Physics of Optoelectronic Devicesphysics of Op toelectronic
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 ~ s in Pure and Applied Optics
T h e Wiley Series in Pure and Applied Optics p~iblislies
outstanding books in the field of optics. T h e nature of these
books may b e basic ("pure" optics) o r practical ("applied"
optics). T h e books are directed towards one o r more of the
following nucliences: researchers in universities. govern-. ment ,
o r industrial laboratories; practitioners o f optics in industry;
o r graduate-level courses in universities. T h e eniphasis is on
the quality of the book anti its impol-tance to the discipline of
optics.
This tc'xt is printed o n acid-free paper.
Copyright @ 1995 by John Wiley & Sons. In(,.
All rights reserved. Published siniultaneously irl C a n l ~ d a
.
Reproduction o r translariun of any part of this work heyond that
prrnlitted by Section 107 o r 108 of the 1976 Unjteii
. . Sti1rt.s Copyright Act without the pernlission of the copyright
owner is unlawful. Requests for pcrmission o r further int
'orn~:~tion should be adclrcssrd to the Permissions Depa r tn~zn t
. Juhn Wilry & Sons, Inc.. 605 Thi rd Ave~luz . New York. NY
10155-0012.
Lihraq of Congress Catuloging-in- Publication D U ~ L I : C h u ~ l
n g , S. L.
Physics of optoe l~at ronic devices / S.L. Chunng.
p. cm. - (Wiley series in pure ancl applied optics)
"Wilzy-Interscience publication." lSHN 0-17 1-10939-8 ( ~ { l k . p
i l p ~ ~ . ) I. Elzctrooptics. 2. Elt.ctl.ooplica1 c\evicr.s. 3.
Si.~!~iic?ncluctori.
I . Title'. I!. Se r~es . C)C673.C4S 19?5 i14-24'70 I 62 1.35
1'(i.15--~lc20
I.'rinteJ i n t he IIniteJ Stares ot't~.1!1*:riit:
Physics of Optoelectronic Devices
This textbook is intended for graduate students and advanced
undergraduate students in electrical engineering. physics, and
materials science. It also pro- vides an overview of the
theoretica1 background for professional researchers in
optoelectronic industries and research organizations. The book
deals with the fundamental principles in semiconductor electronics,
physics, and elec- tromagnetic~, and then systematically presents
practical optoelectronic de- vices, including semiconductor lasers,
optical waveguides, directional couplers, optical modulators, and
photodetectors. Both bulk and quantum- well semico~lductor devices
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 fun- danle~ltal physical principles. Only
through a solid understanding of fun- damental physics are we able
to develop new concepts and design novel devices with superior
performances. The physics ofoptoelectronic devices is a broad field
with interesting applications based on electromagnetics, sen~ ico~
i - d~lctor physics, 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~iconductor optoelectronic 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~ic
devices. 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~ictor physics applied to
optoclectronics, We then disoass t'r~t.p/*ti,r~lg~ltiolz.
yc.rzt*ratiouz, inod~l- Intion, curd rl~~tecriotl ~,J'li,ght. which
z.1 re tllr keys to .ii nclerstanding the p11 ysics behind the
operation cf optoelect.ronic rlevict:~. Fc~r~xanlpl.ct, knowledge
of the ceneration anti propng:;ition of 1 i i ~ I l t is
(:ri!ci;:il for -u~~derstar;ciing how a semi- L.
r. . . concl~iciur laser t,pci-ntrs. 1 11;: t h c o r ~ vf' ga.i:.!
~ ( j z f i i c i e i t l r j f scrujconductor- r . - 1
lvsers shows hotv Iicrl.lt Cc is ii1~1i3jj [ i c ~ ; , ; i i k i
wi1;'t"giii~1? theory shows how light is . -
confincd to the w:~veeuicle in a laser cavi ly. An undel-standing
of the modula- tion of light is useful in designing optical
switches and nlodulators. The absorption coefficient or bulk and
cliianturn-well se~niconcluctors dernon- strates how light is
detected and leads to a discussion on the operating prin- ciples of
photodetectors.
Features
lnlportant topics such as semiconductor heterojunctions and band
struc- ture calculations near the band edges for both bulk and
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 ~ y is discussed and applied
to semicon- ductor 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 tran- sitions in bu lk and quantum-well
semicondtictors. Inlportant se~niconductor lasers such as
double-heterostructure, stripe- geometry gain-guided
sernicond~ictor lasers. quantum-well lasers, dis- tributed feedback
lasers, coupled laser arrays, and s~lrfiice-emittinglasers are
discussed in great detail. High-speed modulation of semiconductor
lasers using both linear and nonlinear gains is investigated
systen:atically. The analytical theory for the laser spectral
linewidth enhancenlerlt factor is derived. New subjects such as
theories on the band structures ofstrained semicon- ductors and
strained quantum-well lasers are investigated. The
electroabsorptions, in bulk (Franz-Keldysh effccts) and quantum-
well sen~iconductors (quantum confined Starkeffects). are discussed
sys- ten~aticalty including exciton effects. Both the bound and
continuum states o f escitons ilsing the hydrogen atom model are d
isc~~ssed . 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~I but is not required, since all of - the essentials a re
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), Chap- .3 . -, - . . . .-
(-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, 10.5- 10.7), Chapter
12, Chapter 14.
The entire book (except for some advanced sections) can also be
used for a two-semester course.
Acknowledgments
After receiving a rigorous training in my Pl1.D. work on
electromag~letics at 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 lun~ber of research results appearing
in the literature, it is difiicult to list all ofthe irnport-tant
contributions in the field. For a textbook. only the f~lndamental
principles are emphasized. I thank those colleagues who granted nle
pern~ission to 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
~naiiuscript. I tha~~kmanyofmyresearch assistants, especially D.
Ahn, C. Y. P chao , 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 and K. C.
Voyles for typ- ing 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~icting research have
beell the stimulus for writing this book; it was an enjoyable
learning experience.
Contents
PART I FUNDAMENTALS -..
2.1 Maxwell's Equations and Boundary Conditions 2.2 Semicondilctor
Electronics Equations 2.3 Generation and Recombination in
Set-niconductors 3.4 Examples and Applications to Optoelectronic
Devices 2.5 Semiconductor p-N and 12-P Heteroj~inutions 2.6
Semiconductor n-N Heter0junf:tion.s and
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 and 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. Theory a j f FI:iechrt!!.;ic Band 5trncfures i a ~
Semiconductors 124
4.1 'The Bloch '-Yhec-;lt-c:ni ;~.!:d t l~c , b . p T~,lsthod
hi-
Simple Bands I 2 4 4.2 Kanc's TvI~dc:! Gir Btir~ii Str~icture: 3r .
p Flettlod . -
).i{i tll [I3:t: c; !.> l l 4 . i ~ 1 3 1 1. fi ! :? iSi,icij
(1) !,.I. 1 29
xii CONTENTS
4.3 Luttinger-Kohn's Model: The k p Method for Degenerate Bands
137
4.4 The Effective Mass 'Tlleory for a S i n ~ l c Band and
Degenerate Bands 141
4.5 Strain Effects on Band Structures 1 44 4.6 Electronic States in
a n Arbitrary One-Dimensiox~al
Potential 157 4.7 Kronig-Penney Model for a Superlattice 166 4.8
Band Stntctures of Sen~iconductor Quantum Wells 175 4.9 Band
Structures of Strained Semiconcluctor
Quantum Wells 185 Problems 1 90 References 195
Chapter 5. Electromagnetics 200
5.1 General Solutions to Maxwell's Equations and Gauge
Transbr~mations -" 200
5.2 Time-Harmonic Fields and Duality Principle 203 5.3 Plane Wave
Reflection From a Layered Medium 205 5.4 Radiation and Far-Field
Pattern 214 Problems 219 References 220
PART I1 PROPAGATION OF LIGHT
Chapter 6. Light Propagation in Various Media
6.1 Plane Wave ~ o l u t i q n s for 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 t r i
c Dielectric Slab Waveguides 7.3 Ray Optics Approach to the
Waveguide Problems 7.4 Rectangular Dielectric Waveg~liclcs 7.5 The
Effective Index Method 7.6 Wiive Guiclance in a Lossy or- Gain
Pvlediurn Problenls Rekrences
I."OI\ITENTS xiii
Chapter 8. kVaveguidc Couplers and Coup1,ed-Mode Theory 253
8.1 bVaveguidc Couplers 8.2 Coupling of Modes in the Time Domain
8.3 Coupled Optical Waveguides 8.4 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
9.1 Optical Transitions Using Fermi's Golden Rule 9.2 Spontaneous
and Stimulated Emissions 9.3 Interband Absorption and Gain 9.4
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 LAaser with
Valence-Band-Mixing Effects Problems References
Chapter 10. Semiconductor Lasers
10.1 Double Hcterojunction Semiconductor Lasers 3 95 10.2
Gain-Guided and Index-Guided Semiconductor Lasers 412 10.3 Quantu
m-Well Lasers 42i 10.4 Strained Quantum-Well Lasers 437 10.5
Co~lpled Laser Arrays 449 10.6 Distributed Feedback Lasers ; 457
10.7 Surface-Emitting Lasers 444 Problems ,: 47 1 References 4'7
3
Chapter I I. Direct Wlodnla ti:,!] of Semicond~e tor Lasers
1 1.1 Rate Equat ic t~s :incl Linear G;rin Analysis
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
G . Light Yn~pagation in 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 sys- tems. 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 gal- lium (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 configbra- tion 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]
3d104s24p3 configuration 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 Ga 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~.rction band. '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:irjn b:in.(:i
3.11~1 I?!, is the band edge of the valerrcc band.
When a light with an, optic;~i entcgv hrj abirve theh;~ndgap E , is
inciderrt on the semicvnductor. c;ptical nbsorptior: is significmt.
Here / I Is tl~e. Planck
Table 1.1 Part of the Periodic Table Containing Group I1 to VI
Elements
-
I
48 Cd 49 In 1 50 ~n 51 Sb 52 Te
[Kr] 46''' [l<r] 4d"' 1 [KT] 4d1(' [Kr] 4d " [k] 4d1?
56 Ba
[Xe] 6s'
5dI06s2
[Ar] = [Ne] 3s'3ph
[Kr] = [Ar] 3d'04s'4p6
[Xe] = [Kr] 4d'05s25p6
figure 1.1. (a) A zinc-blende structure such as those of GaAs and
InP semiconductors. (b) The zinc-blende structure in (a) consists
of two interpenetrating face-centered cubic lattices separated by a
constant vector ( a / 4 ) ( i + 9 + i), where a is the lattice
constant of the semi- conductor.
(a) Bonding diagram -.. (c) Bonding diagram
(b) Band diagram (d) Band diagram
Energy
Empty
Photon
Figure 1.2. (a ) A planar- bonding diagram for a GaAs lattice. Each
bond consists of two valence electrons shared by a gallium and an
arsenic atom. (b! The 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 he 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 tkrent honds
between the Ga and As atoms. ( d ) The e~iergy-b:tnd cli;lgrai:: s
lvuing 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). The 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. The 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. We 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.
The recent pr6gress in modern crystal growth techniques [5] such as
the molecular beam epitaxy (MBE) and the metal-organic chemical
vapor depo- sition (MOCVD), has demonstrated that it is possible to
grow serniconduc- tors of different atomic compositions on top of
another semiconductor substrate Mtith monolayer precision. This
opens up extremely exciting possibil- ities of the so-called
"band-gap engineering." For example, aluminum ar- senide (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. We 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.~:se of 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
ec t i l~n par2llel t o the junction interface
GaAs AlxGa ,-xAs
Figure 1.3. (a) A GaAs/Al,Ga,-,As heterojunction formed with
different band gaps. The band-edge discontinuities in the
conduction band and the valence band a re 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 the heterojunction
surface layer on the GaAs 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 densi- ties, which mearis that a
large amount of current has to be injected before the lasers could
start lasing. With the introduction of the heterojunction semi-
conductor 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 carriers
and photons ~nztkes the ~ t i~~- lu la ied emission process niore
efficient 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~inication systems,
INTRODUCTION
E" - ----.
Figure 1.4. A double-heterojunction sernicon- ductor laser
structure, where the central GaAs
Refractive index profile
region provides both the carrier confinement and optical
confinement because of the conduc- n s
I I
tion ant! valence-band profiles and the refractive index ~ ru f i l
e . This double confinement enhances I
I
stim~~latecl emissiolls and the optical n~oda l gain. -
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
has a 1.424 eV band gap at room temperature. At x - 0.47, the
In,Ga,-,As 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 AB 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 depen- dence, 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 com- monly
used at room temperature:
Most ternary compounds require a quadratic. term. From the above
formu.la, we can calculate the cond~.ictic>n and vall:r!cc:
Garid-edge discontinuities be- tween 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 than
the
(a) F ie ldd (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).
The 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 interest- ing
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, quantum- well 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-two- dimensional 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 poten- tial
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 ( ~ w threshold current quantum-well lasers
[9] has ,
been reported for different m:iterial systems, such as GaAs/AlGaAs,
In- GaAsPI' InP, and InGaAs /' InGr-ASP. .kclva t~ 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- de- vices are based on the
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 smaller than those ( x ) of the outermost A1 ,rGa, -,
As cladding regions. The AI,Ga, -,As Iayers 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 experi- mentally in the 1970s. The idea of using strained
quantum wells [9, 101 by growing senliconductors with different
lattices constants for tunable wave- length 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.
Impor- tant advantages using the strained-layer superlattice or
quantum wells in- clude 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~lce band
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 TM polarized for the transition between
the electron and the light-hole bands and TE polarized for the
transition between the electron and the heavy-hole bands. Many of
these details for valence subband electronic pro~ert ies and
polarization selection rules in quantum-well dcvices are explained
in this book.
1.2 OVERVIEW
This book is divided i ~ ~ t o five parts: I , F;ili:~damentaIs;
11, Propagation; Ill, Generation; IV, hlodulatior,; and V,
r):icoti~l?.oF Light. \Ve start with thc fundame t~tals o 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 spcua)
Optoelectronic Device Characteristics
Maxwell's Equations
(Optical electric and magnetic fields, guided modes and propagarion
constants)
Figure 1.7. Fundamental equations and their applications to
optoelectror~ic device characteris- tics.
Semiconductor Electronics Equations
(Carrier densities, potential. bias voltage, quasi-Fermi levels and
current densities)
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 elec- tronic 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 depen- dent on the static electric bias voltage, which
determines the electron and hole current densities.
' The semiconductor electronic equations governing the electron and
hole concentrations and their corresponding current densities have
to be solved. - 7. 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>ncluctor laser, and the quantum e-fficiency fur
converting cp'iic;ll pol&or to current iil st photodetec- tor,
havc to be investigated 11s;rig tt~esi: sen-!iconductor electronic
equations These fundamental equations xcturil!y are i : ~ ~ ~ p I e
d f.0 eiic11 other, and the
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
per t~~rba t iop theories for various device operation condi-
tions, 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 gov-
erning the electron and hole concentrations and their correspond-
ing 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~aller band-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~lantum Mechanics-(Chapter 3)
What are the eieenvrrlues and eigenfunctions of a rectangular quan-
tum-well potential? These have important applications to semicon-
ductor quantum-well devices.
What are the bound- and continuum-state soll~tions of 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. Thc 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 ' s golderi ru 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!~ation method and
Lijwdin's pcrt~lrbation ~ n e ~ l ~ o c l ' ? 'T'hzsc h;!ve
applications to Kane's medel ;111cl Luttinger-lic~iln's l i l~clcl
io!- *::ilc2~i.-band s t ruc t~~res .
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~iconductor laser
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 c t c ~ ~
Ixsrr st:iic:turz ir-r r e d SpitCc anil ( b ) 11 dist~.il?utecl
lieriit~nck sernicot~ducror- la...:!- .
A directional couplcr modulator
Figure 1.9. A directional coupler modulator wl~ose outptlt light
power Inay be switched by an electric field bias.
How do the propagation constants change in the presence of absorp-
tion 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 feed-
back structure? The 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~ce in the optical
absorption spectrum between a
bulk and a qua'ntum-well semiconductor? What is the difference
between an interband and intersubband transi-
tion in a quantum-well structure'? ~emiconduc to r '~ase r s
(Chapter 10)
What are the operational principles of different types of
semiconduc- tor lasers?
What determines the quantum efficiency and the threshold current of
a diode laser?
f art lV Modulation of Light
Direct Modulation 01 Szn11cond~:ctor Lascrb (Chapter 1 I)
How d o wz clirectly f i~odulatc the I lg i~ t o~ l tpu t from a
semiconductor laser?
Photons 1
Photons from the top or from the 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-well intersubbancl 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 under-
stand 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 ,A , , f o r Si,
GaAs, InAs, lnP, and GaY a t 300 K.
i . Use the [3aj;c'l gap erle;,gil;-s j i : ,A, , i7pf : j l i i1~
K.
(b) Find the optic:ii cjlcri;;: !j.:::Sl.l..:;~~!i!.;j; i-rl !.he
wavelength l.3prn and 1.55j.t m.
1 4 INTRODU! ITION
1.2 (a) Find t 1 1 ~ galliii~n mole i'raction x for In, -,Ga,, As
compound semi- conductor such that its lattice constant equals that
of InP. The Iatticc constants of a few binary compounds are listed
in Appendix K.
(b) Find the allunlinum mole fraclion s for A1,In, -,t As such that
its lattice constant is the same as that of InP.
1.3 Calculate the band edge discontinuities A E , and AE,. for
GaAs/ Al,Ga,-,As heterojunction if x = 0.2 and x = 0.3.
REFERENCES
1. C. Kittel, Introdlrctiorz to Solid Stale Physics, Wiley, New
York, 1976. 2. N. W. Ashcroft arid N. D. Mcrmin, Solici Sftrtc
Physics, Holt, Rillehart & Winston,
Saunders College, Philadelphi;i, 1976. 3. B. G. Streetman, Solicl
State Electroriic Dei~ices, Prentice Hall, Englewood Cliffs,
NJ, 1980. 4. R . F. Pierret. Ser?licond~icto~. F~indtlmelztals,
Vol. 1 in R. F. Pierret and G. W.
Neudeck, Eds., rtlod~ilar Series on Solid State De~iices,
Addison-Wesley, Reading, MA, 1983.
5 . W. T. Tsang, Volume Ed., Lightwaue Corntnliniccrtions
Technology, Vol. 22, Parts A-E, in R. K. Willardson and A. C. Beer,
Eds., Sernicondrictor ariii Setnilrletals, Academic, New York,
1985.
6. H. C. Casey, Jr., and M. B. I'anish, Hetrrostr~ict~ire Lasers,
Parts A and B, Academic, Orlando, FL, 1978.
7. T. Miya, Y. Terunuma, T. Hosaka, and T. Mjyashita, " A n
ultimate low loss single mode fiber at 1.55 pm," Elect~orz. Lett.
15, 106-108 (1979).
8. D . S. Chemla and A. Pinczuk, Guest Ed., Special issues on
Semiconductor Quantum Wells and Superlattices: Physics and
Applications, IEEE J. Qtlczntr~rn - Electron. QE-22 (September
19S6).
9. P. S . Zory, Jr., Ed., Qitnntzirn I.!'ell Lasers, Academic, San
Diego, 1993.
10. T. P. Pearsall, Volumc Ed., S r ra i~ed L n ~ e r
Sziperlcrttices: Physics, Vol. 32, 1990; and Stmiized Layer
S~lperlatt~ces: Miter-ids Scietzce and Technology, Vol. 33, 1991,
in R. K. Willardson and '4. C. Beer, Eds., Senzicondrictor rzrzd
Srmimztals, Academic, New York.
General Semiconductor Optics: arid Electronic Physics
- . I . S. M. Sze, Physics ;?f' Sc1r~rico!.;d!~clr!~- ; ~ L . I I :
I ? . S . Wiley, Ncw York, 1951.
2. S. Wang, F~rr~clilr?~c.nrr~i; (-1:' Serrzicoitt6~~::1c)ii T/~~o
l - ) l ~lrzd Dt.1:ic.e P/~ysics, Prentice I i u l l , Engle~vooc!
C!itTs, N.1. 1989.
3. B. R. Nag, T h e o ~ of Eieorrical Trcrrtspo?.i irl
Scr71icorzdzlctors, Pergamon, Oxford, UK, 1972.
- 7 4. 73. K. Nag, EIe'ctr.oi; I rrrr~j.port i r l Coit y c > ~
u z c f Senticoiltl~~r:t~r~s, Springer, Hcrlin, 19SO.
5. B. K. Ridley, Qz~arrtrlrn Plac~~.ssc,s in Ser~~icc~~clzrcfor.~,
2d ed., Clarendon, Oxford, UK, 1988.
6. N. W. Ashcroft and N. D. Merrnin, Solid Stcrte E'/~y.s i~s ,
Holl, Rinehard & \Vitlsto11, New York, 1976.
7, H. Haken, Iiglzt, Vol. 1, T+'~i~:cs, P/lorotzs, Atorrts,
North-Holland, Amsterdam, 1986.
8. R. Loudon, The Quantum Theory of L,igIqt, 2d ed., Clarendon,
Oxford, UK, 1986. 9. F. Bassani and G. I?. Parravicini,
Electr-orzic States atld Optical Transitiorzs in
solid.^, Pergamon, Oxford, IJK, 1975. 10. G. Bastard, Ware
1?1fcolziz!tics Appliecl to So~tzicortdii~tor Hetc~.ostrz~ct~~res,
Halsted,
New York, 1988. 1 I. 1-1. Maug and S. W. Koct:, Qrln~ltrrrn Tlicury
ofthe Opticizl arzd ELectrorzic Properties
uf' Sertlicondtictors, World Scientific, Singapore, 1990. 12. K.
Hess, Aciucirzced Theory of Scrrricorrd~ictor Deuicds, Yrentice
Hall, Englewood
Cliffs, NJ, 1988. 13. I. M. ~sdi lkovski , B~rncl Str-~ictttre of
Scmicui~d~~crors, Pergamon, Oxford, UK,
1982. 14. K. Seeger, Setniootzd~~(.ror Pllysics, Springer, Berlin,
1982. 15. K. W. Beer, S L L P - L ~ of Serrt i~ui~~i~tctor Physics7
Van Nostrand Reinhold, NEW York,
1990. 16. C. Weisbuch and B. Vinter, Qi~trnlurn Senzicorzdrtctor
Stjzrcticres, Academic, New
York, 1991. 1.7. R. K. Williardson and A. C. Beer, Eds., Opticczl
Properties of I![-V Compo~irzds,
Vol. 3 in Sernico~tdirctors lrrzd Serninletals, Academic, New York,
1967. 18. K. K. Williardson and A. C. Beer, Eds., hfod~ilatiun
TEC/E~L~CILLCS, VOI. 9 in
Sernicoildiictors u~zd Scnzinzetals, Academic, New York, 1972. 19.
M. Cardona, l k fo~ i~~ l~~ t ion \Spcc~roscopy, in Solid State
Phys., Suppl. 11, Academic,
New York, 1969.
General Optical or Quaritum Eiectronics
20. A. Yariv, Qriarrtrtr~r Electr-or&-::, 3d ed., LViley, New
York, 1959.. 21. A. Yariv, Opticczl Elect,-unics, 3d ecl., Holt.
Rinehart Sr Winston. New York, 1985. 211. H. A. Haus, P Y L ~ L : ~
s ~zizd F'ield.~ zil~ OptoeI~~cfrortic'~~, Prentice Hall, Englewood
Cliffs,
NJ, 1983.
2-3. .J. T. Vt:rdeq.cn, La.sr:r Eiectt.:~ir!i:.; f'rentic::~ Elail!
Englewood Cliffs, NJ, 1989. .
--? 23. R. E. A. Saieh and LM. C, l.cic:l:, I''~l~!~/i~t:tc:fi~~~i~
{if Photnrrics, Wiley, New York. 1991..
? ,- 1.3. A. E;. Cihatak ancl K. ?'::yag;t:.:.,jr!!7, .:.',!pllcr:/
.!7!~r,i-;;-~l:i<,s. Cahric lge Univcrsitj- Press, Cambridge,
1-71<. !0::9.
26. W. T. g'sanp, Volume Ed.. Lig17t~vci~'~ C0nzt7zit1zi~atiorn T ~
c h ~ ~ o l o ~ ~ , VOI. 22, Parts A-E in Ser?li~oizllric.to~* C I
~ I C / Serr l in~c~~ls , K. 1;: Willarclson and A. C. Beer, Eds.,
Academic, New York, 1985.
27. R. Dingle, Applicntioru of ~ t l r r l t i q ~ ~ u ~ l t ~ ~ ~
n Wells, Selcctir:e Doping, arid S~iperlrzt- tices, Vol. 24 i n
Semicon~h.lcfor- ctncl Seminletrrls, R. K. Willardson and A. C.
Beer, Eds., Academic, Now York, 1985.
28. J. Wilson and J. F. R. Hawkes, Optoelectronics: Atz
Intr.ocli.iction, Prentice Hall, Englewood Cliffs, NJ, 1983.
29. R. G. Hunsperger, Integrated Opfics: 77reor-y and Technology,
Springer, Berlin, 1984.
30. K. J. Ebeling, Integmltxi Optoelectrotlics, Springer, Berlin,
1993. 31. K. Chnng, Ed., Ha~zdbook of II.Iicrowave arid Optical
Coinponenfs, Vols. 3 and 4,
Wiley, New York, 1991.
Semiconductor Lasers
32. G. -H. R. Thompson, Physics of Setrzicondz~ctor I,u.ser
Der!ices. Wiley, New York, 1980.
33. H. C. Casey, Jr., and M. B. Panish, Hete?~osrri~ct~ir-t.
Lnsers. Parts A and B, Academic, Orlando, FL, 1978.
34. G. P. Agrawal and N. K. Dutta, Long- W~~celettgtlz
Se?nicorrdrictor Lasers, Van Nostrand Reinhold, New York,
1956.
35. P. S. Zory, Jr., Ed., Qlicznt~lm PVefl Lnsers, Academic, San
Diego, 1993. 36. G. A. Evans and J. M. Hammer, Eds., Surfi~ce
Emitting Semicorldlictor Lrisers and
Arrays, Academic, San Diego, 1993. 37. R. K. Williardson and A. C.
Beer, Eds., Lasers, Jilnctiorzs, Tr~~t~spot.t, Vol. 14 in
Senzico~zdrtctors arzd Srrrrirnetals, Academic, New York, 1979. 35.
W. W. Chow, S. LL7. Koch, and M. Sargent 111, Sernicond~rctor-Lnsw
Physics,
Springer, Berlin, 1994.
39. J. K. Butler, Ed., erni icon duct or It~jection Lasers, IEEE
Press, New York, 1980. 40. J. J. Coleman, Ed., Selected Pnpcrs or7
Sernicorzdrictor Diode Lcrsers, SPEE Mile-
stone Series, Vol. MS50. SYIE Optical Engineering Press,
Bellingham, WA, 1992.
Optical Waveguides and Nlodulators
41. H. Nishihara, M. Haruna, and T. Suhara, Optical Irrtegrtr[cd
C'ircriits, McGrslw-Hill. New York. 1989.
43. T. Tamir, I~~re,yr.ated 0-ptics, Springer, Bcr!in, 1979. 43. T.
Ta~nir . Griirlu~I bP'c':r~.e ~?prc~t~ft~crrorlic.r, 2d ecl.,
Springer, Berlin, 1990. 44. D. Marcuse, T/rrory off?ie!rct:-ic
Opiict~l !fi;:~'eg~~icLe.s, A c a c I ~ r n i ~ , NCW York,
1974.
45. A. W. Snyder and J. D. LZ~:)ve, Oi,~itc~;/ !i,'irr.eg~:i:ie
Tlleo~y, Chapman LC Mall, London, 1953.
46. /I-'\. B. 'Buckm:\ri, G:/;llc~i-i-l.321!,,; ?!cv/vriic',~.
S:tiln:!er:; College, Ncw Yurk, 1992.
47. A. Y;\ri*f ar~cl P. E'c.11, il;)fi;.[c! l,ii.l~,cs. i;?
C;.\,.:.l;~!.r., LVilcy. NCLV York, 1984.
45. J. I>. Vincent, F~~rtdcrnrenta1.s qJ' Irzfr(rre~% Ll~fector
r)I>c>r(zii(~n and Testing, Wiley, New York, 1990.
49. R. K. Willardson and A. C. Beer, Eds., Infrared Detcctc~rs,
Vol. 5, in Serrriconduc- tors cind Serniinetnls, Academic, New
York, 1970.
50. R. K. Willardson and A. C. Beer, Eds., Infr~zred Detectors II,
VoI. 12 in - Sernicorzducbors arzci Set-zziilretnls, Academic, New
York, 1977.
51. R. K. Willardson and A. C . Beer, Eds., Mercrlry Cndmi~im
Tell~iride, Vol. 18 i11
Semicond~lctors nnd Seminzetnls, Academic, New York, 1981. 52. A.
Rogalsi, Ed., Selected Papers on Sertzicorzd~~ctor Irzfr~zred
Detectors, SPIE
Milestone Series, Vol. MS66, SPIE Optical Engineering Press,
Bellingham, WA, 1992.
53. M. 0. Manasreh, Ed., Semicondzrctor Q~~nnt~crn Wells ntzd
S~dperlnttices for Lotzg- PVnr!elength Irzfrared Detectors, Artech
House, Boston, MA 1993.
~onl in&r Optics
54. N. Bioembergen, No~zlittear Optics, Addison-Wesley, Redwood
City, CA 1992. (Originally published by W.A. Benjamin, Inc.,
1965).
55. R. W. Boyd, Nonlinear Optics, Academic, San Diego, 1992. 56. A.
C. Newell and J. V. Moloney, Novllinenr Optics, Addison-Wesley,
Redwood
City, CA 1992. 57. Y. R. Shen, The Pi-itrclples of Norzlirlear
Optics, Wiley, New York, 1984. 58. H. M. Gibbs, Oplicnl
Bistnbilify: Controlling Light with Ligt.': Academic, San
Diego, 1985. 59. H. Haug, Ed., Opticnl Nonlirtenl-ifies and
lr~stnbilities ilz Sernicond~~ctors, Academic,
San Diego, 1988.
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 photodetec- tor. 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 elec- tronics equations, and
boundary conditions. We also study the generation and recombination
of carriers in semiconductors. The gen~ra l theory 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 h Hertz 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. He
explained light propagation arci an clectr~nlagnetic wave
phenomenon. Heinrich Hertz demonstrated experimentally the
electromagnetic wave phe- nomenon 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!n thc.ory of i-ight, see
Ref. 3.: for example.
2 2 i3AS!C SEMICC)I\~L>~J<:TORR ELECTL ONlCS
2.1.1 PJlam~ell's Egnatinns in MKS Units
V x E = - - 1 B Faraday's law d t
i)D V x H = J + - Amp?re7s law (2.1.2)
at V - D = p Gauss's law (2.1 -3)
V - B = O Gauss's law (2.1.4)
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). The 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. I t 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 = E E B = p H
In anisotropic media, they may be given by
- D = ~ - E B = P - H
where is the permittivity tensor and is the permeability
tensor:
For electromagnetic fields :it 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,) = O (2.1.10)
fi x (H, - H,) = J, -.\ (2.1.11)
where J, (= limJ,,, a , , J6) is the surface current density (A/m).
Note that the unit normal vector f i 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 , i i ' (B , - Bz) = 0
where p, (= lim ,,,, , , , pa) is the surface charge density (c
/m2) . For an interface across, two dielectric media, where no
surface current or charge
Figure 2.1. Geometr?, f't~r ddrr.i.;Irlg th:? i ) i ; i i r~ ,J i ;
: -y ~ ~ , ! ~ c ~ ! ~ ! c : i i . i ;rrl.oS:; t.hr: intc:rl;,c~ ~
t ' tkvu rnedi:): (a) a rectartgulnr surface: is ct?;;losecl by
tllz i i i i l l l . ) t :r (.' [cl:~shztl line); and (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 slow ( d / d t - 0), we usually have
.. 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 an
electrostatic potential (25:
and
in an isotropic medium. Ecluatio~l (2.1.19) is Poisson's equation.
When the frequency becomes higher, fo r examplc, in a microwave
transistor, one may includc the displacenlcnt c u ~ r en t delis~ty
c ~ ( ~ E ) / d t in the total current density in addition to the
cancl~tcl~or~ c~!rrt'rlt den~i ty J,,,,:
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 on 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 by (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,
Here q = 1.6 x C is the magnitude of a unit charge, p is the hole
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
. 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 V - (J, + J,,) + q- (y - n ) = 0
d l (2.2.8)
Thus, we may separate the above equation into two parts for
electrons and holes:
a V - J,, - q-n = + q K
d r (2.2.9)
where R is the net recombination rate (cmV3 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 = R,, - GI, (2.2.11)
for electrons and
for holes. Thus, we have the current continuity equations for the
carriers:
an 1 - = G, -- R,, + -C J,, d t rl
2.2.3 Carrier Transport Equations
T h e carrier transport equations (assuming Boltzmann
digtrib~rtions for carri- ers) can be written as
where E = -V$ is the electric field, p, and p, a re the electron
and hole nlobiIity, and D,! and U,, are [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 , or nine
scalar components as unltnotvns. We also have (2.2.1), (2.2.13),
(2.2.14]), (2.2.15), and (2.2.16); or nine scalar eclu~tticjns. W e
may also eliminate some af the ~ ~ n k n o w n fui:ctions YLI.L ' [
.~ ;!s J,, ai;(l J,, and reduce the number of
equations to three:
JP 1 -;=: 4 - .Rp -- - V . [-qp,pV+ - ~ D ~ V J I ] (2.2..1.8) 3 t
4
V - ( F V C ~ ) = --q(p - n + C , ) (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 condi- tions 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 ) = r l , exp (2.2.20 j
p ( r ) = rl , exp (2.2.2 1 j
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)
..,
1 he t~vo auxiliary relations are t ' i r r - !:onclegzneratc
sernico.rlr:luctors, for wtlich the hlnswell-Bol tznzann stilt is
ti.t.s ;:re applicable. To Eaki" into ascwnt the effect of
degeneracy, one may rncjilify (2.2.20j and (2.221) simply by using
the
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 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,LLL- with wave
numbers satisfying
2 7 ~ 2 7 23- k =nz-, k , = n P , k , = e- , nz, 1 1 , e -
integers
.t
L x L,, L -
Thus, the number. of available states in a small cubc d k , d k,, d
k - - = d3k in the k space is d3k divided by the a m o ~ l ~ l
t
for each state
d3k c c c ;; /----- I;, A,, k , . ( 2 r ) 3 / ~
Thus
2 d3k 47i-k d k k 2 dk === 1- 2 (2.2.33)
v 4Ti3 Ti- k , k , X-:
If the parabolic band nlodel is used,
where E, is the conduction band edge, we obtain
, k 'dk ,=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 > El, , where E,. is the valence-band
edge. In the nondegenerate limit, it can be sl~own that (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)R EL15CTROKICS
where 1' is Ciamn~a function, we may rewrite (2.2.26) and (2.2.27)
as
and
where
and r (3 /2) = &/ 2 has been used.
Approximate Formula for the Fermi-Dirac Integral. An approximate
ana- lytic 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) VS. 7 and
its asymptotic limits for ,q < - I and 77 x- 1.
The ~naximum errors 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 EF. For a bulk seniicond~~ctor
under thermal equilibrium, the Fermi level EF (= F, = F,) is
determined by the cllarge neutrality condition:
where N;^ is the ionized acceptor concentration and N,f is the
ionized donor concentration. W-e have
where g , is the ground-state dcl.r,!,::?cr:roy factor for acceptor
levels. g . , equals 4 because in Si, Ge, rind Cl;ai';> ;:..ii:h
acceptor I~cvcl car1 accept or:e hole c>E either spin and tile
;lcceptur iavc:! i:; d t ~ t ~ h l y ciegen.=s;lt.e as a res~tlt ol
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]. The horizontal reading of
the intersection point gives the Ferrni energy E,..
In Fig. 2.3b7 we repeat the same procedure for a larger donor
concentra- tion 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; - N i
1.1 f pO = Ni- NL and n o = (2.2.45b)
N; - N,+
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 s to 1-x ~ ~ s e c i i n clc.!crn:~rting L i t .
1 ~ Ferrni level E, using the charge ~l~uti.aliL!/ condi
tivn.
(a)
Energy (eV)
Figure 2.3. (a) A graphir:;it approach to determine the ~ e r r n i
Icvel EF from rhr charge neutrality condition 1.1 = N,T+ p. Both n
and N 2 - t p vs. the horizontal variable EF are plotted and the
intersection of the two curves give the Fcrlni level of the system
N, = 5 X ~ ~ ' ~ / c r n ~ . (b) Same as (a) except that ND = 5 X l
~ ' ' / c r n ~ .
T T ' . r ' r . ' I . . ' . I ' . . ' L ' ' - 4 ~
GaAs (300 K)
An inversion formula is sometimes convenient if we know the carrier
concen (ration r z . Define
1020
10'5
101"
-0.5 0.0 0.5 T .O 1.5 2.0 Energy (eV) ..
The maximum error [1.0, 111 of the above furmilla is 0.006. Further
improve- ment b y two orders of rnagnitctde is possible [12].
A n alternative approach for taking into account the degenerate
semicon- ductor is to replace ni 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
The quasi-Ferrni potentials satisfy
at the ohmic contacts. Here 4,, n,, and y , are the values of the
correspond- ing 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
N, B ?t i - 2.1 x lo6 the ii~trinsic c:iirie; ;oncentration,! we
have the elec- tron concentration
and
Nrl
We can also calculate the band-edge concentratio11 parameters using
HZ: =
0.0665 m, and tnz = 0.50 rrzO and T = 300 K.:
a;, = 2-51 x 10" crn -" r r z , 300
'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~iction band
edgc. If we use 77 = In rl - - 1.458, we obtain El.. EL. = - 37.7
rneV, The error is only 3 meV.
. 1 I E 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 i ~ < t ~ r ~ . A quantui-n
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,,
36 BASIC SEMICONDUCTOR ELECTRONICS
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 nonra diative. 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 momen
tum with another electron or hole. The fundamental mechanisms can
all be described using Fermi's golden rule with the energy and
momentum conser vation 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)
G = G = e n p (2.3.2)
where c is a capture coefficient and e is an emission rate. The net
recombi nation rate is
R=R -G =R -G =cn'P-e n n p p (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 = cno'Po - e (2.3.4)
Ec---'f---
Ev---+--
Ec--+----
Ev--*---
iJ!ure 2.4. The energy band diagrams 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 ne 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 injec- tion, 6n, 6p ( n o + p,), is given by
where S n = bp, since electrons and holes are created in pairs for
interbnnd transitions, and the lifetime
has been used.
.c
r- [here are basicallyL four 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
propor- tional to the density of el'ectrons t i , and the
concentration of the traps
. . ) -1,. . -A-. ., -.., (a) E]ecwo;l captilrc . Z,CC .... 1 311 ,
II?;:,., .,>;: ,(: i : I !id) Hole emission K,,=cnnPI,(1 -ft) c
JI:=-i;2 , .. p,: 4 gf: ,- R : ~ = c , ~ N ~ ~ ~ G,=e p t N (1-ft)
.,
,.\ Figllrc 2.5. 'T'lle erlcrgy l;~;!iliJ tli;lgt,:~~~::, G ;: ~k::
.>!-::.l :!.:i(.:: - i.. . t a ' , , ..I,!--1F:ill '
<e:lc!-ation-recomhinatiiln processes: (:I) crlzl;tron ~ap t~ l
:e , ( f j ) i ! ; ' ~ : ~ i : i ~ ii:. 'l\..i ';i;, t :i i,::;lt'
i:&lpt?irz, arlii (d) hole ernisl;iuil.
N,, rn~lltiplied by the probability that the trap is empty ( 1 - f
; ) , where f, is the occupation probability of the trap,
R,, = c l l nW(1 - f;) (2.3.9)
where c,, is the capture coefficie~lt for the electrons. 2.
Electl-orz en7ission. The generation rate of the electrons due to
this
process is
where e,, is the emission coefficient, and Nf f, - n , is the
density of the traps that are occupied by the electrons.
3. Hole cnptr~re. The 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. The 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 genera- tion-recombination of electrons and holes,
respectively, at thermal equiiib- rium. 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,.
where f o r convenience -;~,e Cjctillc and 0, f r am the t~bove
equations, and
1 , p = = . The r c ! t ~ i l , may be replaced by an effective
concdntr.ation i : , ; , fa!- Ll~ge."ncr:ite ~cmiconductors.
The net recornbination rates for electron and lmles are
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
I by
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):
41) :!ASIC 5EMIC:ONDUCTOR ELECTRONICS
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. The recombination rate is given by
R, = (c,?n + c ,Pp)nN,( 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 (b) Electron emissions
(c) Hole captures (d) HoIe emissions
~ ~ = ( c ; n + c ; p ) p ~ ~ f ~ ~ ~ = ( e ~ n i - e ~ p ) N ~ ( l
-f,)
A t
Figt.rre 2.6. -1'hc energy b:~liil c l i ~ ~ ? r ' : t i ? ? ~ lo:
thz 5;1rlci-t:)-bouncI stntt: A u g e ~ generat ion/ recombin;ltion
processes: ( 2 , ) ;[rctrc,rl c;!pr?lrr, (1)) slt.c:rOn enlission,
(c) hole capture, and (d) hole emission.
3. Hole captur-es with the releasecl energy talcen by an electron
or a hole:
4. Hole emissions with the required energy supplied by an energetic
electron or hole:
G, = ( e i n + 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. el l r rgy h;!rld cl i ;~~iai;:r; i,.i;. tll;:
:>>~ci- io-band t lugcl. grnri-~ttion/rccoml;ination .
.
processes: (3) ~ l e ~ t r ~ n c:ip[ure, I'hj ~ i r : < : : . i
~ i - ~ c'r:iiSric>n, j c i !~..lir c:;~f~turq.:, and i d ) I ~
o i s emission.
,- 7 I his process destroys an electron-hvic pair. The
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
AU G,, = en 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. The
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. The 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,
;\U ' .,ALr = c,, r l ; (2.3.30)
The to t a l net Auger recombir~ation rate is the sum o f the net
rate5 of electrons and holes, s i i l c ~ each procchs creates or
destroys one electron-hole pair, Thcrct'i)~ 2,
2.4 EXAMPLES AND i9P PL.JI-:!;I'iOF!S 'i"3 TIP-t
~.-~~L.Ec'I'RC:NIC: DEW(-:ES . 43
2.3.4 Impart Ionization Generation-Reco~al~ination Process
[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. Microscopi- cally, 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 d c r ;j .~c;K.. ~ i ~ n p f c
e.;iarnplcs to illustrate t.he optical ge neration-recombin;:~tifin
pl-~+:;~r::;;,s;.-: ;ii-~d tl-i;:Ir cfifcctl; cr; photoc!etectors.
More details about ph~todeiec!:{cjri: :.ire ~Si:,ci:ss!;.ct ie
Chapter :L4.
Optical jnjectio~l
homogeneous semiconductor under a uniform optical illumination from
the side.
In the presence of optical injection, the continuity equations
are
a r.1 1 3 - = G,, - R,, + - - J , , ( x ) ;I r . ax-
If a semiconductor is doped p-type with an acceptor concentration
iV,, we know that
nf pO = N 4 and no -- -
%
: 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
The 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~tion process, S ~ ~ I L ' C .
wz 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~l current density is
and the total current density is
v J = J,, + Jp = a.E = u-
e 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 seioicon- dtlctors, ' the electrorl mobility is z-nxcl?
!;irger tha2 tile hi>!e mobility. Since
,L. v c i 3 of tile eiec:crons, we can write the rI i l = ,u,[V/ t
is the :ivex.apt. cli-ift photocurrerlt as
45 BASIC SEMICOP: DlJ(,T3II ELECT'RON[CS
where 7 , = P / L ! , , is the average transit tirn.2 of the
elcctrons across the photoconductor of a length a r~d (C;,,A 1: is
just the total number of 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
illumi- nated 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
qi is unity. The optical energy hv is
where h is in microns. The photon flux (I> is the number of
photons injected per second:
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-'' X 7-81
X ~ O ' ~ A = 1.25 rnA. The average electron trhnsit time is
e 0.1 an T1,? = - = v = 2.56 X l o p 6 s
I., 7 \ - , \
The photocurrent in the photoconductor is
where a photoconductive gain of around 570 occurs in this example.
I
2.4.2 Nonuniform Carrier Generation
If the carrier generation rate is not uniform, carrier diffusion
will be impor- -. tant. 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 I
S/2.
(a) Optical illumination (c)
m O g 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 in to 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.
The 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:
At steady state, d / J t = 0 and
a2 st1
D , , - p a ~ - - == -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. The .. 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 cur ren 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 6n( y )
and .J,,(y) are continuous, we find R i i j ~ d C. The f i n d
expressions for S n ( y ) can be pllt
in the form
Tlle total carrier concentration n( y) = no + an( y) -- art( y);
since the active region is undoped, no 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
com- bined, 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 proper- tics 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~r
e.: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 N D = No+- NA- as the "net7' ionized donor concentration, we
have
-
Depletion Approximation for an Unbiased p-N Junction. Since the
charge density is
and the free carriers y and 17 are depleted in the space charge
region near the junction, we have
where again N,, is the net acceptor concentration in the p side,
and N, is the net donor concentration on the N side.
From Gauss's law, we know that 'T . (EE) = p, which gives
where t,, and E , ~ , are the p~r~ni t t iv i ty in 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-;-' ~ $ E T ' E T T ) . J u N ~ ~ I
O N S
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
5 2 F,\S'C SEMlCOr~DUC'TOri ELECTRONICS
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 EcN,
respectively, before contact (see Fig. 2.10a):
and
BE, = 0.67(EG, - E,,) for the G ~ A S - A , ~ G ~ , . v A s system
(2.5.14)
Before contact -. level
(b) After contact
Figure 2.113. Energy h a n d diugrafll lLjr il IJ-~V l i ~ ' t r '
[ ' ~ j ! ~ l l i t i o i ~ (il) bef(31.e conlact: and ( b ) after
contact.
2.5 ST'M!COI'IDUC'T'OR r - A ' AND i l - :"
HF,~'E~P.OJ~~NCTTCN>;
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)
i -4#(x) p side E,,(x) = - A E , , + q + ( x ) Nside
AE,. -
(d) Electrostatic potential
Figure 2.11. Illustrations of (a) a p-N junc- f V"
tion geometry, (b) the charge distribution, ( c )
t h e e lec~r ic field, and (d) t h e electrostatic po- - - tential
based on the depletion approximation. -x -'XN
=-x P
The conduction band edge E , ( x ) is above E,(.Y) by a n amount
ESP on 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. The final energy
bapd diagram after contact is pIotted in Fig. 2.10(b).
Example A p -GaAs /M-N , , G a , -.i -4s C-v - U.3)
het.erojtinction is formed at thermal equilibrium witho:.lt a n
exterr~al bia:; at room temperature. The 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
AEc=0.67AE,=250.6meV AE,=0 .33AEg=123 .4meV
(b) We calculate the quasi-Fermi levels F, and FN for the bulk
semiconduc- tors for the given N, and N,, separately.
p-GaAs region
(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-GaAs/N-Alo .3Gao . ,As
heterojunction with N,, = 1 x 1 0 I s in the p region and N, = 2 x
lo1' ~ r n - ~ in the N region.
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.:, and
(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,