Physics of Optoelectronic Devices

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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.
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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.
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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.
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 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.
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.
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 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.
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.
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
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
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
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
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
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.
?'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
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'
[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
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,
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
tion ant! valence-band profiles and the refractive index ~ ru f i l e . This double confinement enhances 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, 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.
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;!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 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.
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.
1.2 (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.
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., 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?* 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,' 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.
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.
2.1.1 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. 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:; 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:
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,,,,:
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):
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
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 / ~
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: 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 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
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.
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
-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
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
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,,
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
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)
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.
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
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):
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,
;\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
. -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
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.
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
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 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
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):
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 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,