One-Dimensional Superconductivity in Nanowires

334 Pages·2013·6.102 MB·English

by  Fabio Altomare, Albert M. Chang

FabioAltomareandAlbertM.Chang
One-DimensionalSuperconductivity
inNanowires
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ISBN:978-3-527-40651-7
Fabio Altomare and Albert M. Chang
One-Dimensional Superconductivity
in Nanowires
WILEY-VCH Verlag GmbH & Co. KGaA
TheAuthors AllbookspublishedbyWiley-VCHarecarefully
produced.Nevertheless,authors,editors,and
Dr.FabioAltomare publisherdonotwarranttheinformation
D-WaveSystemsInc. containedinthesebooks,includingthisbook,to
100-4401StillCreekDrive befreeoferrors.Readersareadvisedtokeepin
Burnaby,BC,V5C6G9 mindthatstatements,data,illustrations,
Canada proceduraldetailsorotheritemsmay
inadvertentlybeinaccurate.
Prof.AlbertM.Chang
DukeUniversity LibraryofCongressCardNo.:
DepartmentofPhysics
appliedfor
PhysicsBuilding
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Durham,NC27708-0305 Acataloguerecordforthisbookisavailable
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CoverFigure DeutscheNationalbibliothek
Artisticrenderingofan8nmwide,20μmlong TheDeutscheNationalbibliothekliststhis
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stencil.Electricalcontactstothenanowire(left detailedbibliographicdataareavailableonthe
andrightpads)arerealizedduringthenanowire Internetathttp://dnb.d-nb.de.
deposition.Atthecenter,aQPSjunctionis
currentbiasedbyacurrentsource:Thenumber ©2013WILEY-VCHVerlagGmbH&Co.KGaA,
ofwindingsofthephaseoftheorderparameter Boschstr.12,69469Weinheim,Germany
decreasesbecauseofaphaseslippageevent.The
diagramdoesnotindicatetheactualnanowire Allrightsreserved(includingthoseoftranslation
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ToMariaElena,Mattia,Gabriele,Giuliaandtomyparents:
Thankyouforyourloveandsupport.
Fabio
Tomyfamily:Ying,EmilyandAustin,mysisterMargaret,mybrotherWinston,
myfatherBertrandTsu-Shen,andmylatemother,VirginiaPang-Ying.
Albert
VII
Contents
Preface XI
AbbreviationsandSymbols XV
ColorPlates XXI
PartOne TheoreticalAspectsofSuperconductivityin1DNanowires 1
1 Superconductivity:BasicsandFormulation 3
1.1 Introduction 3
1.2 BCSTheory 4
1.3 Bogoliubov–deGennesEquations–QuasiparticleExcitations 8
1.4 Ginzburg–LandauTheory 10
1.4.1 Time-DependentGinzburg–LandauTheory 12
1.5 GorkovGreen’sFunctions,Eilenberger–Larkin–OvchinnikovEquations,
andtheUsadelEquation 12
1.6 PathIntegralFormulation 19
References 27
2 1DSuperconductivity:BasicNotions 31
2.1 Introduction 31
2.2 ShapeResonances–OscillationsinSuperconductivityProperties 33
2.2.1 EarlyTreatmentsofShapeResonancesin2DFilms 35
2.2.2 Bogoliubov–deGennesEquations,FiniteTemperature,andParabolic-
BandApproximationforRealisticMaterials 40
2.2.3 NumericalSolutionsandThinFilmShapeResonances 42
2.2.4 1DNanowires–ShapeResonancesandSizeOscillations 45
2.3 SuperconductivityinCarbonNanotubes–Single-WalledBundlesand
IndividualMultiwalledNanotubes 48
2.4 PhaseSlips 50
2.4.1 FiniteVoltageinaSuperconductingWireandPhaseSlip 51
2.4.2 PhaseSlipinaJosephsonJunction 52
2.4.3 Langer–Ambegaokar Free EnergyMinimain theGinzburg–Landau
Approximation 55
2.4.4 TransitionRateandFreeEnergyBarrier 59
VIII Contents
2.4.5 FreeEnergyBarrierforaPhaseSlipintheGinzburg–LandauTheory 60
2.4.6 PhysicalScenarioofaThermally-ActivatedPhaseSlip 64
2.4.7 McCumber–HalperinEstimateoftheAttemptFrequency 66
References 71
3 QuantumPhaseSlipsandQuantumPhaseTransitions 75
3.1 Introduction 75
3.2 Zaikin–GolubevTheory 79
3.2.1 DerivationoftheLowEnergyEffectiveAction 80
3.2.2 CoreContributiontotheQPSAction 88
3.2.3 HydrodynamicContributiontothePhase-SlipAction 91
3.2.4 QuantumPhase-SlipRate 92
3.2.5 QuantumPhase-SlipInteractionandQuantum-PhaseTransitions 95
3.2.6 WireResistanceandNonlinearVoltage–CurrentRelations 97
3.3 Short-WireSuperconductor–InsulatorTransition:Büchler,Geshkenbein
andBlatterTheory 100
3.4 Refael,Demler,Oreg,FisherTheory–1DJosephsonJunctionChains
andNanowires 105
3.4.1 DiscreteModelof1DJosephsonJunctionChains 107
3.4.2 ResistanceoftheJosephsonJunctionsandtheNanowire 114
3.4.3 MeanFieldTheoryoftheShort-WireSIT 116
3.5 Khlebnikov–PryadkoTheory–MomentumConservation 121
3.5.1 Gross–PitaevskiiModelandQuantumPhaseSlips 123
3.5.2 Disorder Averaging,QuantumPhaseTransitionandScalingforthe
ResistanceandCurrent–VoltageRelations 126
3.5.3 ShortWires–LinearQPSInteractionandExponentialQPSRate 130
3.6 QuantumCriticalityandPair-Breaking–UniversalConductanceand
ThermalTransportinShortWires 136
References 143
4 Duality 149
4.1 Introduction 149
4.2 Mooij–NazarovTheoryofDuality–QPSJunctions 152
4.2.1 QPS Junction Voltage-Charge Relationship and Shapiro Current
Steps 156
4.2.2 QPSQubits 157
4.3 KhlebnikovTheoryofInteractingPhaseSlipsinShortWires:Quark
ConfinementPhysics 159
References 165
5 ProximityRelatedPhenomena 167
5.1 Introduction 167
5.2 Transport Propertiesof Normal-SuperconductingNanowire-Normal
(N-SCNW-N)Junctions 169
5.2.1 NonequilibriumUsadelEquations 169
5.2.2 ParameterizationoftheUsadelEquations 174
Contents IX
5.2.3 NumericalResults 175
5.3 Superconductor–Semiconductor Nanowire–Superconductor
Junctions 179
5.4 Majorana Fermion in S-SmNW-S Systems with Strong Spin–Orbit
InteractionintheSemiconductor 184
References 188
PartTwo ReviewofExperimentson1DSuperconductivity 193
6 ExperimentalTechniqueforNanowireFabrication 195
6.1 Experimental Technique for the Fabrication of Ultra Narrow
Nanowires 195
6.2 IntroductiontotheTechniques 196
6.2.1 Lithography 197
6.2.2 MetalDeposition 198
6.2.3 Etching 199
6.2.4 PuttingItAllTogether 199
6.3 Step-EdgeLithographicTechnique 201
6.4 MolecularTemplating 202
6.5 SemiconductingStencils 205
6.6 NatelsonandWillet 205
6.7 SNAPTechnique 206
6.8 ChangandAltomare 208
6.9 TemplateSynthesis 209
6.10 OtherMethods 211
6.10.1 IonBeamPolishing 211
6.10.2 AngledEvaporation 213
6.10.3 ResistDevelopment 213
6.11 FutureDevelopments 214
References 216
7 Experimental Review of Experiments on 1D Superconducting
Nanowires 219
7.1 Introduction 219
7.2 Filtering 220
7.3 PhaseSlips 221
7.4 OverviewoftheExperimentalResults 223
7.4.1 Giordano’sExperiments 225
7.4.2 RecentExperimentsonQPS 226
7.4.3 QPSProbedviaSwitchingCurrentMeasurements 239
7.5 OtherEffectsin1DSuperconductingNanowires 248
7.5.1 S-ShapedCurrent–VoltageCharacteristic 248
7.6 AntiproximityEffect 250
7.6.1 StabilizationofSuperconductivitybyaMagneticField 253
7.6.2 ShapeResonanceEffects 255
References 257
X Contents
8 CoherentQuantumPhaseSlips 263
8.1 Introduction 263
8.2 ASingle-ChargeTransistor Based ontheCharge-Phase Dualityofa
SuperconductingNanowireCircuit 263
8.3 QuantumPhase-SlipPhenomenoninUltranarrowSuperconducting
Nanorings 266
8.4 CoherentQuantumPhaseSlip 267
8.5 Conclusion 272
References 273
9 1DSuperconductivityinaRelatedSystem 275
9.1 Introduction 275
9.2 CarbonNanotubes 275
9.2.1 ProximityEffectsinSWNT 276
9.2.2 IntrinsicSuperconductivityinSWNT 278
9.2.3 SuperconductivityinRopesMediatedbytheEnvironment 281
9.3 MajoranaExperiments 286
9.3.1 MajoranaExperimentinSemiconductingNanowires 286
9.3.2 MajoranaExperimentinHybridSuperconductor-TopologicalInsulator
Devices 290
9.4 SuperconductingNanowiresasSingle-PhotonDetectors 292
References 297
10 ConcludingRemarks 301
Index 303
XI
Preface
Thisbookisdevotedtothetopicofsuperconductivityinverynarrowmetallicwires.
Interestinsuchwiresisdrivenbythecontinuingdriveforminiaturizationinthe
electronicsindustry,wherethereductionofheatdissipationbytheuseofintercon-
nectswhicharesuperconductingmaybenecessary.Thishasledtotheinvention
ofnewmethodsofproducingverynarrowwireswithgoodqualityanduniformity
indimensions,andopenedupthepossibilityofnoveldeviceparadigms.
Thesuperconductingstateisastateofcoherentpairsofelectrons,heldtogether
bythemechanismoftheCooperinstability.Whenthelateraldimensionsaresmall,
anewpathwayopensupforthesuperconductortoproducedissipation,associated
withtheenhancedrateatwhichfluctuationscanoccurinthecomplexorderparam-
eter – a quantity which describes the magnitude and phase of the Cooper pairs.
In many regimes, the fluctuations of the phase of the Cooper pairs is often the
dominantprocess,inanalogywithwhatoccursindissipativeJosephsonjunctions.
Thisnewpathwayisrelatedwiththemotionofvorticesintype-IIsuperconductors,
whichgivesrisetodissipationin2D(two-dimensional)andbulksuperconductors,
inwhicharapidchangeofphaseoccursasavortexpassesby.Here,becauseofthe
physicaldimensionofthesystem(comparableto10nm),thevortexquicklypasses
acrosstheentirenarrowwire,producingdissipationintheprocess.
Althoughnanowiresaresmallintheirtransversedimensions,theyarestillmuch
largerthantheFermiwavelength(λ )oftheelectronicsystem,whichisoftheorder
F
ofafewangstrominallmetals.InconventionalsuperconductorssuchasAl,Pb,
Sn,Nb,MoGe,andsoon,inthinfilmornanowireform,thecoherencelengthis
(cid:3) (cid:2)5(cid:3)100nm,typically10–1000timestheFermiwavelength.Thesystemwewill
consideristhereforeinasortofmixed-dimensionalregime.Fromthecondensate
perspective,thesystemis1D(one-dimensional),inthesensethatthetransversedi-
mensionsaresmallerthantheCooperpairsize:forthisreason,thewavefunction,
oralternatively,theorderparameter,describingtheCooperpairs,isuniform,and
thusposition-independent,inthetransversedirections.Fromthefermionicquasi-
particleexcitationperspective,thesystemiseffectively3D(three-dimensional; λ
F
ismuchsmallerthanthelateraldimension)andthereisalargenumberoftrans-
verse channels(from(cid:2) 100 in a multiwalledcarbon nanotube, to (cid:2) 3000 in an
(cid:2) 8nmdiameteraluminumnanowire),analogoustotransversemodesinawave
guide.