Superconductivity in Nanowires: Fabrication and Quantum Transport

273 Pages·2012·3.386 MB·English

by  Alexey Bezryadin

AlexeyBezryadin
SuperconductivityinNanowires
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Alexey Bezryadin
Superconductivity in Nanowires
Fabrication and Quantum Transport
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V
Contents
Preface–SuperconductivityandLittle’sPhaseSlipsinNanowires IX
Abbreviations–ShortList XI
Notations–ShortList XIII
ColorPlates XV
1 Introduction 1
2 SelectedTheoreticalTopicsRelevanttoSuperconductingNanowires 15
2.1 FreeorUsableEnergyofSuperconductingCondensates 15
2.2 HelmholtzandGibbsFreeEnergies 18
2.3 FluctuationProbabilities 23
2.4 Work Performed byaCurrent Sourceon theCondensateina Thin
Wire 27
2.5 HelmholtzEnergyofSuperconductingWires 29
2.6 GibbsEnergyofSuperconductingWires 31
2.7 RelationshipbetweenGibbsandHelmholtzEnergyDensities 35
2.8 RelationshipbetweenThermalFluctuationsandUsableEnergy 36
2.9 CalculusofVariations 38
2.10 Ginzburg–LandauEquations 39
2.11 Little–ParksEffect 46
2.12 KineticInductanceandtheCPRofaThinWire 50
2.13 DrudeFormulaandtheDensityofStates 51
3 Stewart–McCumberModel 53
3.1 KineticInductanceandtheAmplitudeofSmallOscillations 60
3.2 MechanicalAnalogyfortheStewart–McCumberModel 62
3.2.1 DefiningtheSupercurrentThroughHelmholtzFreeEnergy 65
3.2.2 CubicPotential 66
3.2.3 ThermalEscapefromtheCubicPotential 67
3.3 Macroscopic Quantum Phenomena in the Stewart–McCumber
Model 68
3.3.1 MQTinaCubicPotentialatHighBiasCurrents 71
VI Contents
3.4 Schmid–Bulgadaev Quantum Phase Transition in Shunted
Junctions 74
3.5 Stewart–McCumberModelwithNormalizedVariables 76
4 FabricationofNanowiresUsingMolecularTemplates 79
4.1 ChoiceofTemplatingMolecules 86
4.2 DNAMoleculesasTemplates 86
4.3 SignificanceoftheSo-Called“WhiteSpots” 88
5 ExperimentalMethods 91
5.1 SampleInstallation 91
5.2 ElectronicTransportMeasurements 95
6 ResistanceofNanowiresMadeofSuperconductingMaterials 101
6.1 BasicProperties 101
6.2 Little’sPhaseSlips 105
6.3 Little’sFit 108
6.4 LAMHModelofPhaseSlippageatLowBiasCurrents 115
6.5 ComparingLAMHandLittle’sFit 122
7 GolubevandZaikinTheoryofThermallyActivatedPhaseSlips 125
8 StochasticPrematureSwitchingandKurkijärviTheory 131
8.1 StochasticSwitchingRevealedbyV–ICharacteristics 131
8.2 “GeigerCounter”forLittle’sPhaseSlips 135
8.3 MeasuringSwitchingCurrentDistributions 139
8.4 Kurkijärvi–Fulton–Dunkleberger(KFD)Transformation 143
8.5 ExamplesofApplyingKFDTransformations 148
8.6 InverseKFDTransformation 152
8.7 Universal3/2PowerLawforPhaseSlipBarrier 153
8.8 RateofThermallyActivatedPhaseSlipsatHighCurrents 157
8.9 KurkijärviDispersionPowerLawsof2/3and1/3 160
9 MacroscopicQuantumTunnelinginThinWires 163
9.1 GiordanoModelofQuantumPhaseSlips(QPS)inThinWires 165
9.2 ExperimentalTestsoftheGiordanoModel 175
9.3 GolubevandZaikinQPSTheory 183
9.4 KhlebnikovTheory 185
9.5 SpheresofInfluenceofQPSandTAPSRegimes 187
9.6 Kurkijärvi–GargModel 189
9.7 Theorem:InverseRelationshipbetweenDispersionandtheSlopeofthe
SwitchingRateCurve 195
10 Superconductor–InsulatorTransition(SIT)inThinandShortWires 197
10.1 SimpleModelofSITinThinWires 207
Contents VII
11 Bardeen Formula for the Temperature Dependence of the Critical
Current 213
AppendixA SuperconductivityinMoGeAlloys 215
AppendixB VarianceandtheVarianceEstimator 217
AppendixC ProblemsandSolutions 223
References 241
Index 247
IX
Preface – Superconductivityand Little’sPhase Slipsin
Nanowires
Thisbookpresentsanaccountoffabricationtechniques,chargetransportexperi-
ments,andtheoreticalmodelsrelatedtosuperconductingnanowires,thatis,thin
wiresmadeofsuperconductingmaterialsandhavingdimensionsatthenanome-
ter scale.A wireisclassifiedasone-dimensional(1D) ifitsdiameterdissmaller
thanthesuperconductingcoherencelength(cid:1).Insuchwires,thesuperconducting
condensatewavefunctiononlydependsonthepositionalongthewire,whileitis
independentofthepositionwithinthewirecrosssection.Ananowireisclassified
p
as quasi-one-dimensional(quasi-1D) if d < π 2(cid:1). Such conditionensures that
vorticesarenotenergeticallystableinthewire.Therefore,theorderparameteris
approximatelyconstantwithinthewirecrosssection.Since(cid:1) divergesatthecriti-
caltemperature T,itisnottoohardtomakeawirewhichisquasi-1Dnear T.It
c c
ismorechallengingtofabricatenanowireswhichare1Dorquasi-1Deveninthe
limitofzerotemperature.Recentperfectionofnanotechnologyallowsustomake
wireasthinas5nm,whichcanberegardedasquasi-1Devenatzerotemperature.
Thisbookprovidesadetailedanalysisofthetransportpropertiesofsuchultrathin
wires. Amongotherthings,thesmallsizeofthewirescauses themtobehave as
quantumobjects.DuetotheHeisenberg uncertaintyprinciple,suchwiresmight
notbetruesuperconductors.Theirstatebecomesuncertainbecausethewiresexist
inaquantumsuperpositionofnormalandsuperconductingstates.Astheadmix-
tureofthenormalphaseincreases,nominallysuperconductingnanowirescanlose
theirabilitytocarrysupercurrentandcanbecomeeithernormalor,whatitmore
typical,slightlyinsulating.
Evenifquantumfluctuationsarenotsufficientlystrongtodestroysuperconduc-
tivity,theohmicresistance Rofathinwireisalwaysgreater thanzero,provided
that the temperature, T, is above absolute zero. Due to thermal fluctuations,the
phase of the superconducting condensate wavefunction jumps irreversibly, lead-
ingtoadecayofthesupercurrent.Suchjumpsarecalledthermallyactivatedphase
slips(TAPS)orLittle’sphaseslips(LPS).AdetaileddiscussionofLPSisoneofthe
maintopicsofthisbook.
Nanowires made of superconducting materials can generally be classified into
threedifferentcategories:superconductingorS-typewires,normalorN-typewires,
andinsulatingorI-typewires.Assumethat R(T) ! R(0)as T ! 0,whereTis
thetemperatureofthenanowireandR(T)isitselectricresistanceasafunctionof
X Preface–SuperconductivityandLittle’sPhaseSlipsinNanowires
temperature.Also,assume that R denotestheresistanceofthenanowireinthe
n
normal state, that is, at T > T. Then, R(0) D 0 (at T D 0) defines the S-type
c
wires,0 < R(0) < R correspondstotheN-type,and R(0) > R characterizes,by
n n
convention,I-typewires.Thereisnodirectwaytodeterminethetypeofthewire
sincetheconditionT D0isimpossibletosatisfyexperimentally.Thus,theclassi-
ficationisusuallydoneusingindirectapproaches.Forexample,iftheresistanceis
observedtoincreasewithcooling,thenthewireisclassifiedastheI-type.Onthe
otherhand,iftheohmicresistancedecreasesexponentiallywithcooling,thenthe
wireshouldbeclassifiedastheS-type.TheS-typeclassificationalsorequiresthat
thewireexhibitsafinitesupercurrentatsufficientlylowtemperatures.Inthebook,
wesummarizethepropertiesofthetwomostcommontypesofnanowires,name-
ly,theS-typeandI-typewires,andanalyzethequantumphasetransitionbetween
thesetwophases.
Thebookisintendedforundergraduateandbeginninggraduatestudentsinter-
estedinnanoscalesuperconductivity.Thereadersareassumedtohavesomefamil-
iaritywiththebasicsofquantummechanicsandsuperconductivity.Recommend-
ed introductorybooks ongeneral aspects of superconductivityare Introduction to
Superconductivity by M.Tinkham[1]and Superconductivity of Metals and Alloysby
P.G.deGennes[2].Thepresentmonographisfocusedonselectedtopicsrelatedto
thinsuperconductingnanowires.
Itismypleasuretoacknowledgemyindebtednesstothegraduatestudentsand
postdocsIhavesupervisedsince2000.Ofspecialimportanceistheadviceandthe
helpful discussions with Liliya Simpson, Anthony T. Bollinger, Ulas C. Coskun,
CeliaElliott,DavidS.Hopkins,RobertC.DinsmoreIII,MitrabhanuSahu,Sergei
Khlebnikov,ThomasAref,MatthewW.Brenner,PaulGoldbart,AndreyRogachev,
Andrey Belkin, Andrey Zaikin, Konstantin Arutyunov,and Victor Vakaryuk. The
workwassupportedbytheDOEgrantDE-FG02-07ER46453andbytheNSFgrant
DMR10-05645.
Campaign,August2012 AlexeyBezryadin
XI
Abbreviations– Short List
BCS Bardeen,Copper,andSchrieffertheoryofsuperconductivity.
BCScondensate acoherentquantumstatesofamacroscopicnumberofelec-
tronsappearingintheBCStheory.
CP Cooperpair.
CPR Current–phaserelationship.
GL GinzburgandLandauphenomenologicaltheory.
GLAG Ginzburg,Landau,Abrikosov,andGor’kovtheory,whichisa
microscopictheoryderivedfromBCS,butsimilarinformto
theGLtheory.
GZ GolubevandZaikintheory(eitherofTAPSorofQPS).
I-typewire aninsulatingnanowire,i.e.,ananowiremadeofasupercon-
ducting metal, but, due to various reasons, having its zero-
temperatureresistancelargerthanitsnormal-stateresistance.
JJ Josepshonjunction.
KG Kurkijärvi–Gargmodel.
LAMH Langer, Ambegaokar, McCumber, and Halperin combined
model.
LPS Little’sphaseslip.
MT moleculartemplating.
MQT Macroscopicquantumtunneling.
N-typewire Anormalnanowire,thatis,ananowiremadeofasupercon-
ducting metal, but, due to various reasons, having its zero-
temperature resistance larger than zero but lower or equal
compared to its normal-state resistance, which is measured
atT > T.
c
QPS quantum phase slips, usually in quasi-1D superconducting
wires.QPSisaparticularcaseofMQT.
SB Schmid–Bulgadaevquantumphasetransition.
SIS Superconductor–insulator–superconductorjunction.
SNS Superconductor–normalmetal–superconductorjunction.
SIT Superconductor to insulator quantum phase transition. The
meaningofitisthatontheS-sidethewirehaszeroresistance