Solid State and Quantum Theory for Optoelectronics

850 Pages·2009·9.64 MB·English

by  Michael A. Parker

SOLID STATE AND
QUANTUM THEORY FOR
OPTOELECTRONICS
Michael A. Parker
Boca Raton London New York
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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 2009030736
Visit the Taylor & Francis Web site at
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Contents
Preface...........................................................................................................................................xvii
Author............................................................................................................................................xix
Chapter 1 Introduction tothe Solid State....................................................................................1
1.1 Brief Preview....................................................................................................1
1.2 Introduction to Matter and Bonds....................................................................3
1.2.1 Gassesand Liquids..............................................................................3
1.2.2 Solids...................................................................................................4
1.2.3 Bondingand thePeriodic Table..........................................................5
1.2.4 DopantAtoms......................................................................................8
1.3 Introduction to Bands andTransitions.............................................................9
1.3.1 Intuitive Origin of Bands....................................................................9
1.3.2 IndirectBandsand Light- andHeavy-Hole Bands...........................11
1.3.3 Introduction toTransitions................................................................13
1.3.4 Introduction toBand-Edge Diagrams...............................................14
1.3.5 Bandgap States and Defects..............................................................15
1.4 Introduction to thepn Junction......................................................................16
1.4.1 JunctionTechnology.........................................................................17
1.4.2 Band-Edge Diagramsand the pnJunction........................................18
1.4.3 NonequilibriumStatistics..................................................................19
1.5 DeviceTrends.................................................................................................21
1.5.1 Monolithic Integrationof Device Types...........................................21
1.5.2 Year 2000Benchmarks.....................................................................21
1.5.3 SmallOptical Signals........................................................................22
1.5.4 Fabrication Challenges......................................................................23
1.6 VacuumTubes andTransistors......................................................................23
1.6.1 VacuumTube....................................................................................23
1.6.2 Bipolar Transistor..............................................................................24
1.6.3 Field-Effect Transistor.......................................................................25
1.7 Brief Summaryof Some EarlyNanometer-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–BohmEffect 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 Vectorand HilbertSpaces..............................................................................31
2.1.1 Motivation forLinear Algebra inQuantum Theory.........................31
2.1.2 Definition of Vector Space................................................................33
iii
iv Contents
2.1.3 Hilbert Space.....................................................................................34
2.1.4 Comment onthe Length ofa Vector forQuantum Theory..............36
2.1.5 Linear Isomorphism...........................................................................37
2.1.6 Antilinear Isomorphism.....................................................................37
2.2 DiracNotation andEuclidean Vector Spaces................................................37
2.2.1 Kets,Bras, and Brackets for Euclidean Space..................................38
2.2.2 Basisand Completeness forEuclidean Space...................................39
2.2.3 Closure Relation for theEuclidean Vector Space.............................40
2.2.4 Euclidean Dual Vector Space............................................................41
2.2.5 Inner Product and Norm....................................................................44
2.3 Introduction to Coordinateand Vector Representation ofFunctions............45
2.3.1 Initial View of theCoordinate Representation ofFunctions............46
2.3.2 Coordinate Basis Set.........................................................................47
2.3.3 Introduction tothe Inner Product forFunctions...............................49
2.3.4 Representationsof Functions............................................................49
2.4 Function Space with Discrete Basis Sets.......................................................50
2.4.1 Introduction toHilbert Space............................................................50
2.4.2 Hilbert Space ofFunctionswithDiscrete Basis Vectors..................51
2.4.3 Closure Relation for Functions with aDiscrete Basis......................53
2.4.4 Norms and Inner Products forFunction Spaces
with DiscreteBasisSets....................................................................54
2.4.5 Discussion ofWeightFunctions.......................................................55
2.4.6 SomeMiscellaneous Notes on Notation...........................................58
2.5 Function Spaces with Continuous Basis Sets................................................59
2.5.1 Continuous Basis Setof Functions...................................................59
2.5.2 Coordinate Space...............................................................................61
2.5.3 Representationsof theDiracDelta Using Basis Vectors..................64
2.6 Graham–Schmidt OrthonormalizationProcedure...........................................65
2.6.1 Simplest Case of TwoVectors..........................................................65
2.6.2 More than Two Vectors....................................................................66
2.7 Fourier Basis Sets...........................................................................................66
2.7.1 Fourier Cosine Series........................................................................67
2.7.2 Fourier SineSeries............................................................................68
2.7.3 Fourier Series.....................................................................................69
2.7.4 Alternate Basis for theFourierSeries...............................................71
2.7.5 Fourier Transform..............................................................................71
2.8 Closure Relations,Kronecker Delta,and Dirac Delta Functions...................73
2.8.1 Alternate Closure Relationsand Representations
of theKronecker Delta Function forEuclidean Space.....................74
2.8.2 CosineBasis Functions.....................................................................75
2.8.3 Sine Basis Functions.........................................................................77
2.8.4 Fourier Series Basis Functions..........................................................77
2.8.5 SomeNotes........................................................................................78
2.9 Introduction to Direct Product Spaces............................................................79
2.9.1 Overview ofDirect Product Spaces..................................................79
2.9.2 Introduction toDyadicNotation forthe Tensor Product
of Two Euclidean Vectors.................................................................82
2.9.3 Direct Product Spacefrom theFourier Series..................................82
2.9.4 Components and Closure Relation fortheDirect Product
of Functionswith DiscreteBasisSets...............................................84
2.9.5 Noteson theDirect Productsof Continuous Basis Sets...................85
Contents v
2.10 Introduction to MinkowskiSpace..................................................................86
2.10.1 Coordinates andPseudo-Inner Product.............................................86
2.10.2 Pseudo-Orthogonal Vector Notation.................................................86
2.10.3 TensorNotation.................................................................................86
2.10.4 Derivatives.........................................................................................87
2.11 Brief Discussionof Probability andVectorComponents..............................88
2.11.1 Simple2-D Space forStarters...........................................................88
2.11.2 Introduction toApplications ofthe Probability................................90
2.11.3 Discrete and Continuous HilbertSpaces...........................................91
2.11.4 Contrast with RandomVectors.........................................................92
2.12 Review Exercises............................................................................................92
References and Further Readings..............................................................................98
Chapter 3 Operators andHilbert Space.....................................................................................99
3.1 Introduction to Operatorsand Groups............................................................99
3.1.1 Linear Operator...............................................................................100
3.1.2 Transformations of theBasis Vectors Determine
the Linear Operator.........................................................................100
3.1.3 Introduction toIsomorphisms.........................................................101
3.1.4 Comments on Groupsand Operators..............................................101
3.1.5 Permutation Group and a Matrix Representation:
An Example.....................................................................................103
3.2 MatrixRepresentations.................................................................................104
3.2.1 Definition of Matrix for an Operator withIdentical Domain
and RangeSpaces............................................................................105
3.2.2 Matrixof an Operator withDistinctDomain
and RangeSpaces............................................................................106
3.2.3 Dirac Notation forMatrices............................................................107
3.2.4 Operating on anArbitrary Vector...................................................109
3.2.5 MatrixEquation...............................................................................110
3.2.6 Matrices for Function Spaces..........................................................113
3.2.7 Introduction toOperator Expectation Values..................................114
3.2.8 MatrixNotation forAverages.........................................................115
3.3 Common Matrix Operations.........................................................................116
3.3.1 Composition of Operators...............................................................116
3.3.2 Isomorphism between Operatorsand Matrices...............................117
3.3.3 Determinant.....................................................................................118
3.3.4 Introduction tothe Inverseof an Operator......................................120
3.3.5 Trace................................................................................................122
3.3.6 Transpose and Hermitian Conjugate ofa Matrix............................123
3.4 Operator Space.............................................................................................124
3.4.1 Concepts and SectionSummary......................................................124
3.4.2 BasisExpansion ofa Linear Operator............................................126
3.4.3 Introduction tothe Inner Product fora HilbertSpace
of Operators.....................................................................................129
3.4.4 Proof of theInner Product...............................................................131
3.4.5 BasisforMatrices............................................................................132
3.5 Operators and Matrices inDirect Product Space.........................................133
3.5.1 Review ofDirect Product Spaces....................................................133
3.5.2 Operators.........................................................................................134
vi Contents
3.5.3 Matrices of Direct Product Operators.............................................134
3.5.4 MatrixRepresentationof Basis Vectors
for Direct Product Space.................................................................137
3.6 Commutatorsand Algebra ofOperators......................................................138
3.6.1 Initial Discussionof Operator Algebra...........................................139
3.6.2 Introduction toCommutators..........................................................140
3.6.3 SomeCommutator Theorems..........................................................141
3.7 Unitary Operators andSimilarity Transformations......................................143
3.7.1 OrthogonalRotation Matrices.........................................................143
3.7.2 Unitary Transformations..................................................................146
3.7.3 VisualizingUnitary Transformations..............................................147
3.7.4 Traceand Determinant....................................................................148
3.7.5 Similarity Transformations..............................................................148
3.7.6 Equivalent andReducible Representations of Groups....................150
3.8 Hermitian Operators and theEigenvector Equation.....................................151
3.8.1 Adjoint, Self-Adjoint, and Hermitian Operators.............................152
3.8.2 Adjoint and Self-Adjoint Matrices..................................................154
3.9 Relation between Unitaryand Hermitian Operators....................................156
3.9.1 Relation between Hermitian andUnitary Operators.......................156
3.10 Eigenvectors and Eigenvaluesfor Hermitian Operators..............................158
3.10.1 BasicTheorems forHermitian Operators.......................................158
3.10.2 Direct Product Space.......................................................................162
3.11 Eigenvectors, Eigenvalues,and DiagonalMatrices.....................................162
3.11.1 Motivation forDiagonal Matrices...................................................162
3.11.2 Eigenvectors andEigenvalues.........................................................164
3.11.3 Diagonalize a Matrix.......................................................................165
3.11.4 Relation between aDiagonal Operator
and the Change-of-Basis Operator..................................................169
3.12 Theorems forHermitian Operators...............................................................170
3.12.1 Common Theorems.........................................................................171
3.12.2 Bounded Hermitian OperatorsHave Complete Sets
of Eigenvectors................................................................................172
3.12.3 Derivation ofthe Heisenberg Uncertainty Relation........................176
3.13 Raising–Loweringand Creation–Annihilation Operators............................179
3.13.1 Definition of theLadder Operators.................................................179
3.13.2 Matrixand Basis-Vector Representations ofthe Raising
and LoweringOperators..................................................................180
3.13.3 Raising and Lowering Operatorsfor Direct Product Space............182
3.14 Translation Operators...................................................................................183
3.14.1 Exponential Form of theTranslation Operator...............................183
3.14.2 Translation of thePosition Operator...............................................184
3.14.3 Translation of thePosition-CoordinateKet....................................185
3.14.4 Example Using theDirac Delta Function.......................................185
3.14.5 Relation among HilbertSpaceand the 1-D Translation,
and Lie Group.................................................................................186
3.14.6 Translation Operators inThreeDimensions...................................186
3.15 Functions inRotated Coordinates................................................................186
3.15.1 Rotating Functions..........................................................................186
3.15.2 Rotation Operator............................................................................188
3.15.3 Rectangular Coordinates for theGenerator
of Rotations about z.........................................................................189
Contents vii
3.15.4 Rotation ofthe Position Operator...................................................189
3.15.5 StructureConstants and LieGroups...............................................190
3.15.6 StructureConstants forthe Rotation Lie Group.............................191
3.16 DyadicNotation............................................................................................192
3.16.1 Notation...........................................................................................192
3.16.2 Equivalence between the Dyad and theMatrix..............................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 PhaseSpace Coordinates.................................................................204
4.2 Action,Lagrangian, and Lagrange’sEquation.............................................204
4.2.1 Origin ofthe Lagrangian inNewton’sEquations...........................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 thePoissonBracket and Relation
to theCommutator...........................................................................213
4.4.2 BasicProperties forthe PoissonBracket........................................214
4.4.3 Constants ofthe Motion and Conserved Quantities.......................215
4.5 Lagrangian and Normal Coordinates fora Discrete Array ofParticles.......216
4.5.1 Lagrangian and Equationsof Motion..............................................216
4.5.2 Transformationto Normal Coordinates..........................................217
4.5.3 Lagrangian and theNormal Modes.................................................222
4.6 ClassicalField Theory..................................................................................224
4.6.1 Lagrangian and Hamiltonian Density.............................................225
4.6.2 Lagrange Density for1-D WaveMotion........................................227
4.7 Lagrangian and theSchrödinger Equation...................................................230
4.7.1 Schrödinger Wave Equation............................................................230
4.7.2 Hamiltonian Density........................................................................231
4.8 Brief Summaryof theStructureof Space-Time...........................................232
4.8.1 Introduction toSpace-Time Warping..............................................232
4.8.2 Minkowski Space............................................................................233
4.8.3 Lorentz Transformation...................................................................236
4.8.4 SomeExamples...............................................................................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 ofSuperpositionof Basis States
and the Probability Interpretation....................................................249
viii Contents
5.1.4 Probability Interpretation.................................................................250
5.1.5 Averages..........................................................................................252
5.1.6 Motionof theWaveFunction.........................................................254
5.1.7 Collapse ofthe Wave Function.......................................................255
5.1.8 Interpretationsof theCollapse........................................................257
5.1.9 Noncommuting Operatorsand the Heisenberg
Uncertainty Relation........................................................................259
5.1.10 Complete Sets of Observables.........................................................262
5.2 Fundamental Operatorsand Procedures forQuantum Mechanics...............263
5.2.1 Summary ofElementary Facts........................................................263
5.2.2 Momentum Operator.......................................................................264
5.2.3 Hamiltonian Operator and theSchrödinger
Wave Equation................................................................................264
5.2.4 Introduction toCommutation Relationsand Heisenberg
Uncertainty Relations......................................................................266
5.2.5 Derivation ofthe Heisenberg Uncertainty Relation........................267
5.2.6 Program...........................................................................................269
5.3 Examples forSchrödinger’sWaveEquation................................................271
5.3.1 Discussion ofQuantum Wells.........................................................272
5.3.2 Solutions to Schrödinger’s Equationfor theInfinitely
Deep Well........................................................................................273
5.3.3 FinitelyDeep SquareWell..............................................................279
5.4 Harmonic Oscillator......................................................................................285
5.4.1 Introduction toClassical and Quantum
Harmonic Oscillators.......................................................................285
5.4.2 Hamiltonian for theQuantum Harmonic Oscillator........................288
5.4.3 Introduction tothe Ladder Operators
for theHarmonic Oscillator.............................................................288
5.4.4 LadderOperators inthe Hamiltonian..............................................290
5.4.5 Properties of theRaising and Lowering Operators.........................292
5.4.6 EnergyEigenvalues.........................................................................294
5.4.7 EnergyEigenfunctions....................................................................294
5.5 Introduction to Angular Momentum............................................................296
5.5.1 ClassicalDefinition ofAngular Momentum...................................296
5.5.2 Origin ofAngular Momentum in Quantum Mechanics..................297
5.5.3 Angular Momentum Operators.......................................................298
5.5.4 Picturesfor Angular Momentum inQuantum Mechanics..............299
5.5.5 Rotational Symmetry and Conservation
of Angular Momentum....................................................................301
5.5.6 Eigenvalues and Eigenvectors.........................................................303
5.5.7 Eigenvectors asSpherical Harmonics.............................................305
5.6 Introduction to Spinand Spinors..................................................................309
5.6.1 BasicIdea ofSpin...........................................................................309
5.6.2 Link between PhysicalSpace andHilbert Space............................312
5.6.3 Pauli SpinMatrices.........................................................................315
5.6.4 Rotations..........................................................................................317
5.6.5 Direct Product Spacefor a Single Electron....................................318
5.6.6 SpinHamiltonian.............................................................................319
5.7 Angular Momentum for Multiple Systems..................................................323
5.7.1 AddingAngular Momentum...........................................................323
5.7.2 Clebsch–Gordon Coefficients..........................................................326
Contents ix
5.8 Quantum Mechanical Representations.........................................................330
5.8.1 Discussion ofthe Schrödinger,Heisenberg,
and Interaction Representations......................................................331
5.8.2 Schrödinger Representation.............................................................333
5.8.3 Rate of Change of theAverageof an Operator
in theSchrödinger Picture...............................................................334
5.8.4 Ehrenfest’sTheorem for theSchrödinger Representation..............335
5.8.5 Heisenberg Representation..............................................................337
5.8.6 Heisenberg Equation.......................................................................338
5.8.7 Newton’sSecondLaw from the Heisenberg Representation..........339
5.8.8 Interaction Representation...............................................................340
5.9 Time-Independent PerturbationTheory........................................................341
5.9.1 Initial Discussionof Perturbations..................................................341
5.9.2 NondegeneratePerturbationTheory................................................342
5.9.3 Unitary Operator forTime-Independent
Perturbation Theory.........................................................................349
5.10 Time-Dependent Perturbation Theory..........................................................352
5.10.1 Physical Concept.............................................................................353
5.10.2 Time-Dependent Perturbation Theory Formalism
in theSchrödinger Picture...............................................................355
5.10.3 Example forFurther Thought and Questions..................................359
5.10.4 Time-Dependent Perturbation Theory in theInteraction
Representation.................................................................................362
5.10.5 Evolution Operator in theInteraction
Representation.................................................................................364
5.11 Introduction to Optical Transitions..............................................................365
5.11.1 EM Interaction Potential..................................................................365
5.11.2 Integral for theProbability Amplitude............................................367
5.11.3 Rotating WaveApproximation.......................................................369
5.11.4 Absorption.......................................................................................370
5.11.5 Emission..........................................................................................371
5.11.6 Discussion ofthe Results................................................................372
5.12 Fermi’sGolden Rule.....................................................................................373
5.12.1 IntroductoryConcepts on Probability.............................................373
5.12.2 Definition of theDensity ofStates..................................................374
5.12.3 Equations forFermi’s Golden Rule................................................377
5.13 Density Operator...........................................................................................382
5.13.1 Introduction tothe DensityOperator..............................................382
5.13.2 Density Operator andthe Basis Expansion.....................................386
5.13.3 Ensemble and Quantum Mechanical Averages...............................390
5.13.4 Lossof Coherence...........................................................................394
5.13.5 SomeProperties...............................................................................396
5.14 Introduction to Multiparticle Systems..........................................................397
5.14.1 Introduction.....................................................................................397
5.14.2 Permutation Operator......................................................................399
5.14.3 Simultaneous Eigenvectorsof theHamiltonian
and the Interchange Operator..........................................................401
5.14.4 Introduction toFockStates.............................................................403
5.14.5 Origin ofFockStates......................................................................404
5.14.5.1 Bosons..............................................................................406
5.14.5.2 Fermions..........................................................................408