HYBRID PHONONS IN NANOSTRUCTURES

205 Pages·2017·0.842 MB·English

by  RIDLEY, BRIAN K

SERIESONSEMICONDUCTOR
SCIENCEANDTECHNOLOGY
SeriesEditors
R.J.Nicholas UniversityofOxford
H.Kamimura UniversityofTokyo
SERIES ON SEMICONDUCTOR SCIENCE
AND TECHNOLOGY
1. M.Jaros:Physicsandapplicationsofsemiconductormicrostructures
2. V.N.DobrovolskyandV.G.Litovchenko:Surfaceelectronictransport
phenomenainsemiconductors
3. M.J.Kelly:Low-dimensionalsemiconductors
4. P.K.Basu:Theoryofopticalprocessesinsemiconductors
5. N.Balkan:Hotelectronsinsemiconductors
6. B.Gil:GroupIIInitridesemiconductorcompounds:physicsandapplications
7. M.Sugawara:Plasmaetching
8. M.Balkanski,R.F.Wallis:Semiconductorphysicsandapplications
9. B.Gil:Low-dimensionalnitridesemiconductors
10. L.Challis:Electron–phononinteractionsinlow-dimensionalstructures
11. V.Ustinov,A.Zhukov,A.Egorov,N.Maleev:Quantumdotlasers
12. H.Spieler:Semiconductordetectorsystems
13. S.Maekawa:Conceptsinspinelectronics
14. S.D.Ganichev,W.Prettl:Intenseterahertzexcitationofsemiconductors
15. N.Miura:Physicsofsemiconductorsinhighmagneticfields
16. A.V.Kavokin,J.J.Baumberg,G.Malpuech,F.P.Laussy:Microcavities
17. S.Maekawa,S.O.Valenzuela,E.Saitoh,T.Kimura:Spincurrent
18. B.Gil:III-nitridesemiconductorsandtheirmoderndevices
19. A.Toropov,T.Shubina:PlasmonicEffectsinMetal-Semiconductor
Nanostructures
20. B.K.Ridley:HybridPhononsinNanostructures
Hybrid Phonons in Nanostructures
First Edition
B. K. Ridley
ProfessorEmeritusofPhysics,UniversityofEssex,UK
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Preface
Crystalline nanostructures confine electrons and quantize their energies, which
effect has led to the nomenclature quantum wells, quantum wires, and quantum
dots. They also confine lattice waves. In the case of acoustic modes propagating
in wave guides, this confinement is well understood by classical physics. This is
notthecaseforopticalmodesofvibration.Inacousticwaveguidesand,indeed,
innanostructures,thenormalmodesofvibrationaredeterminedbythesatisfac-
tion of the classical connection rules—continuation of particle displacement and
continuityofstress—attheboundaries.Inordertosatisfytheserules,amodemay
havetocombinewithamodeofdifferentpolarizationtoformahybrid.Inmany
cases,alongitudinallypolarizedacoustic(LA)modemustliasewithatransversely
polarizedacoustic(TA)modetoformaviablenormalmodeofthesystem.Such
hybrid modes are commonplace in confining structures and have been known
from the end of the nineteenth century. This is not the case for optical modes.
Relative to acoustic modes, optical modes are but lately come, and their proper-
ties not as well defined or understood. This lacuna has been one of the motives
forwritingthisbook.
An understanding of optical modes in room-temperature devices is of some
importance since they are, in polar structures, the principal source of electrical
resistance.Acousticmodesarealsoimportantinthatrespect,butthesharedfre-
quencies between barrier and well make the confinement of acoustic modes at
once more intricate and more open to simplification. On the one hand, reflec-
tionandtransmissionattheboundaryleadtoarichfamilyofmodepatternsthat
include guided modes and interface waves. On the other hand, as regards the
electron–phononinteraction,itmaybesufficientformanypurposestodisregard
theintricaciesentirelyandtreattheentireacousticspectrumasbulk-like.Thisis
notaneasilyjustifiableoptionforopticalmodes,giventhedisparityoffrequency
between barrier and well that is commonly encountered. In that respect, optical
modes present a problem. What exactly are the mechanical connection rules? In
thecaseofpolarmodes,therearetheusualelectromagneticboundaryconditions
aswellasthosemysteriousrulesassociatedwiththeelasticityofthelattice.Canthe
electromagneticboundaryconditionsbesufficient?Inotherwords,canthecrystal
beregardedsimplyasadielectriccontinuum?Forthoseinterestedsolelyinestim-
atingthestrengthoftheinteractionbetweenelectronsandpolaropticalmodes,the
dielectriccontinuum(DC)modelprovidesasimpleralternativetohybridtheory,
an alternative that is not without some theoretical justification. Nevertheless, a
crystal is not a simple dielectric continuum. If the physics of nanostructures is
toseethesemiconductorasacontinuum,itmustbeacontinuumthatpossesses
both elastic and dielectric properties, inhabited by hybrid lattice vibrations, both
vi Preface
acousticandoptical,alongwithconfinedelectrons.Theseconstitutetheessential
elementsofthenanostructuresandtheirinteractionthatwillbedescribedhere.
Inevitably,suchadescriptiongeneratesmanyequations,whichmanystudents
of nanostructure physics may find somewhat indigestible. As one who prefers
intuition to rigour (for better or worse), and who observes somewhat distantly
the purely formal mathematical approach with some admiration, I have much
sympathywiththisattitude,butthestudentshouldknowthattheequationswould
be much more indigestible were they to portray a truly rigorous reality that took
intoaccountthenaturalanisotropyofsemiconductorcrystals.Forsimplicity,the
hybrid modes that are described here are creatures of purely isotropic solids, in
whichmodesarepolarizedpurelylongitudinallyorpurelytransversely.Moreover,
they are all long-wavelength modes, which allow a clear distinction to be made
betweenopticalandacoustic.SuchapproximationsareacceptablefortheGroups
IV and III-V cubic semiconductors, but not for the hexagonal II-VI materials,
whicharehighlyanisotropicand,moreover,exhibitmorethanoneopticalmode.
Thebookhasbeenwrittenwithcubicsemiconductorsverymuchinmind.
Some parts of this book were written during and after moving house from
Essex to Herefordshire (often to the despair of my wife). I suspect it has kept
me sane during what most think of as one of the most traumatic events of life.
Perhapsphysicsistoberecommendedasabalmintroublesometimes.Mywife,
blessher,doubtsit.
Pembridge2016
Contents
Acknowledgements xi
Introduction 1
PreludetoPart1 6
Part1 Basics
1 AcousticModes 15
1.1 ContinuumTheory 15
1.2 EquationofMotion 16
1.3 Velocities 18
1.4 IsotropicCase 19
1.5 InhomogenousMaterial 19
1.6 Quantization 21
2 OpticalModes 24
2.1 Introduction 24
2.2 MicroscopicTheoryoftheDiamondLattice 25
2.3 DecoupledAcousticandOpticalEquations 29
2.4 Velocities 33
2.5 Isotropy 34
2.6 InhomogeneousSystem 34
3 PolarModesinZincBlende 37
3.1 PolarElements 37
3.2 PolarOpticalModes 38
3.3 InterfaceModes 41
3.4 Velocities 42
3.5 InhomogenousMaterial 43
3.6 Piezoelectricity 43
4 BoundaryConditions 46
4.1 Introduction 46
4.2 AcousticModes 46
4.3 OpticalModes 48
4.4 ElectromagneticBoundaryConditions 54
5 ScalarandVectorFields 55
5.1 Introduction 55
5.2 TheHelmholtzEquation 55
viii Contents
5.3 Cylinder 56
5.4 Sphere 57
Part2 HybridModesinNanostructures
6 Non-PolarSlab 61
6.1 BoundaryConditions 61
6.2 AcousticModes 62
6.3 OpticalModes 67
7 SingleHeterostructure 69
7.1 TheHybridModelforPolarOpticalModes 69
7.2 RemotePhonons 72
7.3 EnergyNormalization 73
7.4 ReducedBoundaryCondition 74
7.5 AcousticHybrids 75
7.6 InterfaceAcousticModes 80
8 QuantumWell 83
8.1 TripleHybrid 83
8.2 EnergyNormalization 88
8.3 ReducedBoundaryCondition 88
8.4 GeneralComments 89
8.5 BarrierModes 90
8.6 AcousticModes 91
8.7 InterfaceAcousticWaves 95
8.8 GuidedAcousticWaves 96
9 QuantumWire 97
9.1 Introduction 97
9.2 CylindricalCoordinates 98
9.3 InterfaceModes 101
9.4 HybridModesinPolarMaterial 103
9.5 AcousticStressesandStrains 106
9.6 FreeSurface 108
10 QuantumDot 110
10.1 Introduction 110
10.2 SphericalCoordinates 110
10.3 PolarDoubleHybrids 114
10.4 QuantumDiscandQuantumBox 115
Contents ix
Part3 Electron–PhononInteraction
11 The Interaction between Electrons and Polar Optical
PhononsinNanostructures:GeneralRemarks 119
11.1 ABriefHistory 119
11.2 Dispersion 121
11.3 CoupledModesandHotPhonons 122
12 Electrons 124
12.1 Confinement 124
12.2 ScatteringRate 128
13 ScatteringRateinaSingleHeterostructure 129
13.1 ScatteringRate 129
14 ScatteringRateinaQuantumWell 135
14.1 Preliminary 135
14.2 ScatteringRateAssociatedwithQuantumWellModes 135
14.3 ScatteringRateAssociatedwithBarrierModes 139
14.4 GeneralRemarks 140
15 ScatteringRateinQuantumWires 142
15.1 GeneralRemarks 142
15.2 ScatteringRate 142
16 TheElectron–PhononInteractioninaQuantumDot 145
16.1 Preamble 145
16.2 Electron–LatticeCoupling 145
16.3 TheExciton 148
17 CoupledModes 153
17.1 Introduction 153
17.2 Long-WavelengthModes 153
17.3 BeyondtheLong-WavelengthApproximation 156
17.4 ScreeninginQuasi-2DStructures 163
17.5 CouplingtoHybrids 170
17.6 Quasi-1DCylindricalStructures 171
17.7 Mobility 172
18 HotPhononLifetime 173
18.1 Introduction 173
18.2 Lifetime 175
18.3 ThermalConductivity 180
References 185
Index 189