Basic Semiconductor Physics
3rd Edition
Published on 8. December 2017
Book
Hardback
XXI, 709 pages
978-3-319-66859-8 (ISBN)
Description
When the ?rst edition ofBasic Semiconductor Physics was published in 2001, there were already many books, review papers and scienti?c journals de- ing with various aspects of semiconductor physics. Since many of them were dealing with special aspects of newly observed phenomena or with very f- damental physics, it was very di?cult to understand the advanced physics of semiconductors without the detailed knowledge of semiconductor physics. For this purpose the author published the ?rst edition for the readers who are involved with semiconductor research and development. Basic Semiconductor Physics deals with details of energy band structures, e?ective mass eq- tion and k·p perturbation, and then describes very important phenomena in semiconductors such as optical, transport, magnetoresistance, and quantum phenomena. Some of my friends wrote to me that the textbook is not only basic but advanced, and that the title of the book does not re?ect the c- tents. However, I am still convinced that the title is appropriate, because the advanced physics of semiconductor may be understood with the knowledge of the fundamental physics. In addition new and advanced phenomena - served in semiconductors at an early time are becoming well-known and thus classi?ed in basic physics. After the publication of the ?rst edition, many typographical errors have been pointed out and the corrected version was published in 2006. The p- lisher and my friends persuade me to revise the book adding new chapters, keeping the subject at the appropriate level.
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Series
Edition
3rd ed. 2017
Language
English
Place of publication
Cham
Switzerland
Publishing group
Springer International Publishing
Target group
College/higher education
Edition type
Revised edition
Illustrations
315
315 s/w Abbildungen
XXI, 709 p. 315 illus.
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Height: 23.5 cm
Width: 15.5 cm
Weight
12114 gr
ISBN-13
978-3-319-66859-8 (9783319668598)
DOI
10.1007/978-3-319-66860-4
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Person
Chihiro Hamaguchi graduated from Electrical Engineering (BS) in 1961, and MS Degree in 1963, and Graduate School of Engineering with Ph.D. Degree 1n 1966, all from Osaka University. He served as a research associate of Engineering Science in 1966, and an associate professor of Electronic Engineering in 1967, of Osaka University. He was a visiting research associate of Physics Department of Purdue University (USA), from 1967 to 1969. He was appointed full professor of Electronic Engineering Department of Osaka University in 1985 and retired in 2001, awarded Professor Emeritus. He was employed as an advisory board member of Sharp Corporation form 2001 till 2014, and also a visiting professor of Kochi University of Technology from 2001 till now. His major field of research is hot carrier transport, magnetotransport, modulation spectroscopy of quantum structures, and quantum transport in heterostructures, energy band calculations of superlattices and nitrides, publishing more than 350 papers. He was elected as 1990: Fellow of the American Physical Society (1990), Fellow of the Institute of Electrical and Electronics Engineers (IEEE) (1992), Fellow of the Institute of Physics, U.K .(1999), and Fellow of the Japan Society of Applied Physics (2008). He received the Award of Gold Rays with Neck Ribbon" (Zuiho-Chuju Shou in Japanese) from Japanese Government and the Emeror in 2016. He published several text books in Japanese, such as Introduction to Electronics and Physics of Materials, Semiconductor Devisce Physics, and Electromagnetic Theory (to be published).
He is a Fellow of American Physical Soviety, Instutute of Physics (U.K.), Institute of Electrical and Electronics Engineers (IEEE, U.S.A), and Japan Society of Applied Physics. He received the Award of Gold Rays with Neck Ribbon" (Zuiho-Chuju Shou in Japanese) in 2016.
Content
1 Energy Band Structures of Semiconductors . . . . . . . . . . . . . . . 1 1.1 Free-Electron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Nearly Free Electron Approximation . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Reduced Zone Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Free{Electron Bands (Empty{Lattice Bands) . . . . . . . . . . . . . . 9 1.5.1 First Brillouin Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.2 Reciprocal Lattice Vectors of fcc Crystal . . . . . . . . . . . . 11 1.5.3 Free Electron Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 Pseudopotential Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6.1 Local Pseudopotential Theory . . . . . . . . . . . . . . . . . . . . . 15 1.6.2 Pseudopotential Form Factors . . . . . . . . . . . . . . . . . . . . . . 19 1.6.3 Nonlocal Pseudopotential Theory . . . . . . . . . . . . . . . . . . . 21 1.6.4 Energy Band Calculation by Local Pseudopotential Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.6.5 Spin{Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6.6 Energy Band Calculations by Nonlocal Pseudopotential Method with Spin{orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.7 k _ p Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.7.1 k _ p Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.7.2 Derivation of the k _ p Parameters . . . . . . . . . . . . . . . . . . 40 1.7.3 15{band k _ p Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.7.4 Antisymmetric Potentials for Zinc Blende Crystals . . . 50 1.7.5 Spin{orbit Interaction Hamiltonian . . . . . . . . . . . . . . . . 51 1.7.6 30{band k _ p Method with the Spin{Orbit Interaction 53 1.8 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2 Cyclotron Resonance and Energy Band Structures . . . . . . . 61 2.1 Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.2 Analysis of Valence Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 IX X Contents 2.3 Spin{Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.4 Non-parabolicity of the Conduction Band. . . . . . . . . . . . . . . . . . 81 2.5 Electron Motion in a Magnetic Field and Landau Levels . . . . . 85 2.5.1 Landau Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.5.2 Density of States and Inter Landau Level Transition . . 90 2.5.3 Landau Levels of a Non-parabolic Band . . . . . . . . . . . . . 92 2.5.4 Landau Levels of the Valence Bands . . . . . . . . . . . . . . . . 96 2.5.5 Magneto{optical Absorption . . . . . . . . . . . . . . . . . . . . . . . 101 2.6 Luttinger Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.7 Luttinger Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3 Wannier Function and Effective Mass Approximation . . . . . 115 3.1 Wannier Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.2 Effective-mass Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.3 Shallow Impurity Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.4 Impurity Levels in Ge and Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.4.1 Valley{Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.4.2 Central Cell Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.5 Electron Motion under an External Field . . . . . . . . . . . . . . . . . 131 3.5.1 Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.5.2 Electron Motion under an External Force . . . . . . . . . . . . 135 3.5.3 Electron Motion and Effective Mass . . . . . . . . . . . . . . . . 138 4 Optical Properties 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.1 Reection and Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.2 Direct Transition and Absorption Coefficient . . . . . . . . . . . . . . . 145 4.3 Joint Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.4 Indirect Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.5 Exciton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.5.1 Direct Exciton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.5.2 Indirect Exciton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.6 Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.6.1 E0, E0 + ?0 Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 <4.6.2 E1 and E1 + <?1 Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4.6.3 E2 Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4.6.4 Exciton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.7 Piezobirefringence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 4.7.1 Phenomenological Theory of Piezobirefringence . . . . . . 177 4.7.2 Deformation Potential Theory . . . . . . . . . . . . . . . . . . . . . 178 4.7.3 Stress-Induced Change in Energy Band Structure . . . . . 181 5 Optical Properties 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5.1 Modulation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5.1.1 Electro-optic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5.1.2 Franz{Keldysh Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Contents XI 5.1.3 Modulation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.1.4 Theory of Electroreectance and Third-Derivative Form of Aspnes . . . . . . . . . . . . . . . 198 5.2 Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.2.1 Selection Rule of Raman Scattering . . . . . . . . . . . . . . . . . 209 5.2.2 Quantum Mechanical Theory of Raman Scattering . . . 214 5.2.3 Resonant Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . 219 5.3 Brillouin Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 5.3.1 Scattering Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 5.3.2 Brillouin Scattering Experiments . . . . . . . . . . . . . . . . . . . 228 5.3.3 Resonant Brillouin Scattering . . . . . . . . . . . . . . . . . . . . . . 231 5.4 Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 5.4.1 Phonon Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 5.4.2 Exciton Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 5.5 Free{Carrier Absorption and Plasmon . . . . . . . . . . . . . . . . . . . . . 241 6 Electron{Phonon Interaction and Electron Transport . . . . . 247 6.1 Lattice Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 6.1.1 Acoustic Mode and Optical Mode . . . . . . . . . . . . . . . . . . 247 6.1.2 Harmonic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.2 Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 261 6.2.1 Collision Term and Relaxation Time . . . . . . . . . . . . . . . . 262 6.2.2 Mobility and Electrical Conductivity . . . . . . . . . . . . . . . . 265 6.3 Scattering Probability and Transition Matrix Element . . . . . . . 270 6.3.1 Transition Matrix Element . . . . . . . . . . . . . . . . . . . . . . . . 270 6.3.2 Deformation Potential Scattering (Acoustic Phonon Scattering) . . . . . . . . . . . . . . . . . . . . . . 273 6.3.3 Ionized Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . . 275 6.3.4 Piezoelectric Potential Scattering . . . . . . . . . . . . . . . . . . . 279 6.3.5 Non{polar Optical Phonon Scattering . . . . . . . . . . . . . . . 282 6.3.6 Polar Optical Phonon Scattering . . . . . . . . . . . . . . . . . . . 283 6.3.7 Inter{Valley Phonon Scattering . . . . . . . . . . . . . . . . . . . . 288 6.3.8 Deformation Potential in Degenerate Bands. . . . . . . . . . 289 6.3.9 Theoretical Calculation of Deformation Potentials . . . . 291 6.3.10 Electron{Electron Interaction and Plasmon Scattering 296 6.3.11 Alloy Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 6.4 Scattering Rate and Relaxation Time . . . . . . . . . . . . . . . . . . . . . 304 6.4.1 Acoustic Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . 308 6.4.2 Non{polar Optical Phonon Scattering . . . . . . . . . . . . . . . 313 6.4.3 Polar Optical Phonon Scattering . . . . . . . . . . . . . . . . . . . 315 6.4.4 Piezoelectric Potential Scattering . . . . . . . . . . . . . . . . . . 317 6.4.5 Inter{Valley Phonon Scattering . . . . . . . . . . . . . . . . . . . . 318 6.4.6 Ionized Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . . 321 6.4.7 Neutral Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . 322 6.4.8 Plasmon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 XII Contents 6.4.9 Alloy Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 6.5 Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 6.5.1 Acoustic Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . 326 6.5.2 Non{Polar Optical Phonon Scattering . . . . . . . . . . . . . . . 327 6.5.3 Polar Optical Phonon Scattering . . . . . . . . . . . . . . . . . . . 330 6.5.4 Piezoelectric Potential Scattering . . . . . . . . . . . . . . . . . . . 332 6.5.5 Inter{Valley Phonon Scattering . . . . . . . . . . . . . . . . . . . . 333 6.5.6 Ionized Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . . 335 6.5.7 Neutral Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . 336 6.5.8 Plasmon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 6.5.9 Alloy Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 6.5.10 Electron Mobility in GaN . . . . . . . . . . . . . . . . . . . . . . . . . 339 7 Magnetotransport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 7.1 Phenomenological Theory of the Hall Effect . . . . . . . . . . . . . . . . 343 7.2 Magnetoresistance Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 7.2.1 Theory of Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . 349 7.2.2 General Solutions for a Weak Magnetic Field . . . . . . . . 350 7.2.3 Case of Scalar Effective Mass . . . . . . . . . . . . . . . . . . . . . . 351 7.2.4 Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 7.3 Shubnikov{de Haas Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 7.3.1 Theory of Shubnikov{de Haas Effect . . . . . . . . . . . . . . . . 357 7.3.2 Longitudinal Magnetoresistance Con_guration . . . . . . . 361 7.3.3 Transverse Magnetoresistance Con_guration . . . . . . . . . 363 7.4 Magnetophonon Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 7.4.1 Experiments and Theory of Magnetophonon Resonance 367 7.4.2 Various Types of Magnetophonon Resonance . . . . . . . . . 375 7.4.3 Magnetophonon Resonance under High Electric and High Magnetic Fields . . . . . . . 380 7.4.4 Polaron Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 8 Quantum Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 8.1 Historical Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 8.2 Two-Dimensional Electron Gas Systems . . . . . . . . . . . . . . . . . . . 390 8.2.1 Two-Dimensional Electron Gas< mos="" inversion="" layer="" .="" 390 8.2.2 Quantum Wells and HEMT. . . . . . . . . . . . . . . . . . . . . . . . 400 8.3 Transport Phenomena in a Two-Dimensional Electron Gas. . . 407 8.3.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 8.3.2 Scattering Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 8.3.3 Mobility of a Two-Dimensional Electron Gas . . . . . . . . . 435 8.4 Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 8.4.1 Kronig{Penney Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 8.4.2 Effect of Brillouin Zone Folding . . . . . . . . . . . . . . . . . . . . 444 8.4.3 Tight Binding Approximation . . . . . . . . . . . . . . . . . . . . . . 447 Contents XIII 8.4.4 sp3s_ Tight Binding Approximation . . . . . . . . . . . . . . . . 450 8.4.5 Energy Band Calculations for Superlattices . . . . . . . . . . 451 8.4.6 Second Nearest-Neighbor sp3 Tight Binding Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 8.5 Mesoscopic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 8.5.1 Mesoscopic Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 8.5.2 De_nition of Mesoscopic Region . . . . . . . . . . . . . . . . . . . . 466 8.5.3 Landauer Formula and Buttiker{Landauer Formula . . . 468 8.5.4 Research in the Mesoscopic Region . . . . . . . . . . . . . . . . . 473 8.5.5 Aharonov{Bohm Effect (AB Effect) . . . . . . . . . . . . . . . . . 474 8.5.6 Ballistic Electron Transport . . . . . . . . . . . . . . . . . . . . . . . 475 8.6 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 8.7 Coulomb Blockade and Single Electron Transistor . . . . . . . . . . 489 8.8 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 8.8.1 Addition Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 8.8.2 Exact Diagonalization Method . . . . . . . . . . . . . . . . . . . . . 499 8.8.3 Hamiltonian for Electrons in a Quantum Dot . . . . . . . . 501 8.8.4 Diagonalization of N Electrons Hamiltonian Matrix . . 504 8.8.5 Electronic States in Quantum Dots . . . . . . . . . . . . . . . . . 505 8.8.6 Quantum Dot States in Magnetic Field . . . . . . . . . . . . . 507 8.8.7 Electronic States in Elliptic and Triangular Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 9 Light Emission and Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 9.1 Einstein Coefficients A and B <. . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 9.2 Spontaneous Emission and Stimulated Emission . . . . . . . . . . . 516 9.3 Band Tail Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 9.4 Luminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 9.4.1 Luminescence due to Band to Band Transition . . . . . . . 527 9.4.2 Luminescence due to Excitons . . . . . . . . . . . . . . . . . . . . . 528 9.4.3 Luminescence via Impurities . . . . . . . . . . . . . . . . . . . . . . . 530 9.4.4 Luminescence in GaP and GaAsP via N traps . . . . . . . . 535 9.4.5 Luminescence from GaInNAs . . . . . . . . . . . . . . . . . . . . . . 538 9.4.6 Light Emitting Diodes (LEDs) in Visible Region . . . . . 540 9.5 Heterostructure Optical Waveguide . . . . . . . . . . . . . . . . . . . . . . . 541 9.5.1 Wave Equations for Planar Waveguide . . . . . . . . . . . . . . 542 9.5.2 Transverse Electric Modes . . . . . . . . . . . . . . . . . . . . . . . . . 546 9.5.3 Transverse Magnetic Modes . . . . . . . . . . . . . . . . . . . . . . . . 547 9.5.4 Effective Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . 548 9.5.5 Con_nement Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 9.5.6 Laser Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 9.6 Stimulated Emission in Quantum Well Structures . . . . . . . . . . 555 9.6.1 Con_nement in Quantum Well . . . . . . . . . . . . . . . . . . . . 557 9.6.2 Optical Transition in Quantum Well Structures . . . . . . 561 9.6.3 Reduced Density of States and Gain . . . . . . . . . . . . . . . . 566 XIV Contents 9.6.4 Strain Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 9.7 Wurtzite Semiconductor Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 9.7.1 Energy Band Structure of Wurtzite Crystals . . . . . . . . . 575 9.7.2 Bowing of the Band Gaps and the Effective Masses in the Ternary Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 9.7.3 Valence Band Structure in the Presence of Strain . . . . . 585 9.7.4 Optical Gain of Nitride Wuantum Well Structures . . . . 593 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 A Delta Function and Fourier Transform . . . . . . . . . . . . . . . . . . . . 595 A.1 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 A.2 Cyclic Boundary Condition and Delta Function . . . . . . 597 A.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 B Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 C Uniaxial Stress and Strain Components in Cubic Crystals . . . 603 D Boson Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 E Random Phase Approximation and Lindhard Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . 611 F Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 G Spontaneous and Stimulated Emission Rates . . . . . . . . . . . . . . . 615 H Spin{Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625