Solid State and Quantum Theory for Optoelectronics pdf - Web Education
850
0
0
Texte intégral
(2) SOLID STATE AND QUANTUM THEORY FOR OPTOELECTRONICS. Michael A. Parker. Boca Raton London New York. CRC Press is an imprint of the Taylor & Francis Group, an informa business.
(3) CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-0-8493-3750-5 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Parker, Michael A. Solid state and quantum theory for optoelectronics / author, Michael A. Parker. p. cm. “A CRC title.” Includes bibliographical references and index. ISBN 978-0-8493-3750-5 (hardcover : alk. paper) 1. Optoelectronics. 2. Quantum theory. 3. Solid state physics. I. Title. TA1750.P3725 2010 621.381’045--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com. 2009030736.
(4) Contents Preface........................................................................................................................................... xvii Author ............................................................................................................................................ xix. Chapter 1. Introduction to the Solid State .................................................................................... 1 1.1 Brief Preview .................................................................................................... 1 1.2 Introduction to Matter and Bonds .................................................................... 3 1.2.1 Gasses and Liquids.............................................................................. 3 1.2.2 Solids ................................................................................................... 4 1.2.3 Bonding and the Periodic Table.......................................................... 5 1.2.4 Dopant Atoms...................................................................................... 8 1.3 Introduction to Bands and Transitions ............................................................. 9 1.3.1 Intuitive Origin of Bands .................................................................... 9 1.3.2 Indirect Bands and Light- and Heavy-Hole Bands ........................... 11 1.3.3 Introduction to Transitions ................................................................ 13 1.3.4 Introduction to Band-Edge Diagrams ............................................... 14 1.3.5 Bandgap States and Defects .............................................................. 15 1.4 Introduction to the pn Junction ...................................................................... 16 1.4.1 Junction Technology ......................................................................... 17 1.4.2 Band-Edge Diagrams and the pn Junction........................................ 18 1.4.3 Nonequilibrium Statistics .................................................................. 19 1.5 Device Trends................................................................................................. 21 1.5.1 Monolithic Integration of Device Types ........................................... 21 1.5.2 Year 2000 Benchmarks ..................................................................... 21 1.5.3 Small Optical Signals ........................................................................ 22 1.5.4 Fabrication Challenges ...................................................................... 23 1.6 Vacuum Tubes and Transistors ...................................................................... 23 1.6.1 Vacuum Tube .................................................................................... 23 1.6.2 Bipolar Transistor .............................................................................. 24 1.6.3 Field-Effect Transistor....................................................................... 25 1.7 Brief Summary of Some Early Nanometer-Scale Devices ............................ 26 1.7.1 Resonant-Tunnel Device ................................................................... 26 1.7.2 Resonant-Tunneling Transistor ......................................................... 26 1.7.2.1 Single-Electron Transistors................................................ 27 1.7.2.2 Quantum Cellular Automation (QCA) .............................. 27 1.7.2.3 Aharanov–Bohm Effect Device......................................... 27 1.7.2.4 Quantum Interference Devices .......................................... 28 1.7.2.5 Josephson Junction ............................................................ 28 1.8 Review Exercises............................................................................................ 28 References and Further Readings.............................................................................. 29. Chapter 2. Vector and Hilbert Spaces......................................................................................... 31 2.1 Vector and Hilbert Spaces .............................................................................. 31 2.1.1 Motivation for Linear Algebra in Quantum Theory ......................... 31 2.1.2 Definition of Vector Space................................................................ 33 iii.
(5) iv. Contents. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.1.3 Hilbert Space ..................................................................................... 34 2.1.4 Comment on the Length of a Vector for Quantum Theory.............. 36 2.1.5 Linear Isomorphism........................................................................... 37 2.1.6 Antilinear Isomorphism ..................................................................... 37 Dirac Notation and Euclidean Vector Spaces ................................................ 37 2.2.1 Kets, Bras, and Brackets for Euclidean Space.................................. 38 2.2.2 Basis and Completeness for Euclidean Space................................... 39 2.2.3 Closure Relation for the Euclidean Vector Space............................. 40 2.2.4 Euclidean Dual Vector Space............................................................ 41 2.2.5 Inner Product and Norm.................................................................... 44 Introduction to Coordinate and Vector Representation of Functions ............ 45 2.3.1 Initial View of the Coordinate Representation of Functions ............ 46 2.3.2 Coordinate Basis Set ......................................................................... 47 2.3.3 Introduction to the Inner Product for Functions ............................... 49 2.3.4 Representations of Functions ............................................................ 49 Function Space with Discrete Basis Sets ....................................................... 50 2.4.1 Introduction to Hilbert Space ............................................................ 50 2.4.2 Hilbert Space of Functions with Discrete Basis Vectors .................. 51 2.4.3 Closure Relation for Functions with a Discrete Basis ...................... 53 2.4.4 Norms and Inner Products for Function Spaces with Discrete Basis Sets .................................................................... 54 2.4.5 Discussion of Weight Functions ....................................................... 55 2.4.6 Some Miscellaneous Notes on Notation ........................................... 58 Function Spaces with Continuous Basis Sets ................................................ 59 2.5.1 Continuous Basis Set of Functions ................................................... 59 2.5.2 Coordinate Space............................................................................... 61 2.5.3 Representations of the Dirac Delta Using Basis Vectors.................. 64 Graham–Schmidt Orthonormalization Procedure........................................... 65 2.6.1 Simplest Case of Two Vectors.......................................................... 65 2.6.2 More than Two Vectors .................................................................... 66 Fourier Basis Sets ........................................................................................... 66 2.7.1 Fourier Cosine Series ........................................................................ 67 2.7.2 Fourier Sine Series ............................................................................ 68 2.7.3 Fourier Series..................................................................................... 69 2.7.4 Alternate Basis for the Fourier Series ............................................... 71 2.7.5 Fourier Transform.............................................................................. 71 Closure Relations, Kronecker Delta, and Dirac Delta Functions................... 73 2.8.1 Alternate Closure Relations and Representations of the Kronecker Delta Function for Euclidean Space ..................... 74 2.8.2 Cosine Basis Functions ..................................................................... 75 2.8.3 Sine Basis Functions ......................................................................... 77 2.8.4 Fourier Series Basis Functions .......................................................... 77 2.8.5 Some Notes........................................................................................ 78 Introduction to Direct Product Spaces............................................................ 79 2.9.1 Overview of Direct Product Spaces .................................................. 79 2.9.2 Introduction to Dyadic Notation for the Tensor Product of Two Euclidean Vectors................................................................. 82 2.9.3 Direct Product Space from the Fourier Series .................................. 82 2.9.4 Components and Closure Relation for the Direct Product of Functions with Discrete Basis Sets............................................... 84 2.9.5 Notes on the Direct Products of Continuous Basis Sets................... 85.
(6) Contents. v. 2.10 Introduction to Minkowski Space .................................................................. 86 2.10.1 Coordinates and Pseudo-Inner Product ............................................. 86 2.10.2 Pseudo-Orthogonal Vector Notation ................................................. 86 2.10.3 Tensor Notation ................................................................................. 86 2.10.4 Derivatives......................................................................................... 87 2.11 Brief Discussion of Probability and Vector Components .............................. 88 2.11.1 Simple 2-D Space for Starters........................................................... 88 2.11.2 Introduction to Applications of the Probability ................................ 90 2.11.3 Discrete and Continuous Hilbert Spaces........................................... 91 2.11.4 Contrast with Random Vectors ......................................................... 92 2.12 Review Exercises............................................................................................ 92 References and Further Readings.............................................................................. 98. Chapter 3. Operators and Hilbert Space ..................................................................................... 99 3.1 Introduction to Operators and Groups............................................................ 99 3.1.1 Linear Operator ............................................................................... 100 3.1.2 Transformations of the Basis Vectors Determine the Linear Operator ......................................................................... 100 3.1.3 Introduction to Isomorphisms ......................................................... 101 3.1.4 Comments on Groups and Operators .............................................. 101 3.1.5 Permutation Group and a Matrix Representation: An Example..................................................................................... 103 3.2 Matrix Representations ................................................................................. 104 3.2.1 Definition of Matrix for an Operator with Identical Domain and Range Spaces............................................................................ 105 3.2.2 Matrix of an Operator with Distinct Domain and Range Spaces............................................................................ 106 3.2.3 Dirac Notation for Matrices ............................................................ 107 3.2.4 Operating on an Arbitrary Vector ................................................... 109 3.2.5 Matrix Equation............................................................................... 110 3.2.6 Matrices for Function Spaces .......................................................... 113 3.2.7 Introduction to Operator Expectation Values.................................. 114 3.2.8 Matrix Notation for Averages ......................................................... 115 3.3 Common Matrix Operations......................................................................... 116 3.3.1 Composition of Operators ............................................................... 116 3.3.2 Isomorphism between Operators and Matrices ............................... 117 3.3.3 Determinant ..................................................................................... 118 3.3.4 Introduction to the Inverse of an Operator...................................... 120 3.3.5 Trace ................................................................................................ 122 3.3.6 Transpose and Hermitian Conjugate of a Matrix............................ 123 3.4 Operator Space ............................................................................................. 124 3.4.1 Concepts and Section Summary...................................................... 124 3.4.2 Basis Expansion of a Linear Operator ............................................ 126 3.4.3 Introduction to the Inner Product for a Hilbert Space of Operators ..................................................................................... 129 3.4.4 Proof of the Inner Product............................................................... 131 3.4.5 Basis for Matrices............................................................................ 132 3.5 Operators and Matrices in Direct Product Space ......................................... 133 3.5.1 Review of Direct Product Spaces.................................................... 133 3.5.2 Operators ......................................................................................... 134.
(7) vi. Contents. 3.5.3 3.5.4. 3.6. 3.7. 3.8. 3.9 3.10. 3.11. 3.12. 3.13. 3.14. 3.15. Matrices of Direct Product Operators ............................................. 134 Matrix Representation of Basis Vectors for Direct Product Space ................................................................. 137 Commutators and Algebra of Operators ...................................................... 138 3.6.1 Initial Discussion of Operator Algebra ........................................... 139 3.6.2 Introduction to Commutators .......................................................... 140 3.6.3 Some Commutator Theorems.......................................................... 141 Unitary Operators and Similarity Transformations ...................................... 143 3.7.1 Orthogonal Rotation Matrices ......................................................... 143 3.7.2 Unitary Transformations.................................................................. 146 3.7.3 Visualizing Unitary Transformations .............................................. 147 3.7.4 Trace and Determinant .................................................................... 148 3.7.5 Similarity Transformations .............................................................. 148 3.7.6 Equivalent and Reducible Representations of Groups.................... 150 Hermitian Operators and the Eigenvector Equation..................................... 151 3.8.1 Adjoint, Self-Adjoint, and Hermitian Operators ............................. 152 3.8.2 Adjoint and Self-Adjoint Matrices .................................................. 154 Relation between Unitary and Hermitian Operators .................................... 156 3.9.1 Relation between Hermitian and Unitary Operators ....................... 156 Eigenvectors and Eigenvalues for Hermitian Operators .............................. 158 3.10.1 Basic Theorems for Hermitian Operators ....................................... 158 3.10.2 Direct Product Space ....................................................................... 162 Eigenvectors, Eigenvalues, and Diagonal Matrices ..................................... 162 3.11.1 Motivation for Diagonal Matrices................................................... 162 3.11.2 Eigenvectors and Eigenvalues......................................................... 164 3.11.3 Diagonalize a Matrix ....................................................................... 165 3.11.4 Relation between a Diagonal Operator and the Change-of-Basis Operator .................................................. 169 Theorems for Hermitian Operators............................................................... 170 3.12.1 Common Theorems ......................................................................... 171 3.12.2 Bounded Hermitian Operators Have Complete Sets of Eigenvectors................................................................................ 172 3.12.3 Derivation of the Heisenberg Uncertainty Relation........................ 176 Raising–Lowering and Creation–Annihilation Operators ............................ 179 3.13.1 Definition of the Ladder Operators ................................................. 179 3.13.2 Matrix and Basis-Vector Representations of the Raising and Lowering Operators.................................................................. 180 3.13.3 Raising and Lowering Operators for Direct Product Space............ 182 Translation Operators ................................................................................... 183 3.14.1 Exponential Form of the Translation Operator ............................... 183 3.14.2 Translation of the Position Operator ............................................... 184 3.14.3 Translation of the Position-Coordinate Ket .................................... 185 3.14.4 Example Using the Dirac Delta Function ....................................... 185 3.14.5 Relation among Hilbert Space and the 1-D Translation, and Lie Group ................................................................................. 186 3.14.6 Translation Operators in Three Dimensions ................................... 186 Functions in Rotated Coordinates ................................................................ 186 3.15.1 Rotating Functions .......................................................................... 186 3.15.2 Rotation Operator ............................................................................ 188 3.15.3 Rectangular Coordinates for the Generator of Rotations about z......................................................................... 189.
(8) Contents. vii. 3.15.4 Rotation of the Position Operator ................................................... 189 3.15.5 Structure Constants and Lie Groups ............................................... 190 3.15.6 Structure Constants for the Rotation Lie Group ............................. 191 3.16 Dyadic Notation............................................................................................ 192 3.16.1 Notation ........................................................................................... 192 3.16.2 Equivalence between the Dyad and the Matrix .............................. 192 3.17 Review Exercises.......................................................................................... 193 References and Further Reading ............................................................................. 199 Chapter 4. Fundamentals of Classical Mechanics .................................................................... 201 4.1 Constraints and Generalized Coordinates..................................................... 201 4.1.1 Constraints ....................................................................................... 201 4.1.2 Generalized Coordinates.................................................................. 202 4.1.3 Phase Space Coordinates................................................................. 204 4.2 Action, Lagrangian, and Lagrange’s Equation ............................................. 204 4.2.1 Origin of the Lagrangian in Newton’s Equations ........................... 205 4.2.2 Lagrange’s Equation from a Variational Principle.......................... 207 4.3 Hamiltonian .................................................................................................. 210 4.3.1 Hamiltonian from the Lagrangian ................................................... 210 4.3.2 Hamilton’s Canonical Equations ..................................................... 211 4.4 Poisson Brackets........................................................................................... 213 4.4.1 Definition of the Poisson Bracket and Relation to the Commutator........................................................................... 213 4.4.2 Basic Properties for the Poisson Bracket ........................................ 214 4.4.3 Constants of the Motion and Conserved Quantities ....................... 215 4.5 Lagrangian and Normal Coordinates for a Discrete Array of Particles....... 216 4.5.1 Lagrangian and Equations of Motion.............................................. 216 4.5.2 Transformation to Normal Coordinates .......................................... 217 4.5.3 Lagrangian and the Normal Modes................................................. 222 4.6 Classical Field Theory .................................................................................. 224 4.6.1 Lagrangian and Hamiltonian Density ............................................. 225 4.6.2 Lagrange Density for 1-D Wave Motion ........................................ 227 4.7 Lagrangian and the Schrödinger Equation ................................................... 230 4.7.1 Schrödinger Wave Equation............................................................ 230 4.7.2 Hamiltonian Density........................................................................ 231 4.8 Brief Summary of the Structure of Space-Time........................................... 232 4.8.1 Introduction to Space-Time Warping.............................................. 232 4.8.2 Minkowski Space ............................................................................ 233 4.8.3 Lorentz Transformation ................................................................... 236 4.8.4 Some Examples ............................................................................... 238 4.9 Review Exercises.......................................................................................... 239 References and Further Readings............................................................................ 243. Chapter 5. Quantum Mechanics................................................................................................ 245 5.1 Relation between Quantum Mechanics and Linear Algebra ....................................................................................... 245 5.1.1 Observables and Hermitian Operators ............................................ 246 5.1.2 Eigenstates ....................................................................................... 247 5.1.3 Meaning of Superposition of Basis States and the Probability Interpretation.................................................... 249.
(9) viii. Contents. 5.1.4 5.1.5 5.1.6 5.1.7 5.1.8 5.1.9. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. Probability Interpretation................................................................. 250 Averages .......................................................................................... 252 Motion of the Wave Function ......................................................... 254 Collapse of the Wave Function....................................................... 255 Interpretations of the Collapse ........................................................ 257 Noncommuting Operators and the Heisenberg Uncertainty Relation........................................................................ 259 5.1.10 Complete Sets of Observables......................................................... 262 Fundamental Operators and Procedures for Quantum Mechanics............... 263 5.2.1 Summary of Elementary Facts ........................................................ 263 5.2.2 Momentum Operator ....................................................................... 264 5.2.3 Hamiltonian Operator and the Schrödinger Wave Equation ................................................................................ 264 5.2.4 Introduction to Commutation Relations and Heisenberg Uncertainty Relations ...................................................................... 266 5.2.5 Derivation of the Heisenberg Uncertainty Relation........................ 267 5.2.6 Program ........................................................................................... 269 Examples for Schrödinger’s Wave Equation................................................ 271 5.3.1 Discussion of Quantum Wells......................................................... 272 5.3.2 Solutions to Schrödinger’s Equation for the Infinitely Deep Well........................................................................................ 273 5.3.3 Finitely Deep Square Well .............................................................. 279 Harmonic Oscillator...................................................................................... 285 5.4.1 Introduction to Classical and Quantum Harmonic Oscillators....................................................................... 285 5.4.2 Hamiltonian for the Quantum Harmonic Oscillator........................ 288 5.4.3 Introduction to the Ladder Operators for the Harmonic Oscillator............................................................. 288 5.4.4 Ladder Operators in the Hamiltonian.............................................. 290 5.4.5 Properties of the Raising and Lowering Operators......................... 292 5.4.6 Energy Eigenvalues ......................................................................... 294 5.4.7 Energy Eigenfunctions .................................................................... 294 Introduction to Angular Momentum ............................................................ 296 5.5.1 Classical Definition of Angular Momentum ................................... 296 5.5.2 Origin of Angular Momentum in Quantum Mechanics.................. 297 5.5.3 Angular Momentum Operators ....................................................... 298 5.5.4 Pictures for Angular Momentum in Quantum Mechanics .............. 299 5.5.5 Rotational Symmetry and Conservation of Angular Momentum.................................................................... 301 5.5.6 Eigenvalues and Eigenvectors......................................................... 303 5.5.7 Eigenvectors as Spherical Harmonics ............................................. 305 Introduction to Spin and Spinors.................................................................. 309 5.6.1 Basic Idea of Spin ........................................................................... 309 5.6.2 Link between Physical Space and Hilbert Space............................ 312 5.6.3 Pauli Spin Matrices ......................................................................... 315 5.6.4 Rotations.......................................................................................... 317 5.6.5 Direct Product Space for a Single Electron .................................... 318 5.6.6 Spin Hamiltonian............................................................................. 319 Angular Momentum for Multiple Systems .................................................. 323 5.7.1 Adding Angular Momentum ........................................................... 323 5.7.2 Clebsch–Gordon Coefficients.......................................................... 326.
(10) Contents. ix. 5.8. 5.9. 5.10. 5.11. 5.12. 5.13. 5.14. Quantum Mechanical Representations ......................................................... 330 5.8.1 Discussion of the Schrödinger, Heisenberg, and Interaction Representations ...................................................... 331 5.8.2 Schrödinger Representation............................................................. 333 5.8.3 Rate of Change of the Average of an Operator in the Schrödinger Picture ............................................................... 334 5.8.4 Ehrenfest’s Theorem for the Schrödinger Representation .............. 335 5.8.5 Heisenberg Representation .............................................................. 337 5.8.6 Heisenberg Equation ....................................................................... 338 5.8.7 Newton’s Second Law from the Heisenberg Representation.......... 339 5.8.8 Interaction Representation ............................................................... 340 Time-Independent Perturbation Theory........................................................ 341 5.9.1 Initial Discussion of Perturbations .................................................. 341 5.9.2 Nondegenerate Perturbation Theory................................................ 342 5.9.3 Unitary Operator for Time-Independent Perturbation Theory......................................................................... 349 Time-Dependent Perturbation Theory .......................................................... 352 5.10.1 Physical Concept ............................................................................. 353 5.10.2 Time-Dependent Perturbation Theory Formalism in the Schrödinger Picture ............................................................... 355 5.10.3 Example for Further Thought and Questions.................................. 359 5.10.4 Time-Dependent Perturbation Theory in the Interaction Representation ................................................................................. 362 5.10.5 Evolution Operator in the Interaction Representation ................................................................................. 364 Introduction to Optical Transitions .............................................................. 365 5.11.1 EM Interaction Potential.................................................................. 365 5.11.2 Integral for the Probability Amplitude ............................................ 367 5.11.3 Rotating Wave Approximation ....................................................... 369 5.11.4 Absorption ....................................................................................... 370 5.11.5 Emission .......................................................................................... 371 5.11.6 Discussion of the Results ................................................................ 372 Fermi’s Golden Rule..................................................................................... 373 5.12.1 Introductory Concepts on Probability ............................................. 373 5.12.2 Definition of the Density of States.................................................. 374 5.12.3 Equations for Fermi’s Golden Rule ................................................ 377 Density Operator........................................................................................... 382 5.13.1 Introduction to the Density Operator .............................................. 382 5.13.2 Density Operator and the Basis Expansion..................................... 386 5.13.3 Ensemble and Quantum Mechanical Averages............................... 390 5.13.4 Loss of Coherence........................................................................... 394 5.13.5 Some Properties............................................................................... 396 Introduction to Multiparticle Systems .......................................................... 397 5.14.1 Introduction ..................................................................................... 397 5.14.2 Permutation Operator ...................................................................... 399 5.14.3 Simultaneous Eigenvectors of the Hamiltonian and the Interchange Operator .......................................................... 401 5.14.4 Introduction to Fock States ............................................................. 403 5.14.5 Origin of Fock States ...................................................................... 404 5.14.5.1 Bosons.............................................................................. 406 5.14.5.2 Fermions .......................................................................... 408.
(11) x. Contents. 5.15 Introduction to Second Quantization............................................................ 408 5.15.1 Field Commutators .......................................................................... 409 5.15.2 Creation and Annihilation Operators .............................................. 410 5.15.3 Introduction to Fock States ............................................................. 412 5.15.4 Interpretation of the Amplitude and Field Operators...................... 414 5.15.5 Fermion–Boson Occupation and Interchange Symmetry ............... 415 5.15.6 Second Quantized Operators ........................................................... 416 5.15.7 Operator Dynamics.......................................................................... 418 5.15.8 Origin of Boson Creation and Annihilation Operators ................... 418 5.16 Propagator..................................................................................................... 422 5.16.1 Idea of the Green Function ............................................................. 422 5.16.2 Propagator for a Conservative System ............................................ 423 5.16.3 Alternate Formulation...................................................................... 424 5.16.4 Propagator and the Path Integral ..................................................... 425 5.16.5 Free-Particle Propagator .................................................................. 426 5.17 Feynman Path Integral.................................................................................. 428 5.17.1 Derivation of the Feynman Path Integral........................................ 428 5.17.2 Classical Limit................................................................................. 430 5.17.3 Schrödinger Equation from the Propagator..................................... 431 5.18 Introduction to Quantum Computing ........................................................... 432 5.18.1 Turing Machines.............................................................................. 432 5.18.2 Block Diagrams for the Quantum Computer .................................. 434 5.18.3 Memory Register with Multiple Spins............................................ 435 5.18.4 Feynman Computer for Negation without a Program Counter .......................................................................... 436 5.18.5 Example Physical Realizations of Quantum Computers ................ 439 5.19 Introduction to Quantum Teleportation........................................................ 440 5.19.1 Local versus Nonlocal ..................................................................... 440 5.19.2 EPR Paradox.................................................................................... 441 5.19.3 Bell’s Theorem ................................................................................ 442 5.19.4 Quantum Teleportation.................................................................... 443 5.20 Review Exercises.......................................................................................... 445 References and Further Reading ............................................................................. 458 Chapter 6. Solid-State: Structure and Phonons......................................................................... 461 6.1 Origin of Crystals ......................................................................................... 461 6.1.1 Orbitals and Spherical Harmonics................................................... 461 6.1.2 Hybrid Orbital ................................................................................. 463 6.2 Crystal, Lattice, Atomic Basis, and Miller Notation.................................... 464 6.2.1 Lattice .............................................................................................. 464 6.2.2 Translation Operator........................................................................ 465 6.2.3 Atomic Basis ................................................................................... 467 6.2.4 Unit Cells......................................................................................... 467 6.2.5 Miller Indices................................................................................... 468 6.3 Special Unit Cells ......................................................................................... 469 6.3.1 Body-Centered Cubic Lattice .......................................................... 469 6.3.2 Face-Centered Cubic Lattice ........................................................... 470 6.3.3 Wigner–Seitz Primitive Cell............................................................ 470 6.3.4 Diamond and Zinc Blende Lattice .................................................. 471 6.3.5 Tetrahedral Bonding and the Diamond Structure ........................... 472.
(12) Contents. xi. 6.4. Reciprocal Lattice ......................................................................................... 472 6.4.1 Primitive Reciprocal Lattice Vectors .............................................. 473 6.4.2 Discussion of Reciprocal Lattice Vector in the Fourier Series ........................................................................ 474 6.4.3 Fourier Series and General Lattice Translations ............................. 475 6.4.4 Application to X-Ray Diffraction.................................................... 476 6.4.5 Comment on Band Diagrams and Dispersion Curves .................... 478 6.5 Comments on Crystal Symmetries ............................................................... 479 6.5.1 Space and Point Groups .................................................................. 479 6.5.2 Rotations.......................................................................................... 481 6.5.3 Defects ............................................................................................. 484 6.5.4 Introduction to Symmetries in Quantum Mechanics ...................... 484 6.6 Phonon Dispersion Curves for Monatomic Crystal ..................................... 486 6.6.1 Introduction to Normal Modes for Monatomic Linear Crystal .................................................................................. 487 6.6.2 Equations of Motion........................................................................ 491 6.6.3 Phonon Group Velocity for Monatomic Crystal............................. 494 6.6.4 Three-Dimensional Monatomic Crystals......................................... 496 6.6.5 Longitudinal Vibration of a Rod and Young’s Modulus................ 496 6.7 Classical Phonons in Diatomic Linear Crystal............................................. 498 6.7.1 The Dispersion Curves .................................................................... 498 6.7.2 Approximation for Small Wave Vector .......................................... 500 6.7.3 Discussion........................................................................................ 500 6.8 Phonons and Modes ..................................................................................... 502 6.8.1 Modes in Monatomic 1-D Finite Crystal with 1-D Motion and Fixed-Endpoint Boundary Conditions....................................................................... 502 6.8.2 Periodic Boundary Conditions ........................................................ 505 6.8.3 Modes for 2-D and 3-D Waves on Linear Monatomic Array ........ 507 6.8.4 Modes for the 2-D and 3-D Crystal ................................................ 508 6.8.5 Amplitude and Phonons .................................................................. 509 6.9 The Phonon Density of States ...................................................................... 510 6.9.1 Introductory Discussion................................................................... 510 6.9.2 The Density of States in ~ k-Space .................................................... 512 6.9.3 Density of States for 2-D Crystal Near k ¼ 0 for the Acoustic Branch .................................................................. 514 6.9.4 Summary of Technique ................................................................... 515 6.9.5 3-D Crystal in Long-Wavelength Limit .......................................... 516 6.10 Comments on Phonon Crystal Momentum .................................................. 517 6.10.1 Anticipations for Momentum .......................................................... 517 6.10.2 Conservation of Momentum in Crystals ......................................... 518 6.11 The Phonon Bose–Einstein Probability Distribution ................................... 519 6.11.1 Discussion of Reservoirs and Equilibrium...................................... 519 6.11.2 Equilibrium Requires Equal Temperatures ..................................... 521 6.11.3 Discussion of Boltzmann Factor ..................................................... 522 6.11.4 Bose–Einstein Probability Distribution for Phonons ...................... 523 6.11.5 Statistical Moments for Phonon Bose–Einstein Distribution.......... 524 6.12 Introduction to Specific Heat........................................................................ 526 6.12.1 Discussion of Specific Heat ............................................................ 526 6.12.2 Einstein Model for Specific Heat .................................................... 528 6.12.3 Debye Model for Specific Heat....................................................... 528.
(13) xii. Contents. 6.13 Quantum Mechanical Development of Phonon Fields ................................ 530 6.13.1 Basis States for Fourier Series with Periodic Boundary Conditions....................................................................... 531 6.13.2 Lagrangian for Line of Atoms ........................................................ 532 6.13.3 Classical Hamiltonian...................................................................... 535 6.13.4 Introduction to Quantizing Phonon Field and Hamiltonian .............................................................................. 536 6.13.5 Introduction to Phonon Fock States ................................................ 538 6.14 Phonons and Continuous Media................................................................... 539 6.14.1 Wave Equation and Speed .............................................................. 540 6.14.2 Hamiltonian for One-Dimensional Wave Motion........................... 542 Review Exercises .................................................................................................... 543 References and Further Readings............................................................................ 548 Chapter 7. Solid-State: Conduction, States, and Bands............................................................ 551 7.1 Equation of Continuity ................................................................................. 551 7.1.1 Classical DC Conduction ................................................................ 551 7.1.2 Collisions and Drift Mobility .......................................................... 553 7.1.3 Classical Equation of Continuity..................................................... 555 7.1.4 Equation of Continuity for Quantum Particles ............................... 557 7.2 Scattering Matrices ....................................................................................... 560 7.2.1 Introduction to Scattering Theory ................................................... 560 7.2.2 Amplitudes ...................................................................................... 562 7.2.3 Reflectivity and Transmissivity ....................................................... 563 7.2.4 Modifications for Heterostructure ................................................... 567 7.2.5 Reflectance and Transmittance........................................................ 568 7.2.6 Current-Density Amplitudes............................................................ 569 7.3 The Transfer Matrix...................................................................................... 570 7.3.1 Simple Interface............................................................................... 572 7.3.2 Simple Electronic Waveguide ......................................................... 573 7.3.3 Transfer Matrix for Electron-Resonant Device............................... 574 7.3.4 Resonance Conditions for Electron Resonance Device .................. 575 7.3.5 Quantum Tunneling......................................................................... 579 7.3.6 Tunneling and Electrical Contacts .................................................. 580 7.4 Introduction to Free and Nearly Free Quantum Models .............................. 581 7.4.1 Potential in Cubic Monatomic Crystal............................................ 582 7.4.2 Free Electron Model........................................................................ 582 7.4.3 Nearly Free Electron Model ............................................................ 584 7.4.4 Bragg Diffraction and Group Velocity ........................................... 587 7.4.5 Brief Discussion of Electron Density and Bandgaps...................... 588 7.5 Bloch Function ............................................................................................. 589 7.5.1 Introduction to Bloch Wave Function............................................. 589 7.5.2 Proof of Bloch Wave Function ....................................................... 592 7.5.3 Orthonormality Relation for Bloch Wave Functions ...................... 594 7.6 Introduction to Effective Mass and Band Current ....................................... 596 7.6.1 Mass, Momentum, and Newton’s Second Law .............................. 596 7.6.2 Electron and Hole Current .............................................................. 599 7.7 3-D Band Diagrams and Tensor Effective Mass ......................................... 602 7.7.1 E–k Diagrams for 3-D Crystals ....................................................... 602 7.7.2 Effective Mass for Three-Dimensional Band Structure .................. 604 7.7.3 Introduction to Band-Edge Diagrams ............................................. 609.
(14) Contents. xiii. 7.8. 7.9. 7.10. 7.11. 7.12. 7.13. 7.14. The Kronig–Penney Model for Nearly Free Electrons ................................ 611 7.8.1 Model............................................................................................... 611 7.8.2 Bands ............................................................................................... 614 7.8.3 Bandwidth and Periodic Potential ................................................... 616 Tight Binding Approximation ...................................................................... 617 7.9.1 Introduction ..................................................................................... 617 7.9.2 Bloch Wave Functions .................................................................... 619 7.9.3 Dispersion Relation and Bands ....................................................... 620 Introduction to Effective Mass Equation...................................................... 623 7.10.1 Thesis............................................................................................... 623 7.10.2 Discussion of the Single-Band Effective-Mass Equation ................................................................. 625 7.10.3 Envelope Approximation................................................................. 628 7.10.4 Diagonal Matrix Elements of VE ..................................................... 629 7.10.5 Summary.......................................................................................... 630 Introduction to ~ k ~ p Band Theory ................................................................ 632 7.11.1 Brief Reminder on Bloch Wave Function ...................................... 632 7.11.2 ~ k ~ p Equation for Periodic Bloch Function..................................... 633 7.11.3 Nondegenerate Bands...................................................................... 634 7.11.4 ~ k ~ p Theory for Two Nondegenerate Bands ................................... 637 Introduction to ~ k ~ p Theory for Degenerate Bands...................................... 638 7.12.1 Summary of Concepts and Procedure ............................................. 638 7.12.2 Hamiltonian for Kane’s Model........................................................ 640 7.12.3 Eigenequation for Periodic Bloch States......................................... 641 7.12.4 Initial Basis Set................................................................................ 642 7.12.5 Matrix of Hamiltonian..................................................................... 643 7.12.6 Eigenvalues...................................................................................... 646 7.12.7 Effective Mass ................................................................................. 647 7.12.8 Wave Functions............................................................................... 648 Introduction to Density of States.................................................................. 649 7.13.1 Introduction to Localized and Extended States............................... 649 7.13.2 Definition of Density of States........................................................ 650 7.13.3 Relation between Density of Extended States and Boundary Conditions................................................................ 653 7.13.4 Fixed-Endpoint Boundary Conditions............................................. 654 7.13.5 Periodic Boundary Condition.......................................................... 655 7.13.6 Density of k-States........................................................................... 657 7.13.7 Electron Density of Energy States for Two-Dimensional Crystal.......................................................... 659 7.13.8 Electron Density of Energy States for Three-Dimensional Crystal........................................................ 661 7.13.9 General Relation between k and E Mode Density .......................... 662 7.13.10 Tensor Effective Mass and Density of States ................................. 663 7.13.11 Overlapping Bands .......................................................................... 665 7.13.12 Density of States from Periodic and Fixed-Endpoint Boundary Conditions....................................................................... 667 7.13.13 Changing Summations to Integrals ................................................. 668 7.13.14 Comment on Probability ................................................................. 669 Infinitely Deep Quantum Well in a Semiconductor..................................... 671 7.14.1 Envelope Function Approximation for Infinitely Deep Well........................................................................................ 672.
(15) xiv. Contents. 7.14.2 Solutions for Infinitely Deep Quantum Well in 3-D Crystal .................................................................................. 673 7.14.3 Introduction to the Density of States .............................................. 676 7.15 Density of States for Reduced Dimensional Structures ............................... 677 7.15.1 Envelope Function Approximation ................................................. 678 7.15.2 Density of Energy States for Quantum Well .................................. 680 7.15.3 Density of Energy States for Quantum Wire .................................. 685 7.16 Review Exercises.......................................................................................... 689 References and Further Readings............................................................................ 694 Chapter 8. Statistical Mechanics ............................................................................................... 695 8.1 Introduction to Reservoirs ............................................................................ 695 8.1.1 Definition of Reservoir.................................................................... 696 8.1.2 Example of the Fluctuation-Dissipation Theorem .......................... 697 8.1.3 Reservoirs for Optical Emitter ........................................................ 698 8.1.4 Comment ......................................................................................... 698 8.2 Statistical Ensembles and Introduction to Statistical Mechanics ................. 699 8.2.1 Microcanonical Ensemble, Entropy, and States.............................. 699 8.2.2 Canonical Ensemble ........................................................................ 702 8.2.3 Grand Canonical Ensemble ............................................................. 704 8.3 The Boltzmann Distribution ......................................................................... 704 8.3.1 Preliminary Discussion of States and Probability ........................... 704 8.3.2 Derivation of Boltzmann Distribution Using a Thermal Reservoir ........................................................................ 707 8.3.3 Derivation of Boltzmann Distribution Using an Ensemble.......................................................................... 708 8.3.4 Counting Degenerate States ............................................................ 711 8.3.5 Boltzmann Distribution for Distinguishable Boson-Like Particles........................................................................ 712 8.3.6 Independent, Distinguishable Subsystems ...................................... 717 8.4 Introduction to Fermi–Dirac Distribution..................................................... 718 8.4.1 Fermi–Dirac Distribution................................................................. 719 8.4.2 Density of Carriers .......................................................................... 720 8.4.3 Comments........................................................................................ 722 8.5 Derivation of Fermi–Dirac Distribution ....................................................... 722 8.5.1 Pauli Exclusion Principle ................................................................ 722 8.5.2 Brief Review of Maxwell–Boltzmann Distribution ........................ 724 8.5.3 Fermi–Dirac and Bose–Einstein Distributions ................................ 725 8.6 Effective Density of States, Doping, and Mass Action ............................... 729 8.6.1 Carrier Concentrations..................................................................... 730 8.6.2 Law of Mass Action ........................................................................ 732 8.6.3 Electric Fields .................................................................................. 732 8.6.4 Some Comments.............................................................................. 734 8.7 Dopant Ionization Statistics.......................................................................... 734 8.7.1 Dopant Fermi Function ................................................................... 734 8.7.2 Derivation ........................................................................................ 735.
(16) Contents. xv. 8.8. pn Junction at Equilibrium ........................................................................... 736 8.8.1 Introductory Concepts ..................................................................... 736 8.8.2 Quick Calculation of Built-in Voltage of pn Junction.................... 739 8.8.3 Junction Fields................................................................................. 741 8.9 Review Exercises.......................................................................................... 743 References and Further Readings............................................................................ 745 Appendix A. Growth and Fabrication Methods......................................................................... 747. Appendix B. Dirac Delta Function ............................................................................................ 763. Appendix C. Fourier Transform from the Fourier Series .......................................................... 775. Appendix D. Brief Review of Probability ................................................................................. 779. Appendix E. Review of Integrating Factors .............................................................................. 787. Appendix F. Group Velocity ..................................................................................................... 789. Appendix G. Note on Combinatorials ....................................................................................... 797. Appendix H. Lagrange Multipliers ............................................................................................ 799. Appendix I. Comments on System Return to Equilibrium ...................................................... 805. Appendix J. Bose–Einstein Distribution................................................................................... 809. Appendix K. Density Operator and the Boltzmann Distribution .............................................. 811. Appendix L. Coordinate Representations of Schrödinger Wave Equation ............................... 813. Index............................................................................................................................................. 815.
(17)
(18) Preface Commercialization has brought rapid change to technology using well-established physical principles such as infrastructure. Separating the physical principles from their device applications leads to a convenient division in a book such as this one since physical principles, concepts, and mathematical theory require only moderate revision over many years whereas the devices and processes inherent to new technology require more rapid and extensive change. However, the reader should not adopt the position that meaningful experimental work cannot be performed without first exhaustively modeling a new device. In fact, either appropriate models or the relevant parameters for existing models might not be available, and therefore the researcher would need to be guided by ‘‘informed intuition’’ gleaned from formal courses and experiment in the laboratory. Optoelectronics and photonics implement and apply various forms of the ‘‘matter–light’’ interaction. This book primarily introduces the solid-state and quantum theory for ‘‘matter’’ but postpones a discussion of ‘‘light’’ and its interaction with matter to the companion volume Physics of Optoelectronics. The present book covers in some detail many of the transitional topics from the intermediate=elementary to advanced levels. Chapter 1 structures the general conceptual framework for the book regarding bonding, bands and devices. However, the concepts of some topical areas will be accessible to the reader only after digesting later chapters. Chapters 2 and 3 cover the mathematics of Hilbert spaces with the philosophy of providing conceptual pictures and an operational basis for computation without overburdening the reader with the ‘‘definition– theorem–proof’’ format often expected in mathematics texts. These mathematical foundations focus on the abstract form of the linear algebra for vectors and operators, and supply the ‘‘pictures’’ that are often lacking in studies of the quantum theory that would otherwise make the subject more intuitive. A picture does not always accurately represent the mathematics of a concept but does help in conveying the meaning or ‘‘way of thinking’’ about the concept. This book provides several lead-ins to the quantum theory including a brief review of Lagrange and Hamilton’s approach to classical mechanics, a discussion of the link with Hilbert space, and an introduction to the Feynman path integral. Chapter 4 summarizes the Hamiltonian and Lagrangian formalism necessary for the proper development of the quantum theory. However, Chapter 5 provides the more fundamental connection between the Hilbert space and quantum theory as well as demonstrating the Schrödinger wave equation from the Feynman path integer. Chapter 5 discusses standard topics such as the quantum well, harmonic oscillator, representations, perturbation theory, and spin and expands into the density operator and applications to quantum computing and teleportation. Chapter 6 provides an introduction to the solid state with an emphasis on the crystalline form of matter and its implications for phonon and electronic properties required for a follow-on course in optoelectronics. Chapter 7 introduces effective mass (scalar and tensor), three different band theories (Kronig-Penney, Tight Binding, and k-p), and density of states for bulk and reduced dimensional structures. Chapter 8 provides the concepts for ensembles and microstates in detail with an emphasis on the derivation of particle population distributions across energy levels. These derivations start with entropy and incorporate indistinguishability and spin (Boson, Fermion) properties while providing clear pictures to illustrate the development. The material has been taught for seven years in various formats to graduate research students and to undergraduates. The students come from a variety of departments but primarily from electrical and computer engineering, physics, and materials science. Beginning graduate students and advanced undergraduates can cover significant portions of this book in about 26–28 classes with 1.4 h of lecture per class. The number of classes devoted to the various topics often needs some adjustment depending on the pace of the course and the background of the students. The course devotes at least six or seven classes to the Hilbert spaces (discrete and continuous basis vectors, xvii.
(19) xviii. Preface. projection operators, orthonormal expansions, commutators, Hermitian and unitary operators, eigenvectors, and eigenvalues), at least six or seven classes to the introductory quantum theory (quantum wells, harmonic oscillator, time-independent perturbation theory, density operator), approximately four or five classes to phonons (direct and reciprocal lattices, dispersion curves and group velocity, and density of states), five or six classes to conduction and bands (quantum equation of continuity, effective mass, band diagrams, density of states, and, most importantly, the Bloch theorem), and at least four or five classes covering statistical mechanics and its application to carrier concentration (Lagrange multipliers, Boltzmann and Fermi distributions, Fermi functions, and diodes). More advanced classes cover all of the mathematics, the classical mechanics, quantum mechanical spin and angular momentum, propagators and the Feynman path integral, tensor mass, tight-binding, and k-p band theory. However, these additional topics are not necessary to read Physics of Optoelectronics as a follow-on course for semiconductor emitters and detectors, and as an introduction to quantum optics. The undergraduate reader (junior–senior) will find the Hilbert space and matrices accessible along with select sections on the quantum theory including the quantum well material, the electron spin, the harmonic oscillator, and the time-independent perturbation theory, as well as all of the material on phonons. The average undergraduate will be able to handle the conduction processes, the scalar effective mass, the Kronig–Penney model, and the electron density of states. A comment regarding the end-of-chapter review exercises should be made. The problems help one to understand and internalize the material contained in the chapter. The reader should make an effort to work through some of them. None of the problems are very difficult. However, some of the information or starting assumptions for a few of the problems have been omitted. As a result, the reader will need to understand the problem, develop a solution if possible, and then determine the range=conditions of validity. The programs at Cornell University, Rutgers University, Syracuse University, and Rome Laboratory (AFRL) along with many publications have help mold the views presented within the text. A number of people deserve mention for assistance in various capacities over the years: Eun-Hyeong Yi, P.D. Swanson, C.L. Tang, and E.A. Schiff for research, publications, and advice; S. Thai, D.G. Daut, and R.J. Michalak for assistance with programs, committees, and funding; Z. Gajic, R.L. Liboff, J. Scafidi, M. Sussman, D. Parker, and P. Kornreich for their advice and helpful discussions; and Y. Lu, S. McAffee, P. Panayotatos, M.F. Caggiano, and J. Zhao for committee participation and discussion. Special recognition goes to the staff at Taylor & Francis for their advice and efforts to bring the text to publication while providing a sufficiently flexible schedule. I am especially grateful to my wife Carol for her constant support, encouragement, and suggestions on various aspects of the book, and career advice. She has grown accustomed to the everpresent travel computer on many trips as well as the stacks of papers and books, reams of notes and calculation, and the long hours devoted to research and laboratory issues. I am also thankful to my students who have attended the courses and have applied the material to their research while posing challenging questions, interesting solutions, and helpful suggestions. Michael A. Parker.
(20) Author Dr. Michael A. Parker has developed optoelectronic theory and devices for the past several decades, taught graduate and undergraduate classes in physics and engineering at leading universities, served as a technical advisor and research scientist at a government laboratory, and founded a local firm for consulting, research, and development. He earned a PhD in physics for research in condensed matter physics with foundational work in the theory of particle physics and mathematics. He was especially interested in the quantum vacuum rich in ‘hidden’ intrinsic mechanisms with noise as the ‘rule’ rather than the ‘exception’. His post doctoral work branched into optical= photonic experiment, theory and fabrication. Dr. Parker’s research includes applications of quantum optics (a close relative of quantum electrodynamics) in the area of noise as a conveyor of information, along with the associated areas of fabrication, experiment and theory for semiconductor emitters and novel optical logic components, optically controlled molecular processes for photodissolution, and optical processes in semiconductors and amorphous materials. Dr. Parker has publications ranging from high-impact journals to general-interest reading, patents and disclosures, conferences, and software.. xix.
(21)
(22) 1 Introduction to the Solid State Matter, fields, and their interactions produce the world we know. Matter takes on various forms including gasses, liquids, and solids although the study of ‘‘solid state’’ traditionally focuses on solids and often specifically crystals. The present chapter overviews and summarizes important topics in the study of the solid state such as the origin of bands and the nature of transitions between bands. The discussion shows the transition of devices from tubes to bipolar junction transistors (BJTs) and field-effect transistors (FETs) to nanodevices.. 1.1 BRIEF PREVIEW The invention and development of new devices requires not only a clear understanding of present engineering and science practice, but also sufficient theoretical background to understand new discoveries in a variety of fields. For these reasons, we develop quantum theory from the start and then apply it to areas such as energy band theory and electrical transport. Our study concentrates on the electronic properties of solids (as opposed to gases and liquids). Modern technology primarily relies on the crystalline materials and secondarily on amorphous materials and polymers. The present chapter introduces the various forms of matter including solids, liquids, and gases. The earliest studies of the solid state have focused on homostructures consisting of identical molecules arranged in a periodic array; these materials can be doped to enhance the electrical conduction. In contrast, heterostructures have layers of dissimilar materials. In all cases of crystalline solids, the atoms and molecules form a periodic array. The periodic structure is described by the lattice as a mathematical object consisting of a periodic array of points. The crystal is formed by adding a ‘‘cluster of atoms’’ (a.k.a, an atomic basis) to each lattice point—the cluster can have as few as one atom. The crystal structure has importance for the conduction properties of the material as well as many of the physical material properties such as ‘‘material hardness’’ and mass density, and for semiconductor processing such as for the possible cleave and etching planes. Every lattice has a reciprocal lattice that represents the k-vectors in spatial Fourier transforms. The reciprocal lattice vectors provide zone boundaries for phonon and carrier band diagrams. The operation of the vast majority of modern electronic components can only be explained through band theory. The crystalline material structure immediately leads to the electron and hole bands. The relation between bands and crystalline structure can most easily be demonstrated by the Kronig–Penney model. This model makes explicit use of the wave nature of electrons and shows how bands arise from a one-dimensional (1-D) array of atoms. On the other hand, the K–P theory (as distinct from the Kronig–Penney model) provides a more predictive model for band structure and effective mass. The band structure produces an effective mass for the electron and hole, which can be many orders of magnitude smaller than a free-electron mass. The effective mass can most simply be calculated from the curvature of the conduction or valence band. Evidently, the effective mass has very important consequences for electrical conduction and the high-frequency performance of many devices. The bands themselves consist of very closely spaced discrete states usually termed extended states because they correspond to traveling plane waves. Purely crystalline materials do not have states in the energy bandgap. However, defects and doping result in localized states within the gap that can trap the electrons and holes in a specific region of the material. The band structure of conventional electronic devices can only be fully described by resorting to the quantum theory, which is the study of the wave nature of material particles. Nanoscale and optoelectronic devices make extensive use of the quantum theory. Nanoscale devices have 1.
Figure
+7
Documents relatifs
