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Quantum Mechanics for Nanostructures
Thepropertiesofnewnanoscalematerials,theirfabricationandapplica-
tions,aswellastheoperationalprinciplesofnanodevicesandsystems,are
solelydeterminedbyquantum-mechanicallawsandprinciples.Thistextbook
introducesengineerstoquantummechanicsandtheworldofnanostructures,
enablingthemtoapplythetheoriestonumerousnanostructureproblems.
Thebookcoversthefundamentalsofquantummechanics,includinguncer-
taintyrelations,theSchro¨dingerequation,perturbationtheory,andtunneling.
Thesearethenappliedtoaquantumdot,thesmallestartificialatom,andcom-
paredwiththecaseofhydrogen,thesmallestatominnature.Nanoscaleobjects
withhigherdimensionality,suchasquantumwiresandquantumwells,are
introduced,aswellasnanoscalematerialsandnanodevices.Numerousexam-
plesthroughoutthetexthelpstudentstounderstandthematerial.
VLADIMIRV.MITINisSUNYDistinguishedProfessorattheDepartmentof
Electrical Engineering and Adjunct Professor of Physics at the University
atBuffalo,TheStateUniversityofNewYork.Heistheauthorofeighttext-
booksandmonographsandmorethan490professionalpublicationsand
presentations.
DMITRYI.SEMENTSOVisProfessorofPhysicsatUlyanovskStateUniversity,
Russia. He is the author of more than 420 papers in peer-reviewed jour nals.
NIZAMIZ.VAGIDOVisResearchAssistantProfessorofElectricalEngineeringat
theUniversityatBuffalo,TheStateUniversityofNewYork.Heistheauthor
ofabout90professionalpublicationsinthefieldsofsolid-stateelectronics,
nanoelectronics,andnanotechnology.
Quantum Mechanics
for Nanostructures
Vladimir V. Mitin
University at Buffalo, The State University of New York
Dmitry I. Sementsov
Ulyanovsk State University
Nizami Z. Vagidov
Uni ver sity at Buffalo, The State Uni ver sity of New York
CAMBRIDGEUNIVERSITYPRESS
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São Paulo, Delhi, Dubai, Tokyo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521763660
© V. Mitin, D. Sementsov and N. Vagidov 2010
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2010
ISBN-13 978-0-511-72953-9 eBook (NetLibrary)
ISBN-13 978-0-521-76366-0 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
Contents
Preface pageix
Listofnotation xiii
1 Thenanoworldandquantumphysics 1
1.1 Areviewofmilestonesinnanoscienceandnanotechnology 1
1.2 Nanostructuresandquantumphysics 4
1.3 Layerednanostructuresandsuperlattices 8
1.4 Nanoparticlesandnanoclusters 10
1.5 Carbon-basednanomaterials 14
2 Wave–particledualityanditsmanifestationinradiation
andparticlebehavior 19
2.1 Blackbodyradiationandphotongas 19
2.2 Thequantumcharacteroftheinteractionofradiation
withmatter 31
2.3 Wavepropertiesofparticles 39
2.4 Theuncertaintyrelations 47
2.5 Theworldofthenanoscaleandthewavefunction 52
2.6 TheSchro¨dingerequation 56
2.7 Summary 63
2.8 Problems 63
3 Layerednanostructuresasthesimplestsystemstostudy
electronbehaviorinaone-dimensionalpotential 65
3.1 Themotionofafreeelectroninvacuum 66
3.2 Anelectroninapotentialwellwithinfinitebarriers 69
3.3 Anelectroninapotentialwellwithfinitebarriers 75
3.4 Propagationofanelectronabovethepotentialwell 84
3.5 Tunneling:propagationofanelectronintheregion
ofapotentialbarrier 89
3.6 Summary 101
3.7 Problems 101
v
vi Contents
4 Additionalexamplesofquantizedmotion 105
4.1 Anelectroninarectangularpotentialwell(quantumbox) 105
4.2 Anelectroninaspherically-symmetricpotentialwell 109
4.3 Quantumharmonicoscillators 115
4.4 Phonons 126
4.5 Summary 133
4.6 Problems 134
5 Approximatemethodsoffindingquantumstates 136
5.1 Stationaryperturbationtheoryforasystemwith
non-degeneratestates 136
5.2 Stationaryperturbationtheoryforasystemwith
degeneratestates 141
5.3 Non-stationaryperturbationtheory 142
5.4 Thequasiclassicalapproximation 148
5.5 Summary 151
5.6 Problems 152
6 Quantumstatesinatomsandmolecules 155
6.1 Thehydrogenatom 155
6.2 Theemissionspectrumofthehydrogenatom 166
6.3 Thespinofanelectron 169
6.4 Many-electronatoms 173
6.5 Thewavefunctionofasystemofidenticalparticles 181
6.6 Thehydrogenmolecule 184
6.7 Summary 190
6.8 Problems 191
7 Quantizationinnanostructures 193
7.1 Thenumberanddensityofquantumstates 193
7.2 Dimensionalquantizationandlow-dimensionalstructures 199
7.3 Quantumstatesofanelectroninlow-dimensionalstructures 204
7.4 Thenumberofstatesanddensityofstatesfornanostructures 210
7.5 Double-quantum-dotstructures(artificialmolecules) 218
7.6 Anelectroninaperiodicone-dimensionalpotential 229
7.7 Aone-dimensionalsuperlatticeofquantumdots 241
7.8 Athree-dimensionalsuperlatticeofquantumdots 250
7.9 Summary 254
7.10 Problems 255
8 Nanostructuresandtheirapplications 258
8.1 Methodsoffabricationofnanostructures 258
Contents vii
8.2 Toolsforcharacterizationwithnanoscaleresolution 269
8.3 Selectedexamplesofnanodevicesandsystems 282
AppendixA Classicaldynamicsofparticlesandwaves 310
A.1 Classicaldynamicsofparticles 311
A.2 Oscillatorymotionofaparticle 321
A.3 Summary 334
A.4 Problems 335
AppendixB Electromagneticfieldsandwaves 338
B.1 Equationsofanelectromagneticfield 338
B.2 Electromagneticwaves 345
B.3 Reflectionofaplanewavefromtheinterfacebetweentwomedia 353
B.4 Lightanditswaveproperties 362
B.5 Summary 374
B.6 Problems 375
AppendixC Crystalsasatomiclattices 378
C.1 Crystallinestructures 379
C.2 Thenatureofattractionandrepulsionforces 385
C.3 Degenerateelectrongas 392
C.4 Wavesinacrystallinelatticeandnormalcoordinates 396
C.5 Theenergyspectrumofanelectroninacrystal 400
C.6 Electronsinsemiconductors 411
C.7 Summary 420
C.8 Problems 421
AppendixD Tablesofunits 423
Index 427
