Quantum Transport: Introduction to Nanoscience

591 Pages·2009·10.105 MB·English

by  Nazarov Y.V., Blanter Y.M.

Quantum Transport
Introduction to Nanoscience
Quantum transport is a diverse field, sometimes combining seemingly contradicting
concepts–quantumandclassical,conductingandinsulating–withinasinglenano-device.
Quantumtransportisanessentialandchallengingpartofnanoscience,andunderstanding
itsconceptsandmethodsisvitaltothesuccessfuldesignofdevicesatthenano-scale.
Thistextbookisacomprehensive introductiontotherapidlydeveloping fieldofquan-
tumtransport.Theauthorspresentthecomprehensivetheoreticalbackground,andexplore
the groundbreaking experiments that laid the foundations of the field. Ideal for graduate
students,eachsectioncontainscontrolquestionsandexercisestocheckthereader’sunder-
standingofthetopicscovered.Itsbroadscopeandin-depthanalysisofselectedtopicswill
appealtoresearchersandprofessionalsworkinginnanoscience.
YuliV.NazarovisatheoristattheKavliInstituteofNanoscience,DelftUniversityofTech-
nology.HeobtainedhisPh.D.fromtheLandauInstituteforTheoreticalPhysicsin1985,
andhasworkedinthefieldofquantumtransportsincethelate1980s.
YaroslavM.BlanterisanAssociateProfessorintheKavliInstituteofDelftUniversityof
Technology.PrevioustothisNeuroscience,hewasaHumboldtFellowattheUniversityof
KarlsruheandaSeniorAssistantattheUniversityofGeneva.

Quantum Transport
Introduction to Nanoscience
YULI V. NAZAROV
DelftUniversityofTechnology
YAROSLAV M. BLANTER
DelftUniversityofTechnology
CAMBRIDGE UNIVERSITY PRESS
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CambridgeUniversityPress
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Informationonthistitle:www.cambridge.org/9780521832465
(cid:2)c Y.NazarovandY.Blanter2009
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2009
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloginginPublicationdata
Nazarov,YuliV.
Quantumtransport:introductiontonanoscience/YuliV.Nazarovand
YaroslavM.Blanter.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-83246-5
1. Electrontransport. 2. Quantumtheory. 3. Nanoscience. I. Blanter,
YaroslavM.,1967– II. Title.
QC176.8.E4N398 2009
(cid:3)
620.5–dc22
2008054041
ISBN978-0-521-83246-5hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Contents
Preface pagevii
Introduction 1
1 Scattering 7
1.1 Wavepropertiesofelectrons 7
1.2 Quantumcontacts 17
1.3 ScatteringmatrixandtheLandauerformula 29
1.4 Countingelectrons 41
1.5 Multi-terminalcircuits 49
1.6 Quantuminterference 63
1.7 Time-dependenttransport 81
1.8 Andreevscattering 98
1.9 Spin-dependentscattering 114
2 Classicalandsemiclassicaltransport 124
2.1 Disorder,averaging,andOhm’slaw 125
2.2 Electrontransportinsolids 130
2.3 Semiclassicalcoherenttransport 137
2.4 CurrentconservationandKirchhoffrules 155
2.5 Reservoirs,nodes,andconnectors 165
2.6 Ohm’slawfortransmissiondistribution 175
2.7 Spintransport 187
2.8 Circuittheoryofsuperconductivity 193
2.9 Fullcountingstatistics 205
3 Coulombblockade 211
3.1 Chargequantizationandchargingenergy 212
3.2 Single-electrontransfers 223
3.3 Single-electrontransportandmanipulation 237
3.4 Co-tunneling 248
3.5 Macroscopicquantummechanics 264
3.6 Josephsonarrays 278
3.7 SuperconductingislandsbeyondtheJosephsonlimit 287
vi Contents
(cid:2)
4 Randomnessandinterference 299
4.1 Randommatrices 299
4.2 Energy-levelstatistics 309
4.3 Statisticsoftransmissioneigenvalues 324
4.4 Interferencecorrections 336
4.5 Stronglocalization 363
5 Qubitsandquantumdots 374
5.1 Quantumcomputers 375
5.2 Quantumgoodies 386
5.3 Quantummanipulation 397
5.4 Quantumdots 406
5.5 Chargequbits 427
5.6 Phaseandfluxqubits 436
5.7 Spinqubits 445
6 Interaction,relaxation,anddecoherence 457
6.1 Quantizationofelectricexcitations 458
6.2 Dissipativequantummechanics 470
6.3 Tunnelinginanelectromagneticenvironment 487
6.4 Electronsmovinginanenvironment 499
6.5 Weakinteraction 513
6.6 Fermionicenvironment 523
6.7 Relaxationanddecoherenceofqubits 538
6.8 Relaxationanddephasingofelectrons 549
AppendixA Survivalkitforadvancedquantummechanics 562
AppendixB Survivalkitforsuperconductivity 566
AppendixC Unitconversion 569
References 570
Index 577
Preface
This book provides an introduction to the rapidly developing field of quantum transport.
Quantum transport is an essential and intellectually challenging part of nanoscience; it
comprises a major research and technological effort aimed at the control of matter and
devicefabricationatsmallspatialscales.Thebookisbasedonthemastercoursethathas
beengivenbytheauthorsatDelftUniversityofTechnologysince2002.Mostofthemat-
erial is at master student level (comparable to the first years of graduate studies in the
USA). The book can be used as a textbook: it contains exercises and control questions.
Theprogramofthecourse,readingschemes,andeducation-relatedpracticalinformation
canbefoundatourwebsitewww.hbar-transport.org.
We believe that the field is mature enough to have its concepts – the key principles
thatareequallyimportantfortheoristsandforexperimentalists–taught.Wepresentata
comprehensive level a number of experiments that have laid the foundations of the field,
skipping the details of the experimental techniques, however interesting and important
they are. To draw an analogy with a modern course in electromagnetism, it will discuss
the notions of electric and magnetic field rather than the techniques of coil winding and
electricisolation.
We also intended to make the book useful for Ph.D. students and researchers, includ-
ing experts in the field. We can liken the vast and diverse field of quantum transport to a
mountainrangewithseveralhighpeaks,anumberofsmallermountainsinbetween,and
manyhillsfillingthespacearoundthemountains.Therearecurrentlymanygoodreviews
concentrating on one mountain, a group of hills, or the face of a peak. There are several
booksgivingaviewofacoupleofpeaksvisiblefromaparticularpoint.Withthisbook,we
attempttoperformanoverviewofthewholemountainrange.Thiscomesattheexpense
ofdetail:ourbookisnotatamonographlevelandomitssometoughderivations.Thelevel
ofdetailvariesfromtopictotopic,mostlyreflectingourtastesandexperiencesratherthan
theimportanceofthetopic.
Weprovideasignificantnumberofreferencestocurrentresearchliterature:morethana
commontextbookdoes.Wedonotgivearepresentativebibliographyofthefield.Nordo
thereferencesgivenindicatescientificprecedences,priorities,andrelativeimportanceof
thecontributions.Thepresenceorabsenceofcertaincitationsdoesnotnecessarilyreflect
ourviewsontheseprecedencesandtheirrelativeimportance.
This book results from a collective effort of thousands of researchers and students
involvedinthefieldofquantumtransport,andwearepleasedtoacknowledgethemhere.
WearedeeplyandpersonallyindebtedtoourPh.D.supervisorsandtodistinguishedsenior
colleagues who introduced us to quantum transport and guided and helped us, and to
comrades-in-researchworkinginuniversitiesandresearchinstitutionsallovertheworld.
viii Preface
(cid:2)
Thisbookwouldneverhavegotunderwaywithoutfruitfulinteractionswithourstudents.
PartsofthebookwerewrittenduringourextendedstaysatWeizmannInstituteofScience,
ArgonneNationalLaboratory,AspenCenterofPhysics,andInstituteofAdvancedStudies,
Oslo.
It is inevitable that, despite our efforts, this book contains typos, errors, and less com-
prehensivediscourses.Wewouldbehappytohaveyourfeedback,whichcanbesubmitted
viathewebsitewww.hbar-transport.org.Wehopethatitwillbepossibletherebytoprovide
somelimited“technical”support.
Introduction
It is an interesting intellectual game to compress an essence of a science, or a given
scientific field, to a single sentence. For natural sciences in general, this sentence would
probably read: Everything consists of atoms. This idea seems evident to us. We tend to
forgetthattheideaisratherold:itwasputforwardinAncientGreecebyLeucippusand
Democritus,anddevelopedbyEpicurus,morethan2000yearsago.Formostofthistime,
theidearemainedatheoreticalsuggestion.Itwasexperimentallyconfirmedandestablished
asacommonpointofviewonlyabout150yearsago.
Those 150 years of research in atoms have recently brought about the field of
nanoscience,aimingatestablishingcontrolandmakingusefulthingsattheatomicscale.
It represents the common effort of researchers with backgrounds in physics, chemistry,
biology, material science, and engineering, and contains a significant technological com-
ponent.Itistechnologythatallowsustoworkatsmallspatialscales.Theultimategoalof
nanoscienceistofindmeanstobuildupusefulartificialdevices–nanostructures–atomby
atom.ThebenefitsandgreatprospectsofthisgoalwouldbeobviouseventoDemocritus
andEpicurus.
This book is devoted to quantum transport, which is a distinct field of science. It is
also a part of nanoscience. However, it is a very unusual part. If we try to play the same
gameofputtingtheessenceofquantumtransportintoonesentence,itwouldread:Itisnot
important whether a nanostructure consists of atoms. The research in quantum transport
focuses on the properties and behavior regimes of nanostructures, which do not immedi-
ately depend on the material and atomic composition of the structure, and which cannot
beexplainedstartingbyclassical(thatis,non-quantum)physics.Mostimportantly,ithas
been experimentally demonstrated that these features do not even have to depend on the
size of the nanostructure. For instance, the transport properties of quantum dots made of
a handful of atoms may be almost identical to those of micrometer-size semiconductor
devicesthatencompassbillionsofatoms.
Thetwomostimportantscalesofquantumtransportareconductanceandenergyscale.
Themeasureofconductance,G,istheconductancequantumG ≡e2/π(cid:2),thescalemade
Q
offundamentalconstants:electronchargee(mostofquantumtransportisthetransportof
electrons) and the Planck constant (cid:2) (this indicates the role of quantum mechanics). The
energyscaleisdeterminedbyflexibleexperimentalconditions:bythetemperature,k T,
B
and/orthebiasvoltageappliedtoananostructure,eV.Thebehaviorregimeisdetermined
bytherelationofthisscaletointernalenergyscalesofthenanostructure.Whereasphysical
principles,asstressed,donotdepend onthesizeofthenanostructure, theinternalscales
do.Ingeneral,theyarebiggerforsmallernanostructures.