancement factor
[13-15]. No te th at the sum ove r th e sta tes x becomes th e sum over a
continuum di stribution o f state s k; and k y '
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<PnCO)12 as the density of states. We start with
03.1.25).
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
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7. J. D. Dow and D. Redfield, "Electroabsorption in semiconductors: The excitonic
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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
<T )
=
JOT p( T) d-r =
Tn'
we find
( 14.1.53)
Since the average photocurrent with the photoconductive gain
<N ) injected (primary) electrons per second is
Tn
1p == (I) = <N )q1"
Tn
IT
l
due to
(14.1.54)
1
the spectral density function due to <N ) injected electrons per second is
obtained from (14.1.44), (14.1.53), and (14.1.54)
( 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 <P (number of photons per
second per unit area) of monochromatic radiation at a wavelength A, we can
write the rms photocurrent
(14.2.16)
where A is the photodetector illumination area, <P = PI. l.(hvA), PI. is the
root-mean-square input optical power at A (PI. = P rm s = mPo p t /12), and 7] is
the quantum efficiency of the photodiode including the effects of the internal
quantum efficiency, the reflection, and the absorption depth. The current
responsivity is therefore
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_
2<f>B
(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<p(X)jX dx 'l3 p(x')e-<P(X') ., qe'P( X)jxG(x')e-'P(X')dx' + I n( O) e'P (X)
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 - <peW)' + q It d x' G( x') e - / ,([a,,(x") -Pp(x"»)dx"
1 __ I~v dX'a x')e- /J[a,,(x"I-t3 p ex")]dx"
=
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<O
(b) Charge distribution
,...-+qN A
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<<t»2Izl<t»t>1 ; therefore, a
rotation of the polarization as a function of the polarization angle <t» measured from the TM polarization waveincident at a fixed angle of incidence
set at the Brewster angle 0 B = 73° shows the polarization dependence [95,
96J in Fig. 14.16 a cos 2 <t» , as expected from the theory. At <t» = 90°, the
incident light is polarized along the'TE direction (i.e., the polarization of the
incident radiation is perpendicular to the plane of incidence; therefore, it is
parallel to the quantum-well plank of the QWIP), and no absorption occurs
for this polarization.
Photoconductive Gain in Quantum Wells. For the bound-to-bound state
transition shown in Fig. 14.17a, the photoexcited electrons in the E 2 level
have to get out of the well either by tunneling or by thermionic emission (or
by other phonon or impurity scattering processes). It is therefore desirable to
design the £2 level to be close to the barrier energy such that the absorption
PHOTODE rL CTOR.~
618
1.4r-------~----------------_,
1.2
1.0
WAVEGUIDE
w 0.8
u
z
<t
CD
~ 0.6
(f)
CD
<t
0.4
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 <P is the photon flux density (number of photons p er second per unit
area), and A is the area of the photon illumination area and 7Jw is the net
quantum efficiency of a single well including the effect of th e escape probability from the w ell.
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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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 <P is simply
<P( ~)
1
.
= - - e 1m cp
(A.20)
J2rr
where m is an integer, since <p(~) should be periodic in
A change of variable from e to g,
~
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<Ga1_xAs heterostructures for growth on (OODlnP," 1. Appl. Phys. 62, 2551-2553 (987).
9. G. Ji, D.Huang, U. K. Reddy, T. S. Henderson, R. Houdre, and H. Morko<,S,
"Optical investigation of highly strained InGaAs -GaAs multiple quantum wells,"
J. Appl. Phvs. 62, 3366-3373 (1987).
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<D CI-I/\RACTERJSTIC EQUATION
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) [<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!<Pn( r) > =
J¢~(r) P<Pn( r) d' r
Pnm = <rP,Jpl4>m>
= (
(1.4 )
<P nll/J) <l/J l<Pm)
= 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 <P':n(r) and integrated over the whole space, the above equation
becomes
. 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 <l_ ab
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
<l {
+ 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'<D Inl:
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.+