Basic Semiconductor Physics (3^rd Ed., Softcover reprint of the original 3rd ed. 2017) Graduate Texts in Physics Series

The new edition of this textbook presents a detailed description of basic semiconductor physics. The text covers a wide range of important phenomena in semiconductors, from the simple to the advanced. Four different methods of energy band calculations in the full band region are explained: local empirical pseudopotential, non-local pseudopotential, KP perturbation and tight-binding methods. The effective mass approximation and electron motion in a periodic potential, Boltzmann transport equation and deformation potentials used for analysis of transport properties are discussed.
Further, the book examines experiments and theoretical analyses of cyclotron resonance in detail. Optical and transport properties, magneto-transport, two-dimensional electron gas transport (HEMT and MOSFET) and quantum transport are reviewed, while optical transition, electron-phonon interaction and electron mobility are also addressed.
Energy and electronic structure of a quantum dot (artificial atom) are explained with the help of Slater determinants. The physics of semiconductor lasers is also described, including Einstein coefficients, stimulated emission, spontaneous emission, laser gain, double heterostructures, blue lasers, optical confinement, laser modes, and strained quantum well lasers, offering insights into the physics of various kinds of semiconductor lasers.
In this third edition, energy band calculations in full band zone with spin-orbit interaction are presented, showing all the matrix elements and equipping the reader to prepare computer programs of energy band calculations. The Luttinger Hamiltonian is discussed and used to analyze the valence band structure. Numerical calculations of scattering rate, relaxation time, and mobility are presented for typical semiconductors, which are very helpful for understanding of transport. Energy band structures and effective masses of nitrides such as GaN, InN, AlN and their ternary alloys are discussed because they are very important materials for the blue light emission, and high power devices with and high frequency.
Learning and teaching with this textbook is supported by problems and solutions in the end of the chapters.
The book is written for bachelor and upper undergraduate students of physics and engineering.

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

1.5.1 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 Theo

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

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

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 . . . . . . . . . . . . . . . . . . . . . . .

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

2.5.4 Landau Levels of the Valence Bands . . . . . . . . . . . . . . . . 96

2.5.5 Magneto{optical Absorption . . . . . . . . . . . . . . . . . . . . . . . 101

2.7 Luttinger Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3 Wannier Function and Effective Mass Approximation . . . . . 115

3.1 Wannier Function . . . . . . . . . . . . . . . . . . . . . . .

3.2 Effective-mass Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.4 Impurity Levels in Ge and Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.4.1 Valley{Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 127

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Direct Transition and Absorption Coefficient . . . . . . . . . . . . . . . 145

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 + Δ0 Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.6.3 E2 Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.6.4 Exciton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.7 Piezobirefringence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

4.7.2 Deformation Potential Theory . . . . . . . . . . . . . . . . . . . . . 178

4.7.3 Stress-Induced Change in Energy Band Structure . . . . . 181

5.1 Modulation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.1.1 Electro-optic Effect . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1.2 Franz{Keldysh Effe

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

5.3 Brillouin Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

5.3.1 Scattering Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

5.3.2 Brillouin Scattering Experiments . . . . . . . . . . . . . . . . .

5.3.3 Resonant Brillouin Scattering .

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

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

6.3.6 Polar Optical Phonon Scattering . . . . . . . . . . . . . . . . . . . 283

6.3.7 Inter{Valley Phonon Scattering . . . . . . . . . . . . . . . . . . . . 288

6.3.9 Theoretical Calculation of Deformation Potentials . . .

6.3.10 Electron{Electron Interaction and Plasmon Scattering 296

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

XII Contents

6.4.9 Alloy Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

6.5 Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

6.5.1 Acoustic Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . 326

6.5.3 Polar Optical Phonon Scattering . . . . . . . . . . . . . . . . . . . 330

6.5.4 Piezoelectric Potential Scattering . . . . . . . . . . . . . . . . . . . 332

6.5.6 Ionized Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . . 335

6.5.7 Neutral Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . 336

6.5.8 Plasmon Scattering . . . . . .

6.5.9 Alloy Scatte

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 . . . . . . . . . . . . . . . . . . . . . .

7.3.1 Theory of Shubnikov{de Haas Effect . . . . . . . .

7.3.2 Longitudinal Magnetoresistance Con_guration . . . . . . . 361

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

8.2 Two-Dimensional Electron Gas Systems .

8.2.1 Two-Dimensional Electron Gas<

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 sp3

8.4.6 Second Nearest-Neighbor sp3 Tight Binding

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 . . . . . . . .

8.7 Coulomb Blockade and Single El

8.8 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

8.8.2 Exact Diagonalization Method . . . . . . . . . . . . . . . . . . . . . 499

8.8.3 Hamiltonian for Electrons in a Quantum Dot . . . . . . . . 501

8.8.5 Electronic States in Quantum Dots . . . . . . . . . . . . . . . . . 505

8.8.7 Electronic States in Elliptic and Triangular Quantum

Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

9.1 Einstein Coefficients A and B &

9.5.1 Wave Equations f

9.5.2 Transverse Electric Modes . . . . . . . . . . . . . . . . . . . . . . . . . 546

9.5.4 Effective Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . 548

9.5.5 Con_nement Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

9.6 Stimulated Emission in Quantum Well Structures . . . . . . . . . . 555

9.6.1 Con_nement in Quantum Well . . . . . . . . . . . . . . . . . . . . 557

9.6.3 Reduced Density of States and Gain . . . . . . . . . . . . . . . . 566

9.7

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.1 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

A.2 Cyclic Boundary Condition and Delta Function . . . . . . 59

B Gamma Function . . . . . . .

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 . . . . . . . . . .

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.

Comprehensive, successfully tested textbook now in 3rd edition

Discusses semiconductor lasers in detail

Only textbook with perturbation theory to explain the exact picture of the electron-electron interaction and the physics of the effective mass theory for transport

Includes supplementary material: sn.pub/extras